[ { "id": "Linear_algebra_0000", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "For what value(s) of $h$ and $k$ does the linear system have infinitely many solutions?\n\\begin{array}{rcrcr} 6x_1 &+& 5x_2 &=& 2\\\\ h x_1 &+& k x_2 &=&-2 \\end{array} $h$ [ANS] [ANS] and $k$ [ANS] [ANS]\nNote: $h$ and $k$ must be integers or fractions reduced to lowest terms.", "answer_v1": [ "=", "-6", "=", "-5" ], "answer_type_v1": [ "MCS", "NV", "MCS", "NV" ], "options_v1": [ [ "=", "≠" ], [], [ "=", "≠" ], [] ], "problem_v2": "For what value(s) of $h$ and $k$ does the linear system have infinitely many solutions?\n\\begin{array}{rcrcr} 2x_1 &-& 7x_2 &=&-2\\\\ h x_1 &+& k x_2 &=& 5 \\end{array} $h$ [ANS] [ANS] and $k$ [ANS] [ANS]\nNote: $h$ and $k$ must be integers or fractions reduced to lowest terms.", "answer_v2": [ "=", "-5", "=", "35/2" ], "answer_type_v2": [ "MCS", "NV", "MCS", "NV" ], "options_v2": [ [ "=", "≠" ], [], [ "=", "≠" ], [] ], "problem_v3": "For what value(s) of $h$ and $k$ does the linear system have infinitely many solutions?\n\\begin{array}{rcrcr} 3x_1 &-& 5x_2 &=& 1\\\\ h x_1 &+& k x_2 &=&-3 \\end{array} $h$ [ANS] [ANS] and $k$ [ANS] [ANS]\nNote: $h$ and $k$ must be integers or fractions reduced to lowest terms.", "answer_v3": [ "=", "-9", "=", "15" ], "answer_type_v3": [ "MCS", "NV", "MCS", "NV" ], "options_v3": [ [ "=", "≠" ], [], [ "=", "≠" ], [] ] }, { "id": "Linear_algebra_0001", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "For what value(s) of $h$ is the linear system consistent?\n\\begin{array}{rcrcr} 5x_1 &+& 4x_2 &=& h\\\\-10x_1 &-& 8x_2 &=&-1 \\end{array} $h$ [ANS] [ANS]", "answer_v1": [ "=", "1/2" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "=", "≠" ], [] ], "problem_v2": "For what value(s) of $h$ is the linear system consistent?\n\\begin{array}{rcrcr} 15x_1 &+& 6x_2 &=& h\\\\-5x_1 &-& 2x_2 &=& 2 \\end{array} $h$ [ANS] [ANS]", "answer_v2": [ "=", "-6" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "=", "≠" ], [] ], "problem_v3": "For what value(s) of $h$ is the linear system consistent?\n\\begin{array}{rcrcr}-6x_1 &+& 15x_2 &=& h\\\\ 4x_1 &-& 10x_2 &=&-1 \\end{array} $h$ [ANS] [ANS]", "answer_v3": [ "=", "3/2" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "=", "≠" ], [] ] }, { "id": "Linear_algebra_0002", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Find all solutions to the system\n\\begin{array}{rcrcr}4x_1 &+& 3x_{2} &=&-14 \\\\3x_1 &-& 3x_{2} &=& 0\\end{array} by eliminating one of the variables. $(x_1,x_2)=$ [ANS]\nHelp: If there is a solution $(a,b)$, enter your answer as a point (a,b). If there is a free variable in the solution, use s1 to denote the variable $s_1$. If there is no solution, type no solution.", "answer_v1": [ "(-2,-2)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all solutions to the system\n\\begin{array}{rcrcr}2x_1 &+& 5x_{2} &=&-2 \\\\2x_1 &-& 4x_{2} &=& 16\\end{array} by eliminating one of the variables. $(x_1,x_2)=$ [ANS]\nHelp: If there is a solution $(a,b)$, enter your answer as a point (a,b). If there is a free variable in the solution, use s1 to denote the variable $s_1$. If there is no solution, type no solution.", "answer_v2": [ "(4,-2)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all solutions to the system\n\\begin{array}{rcrcr}2x_1 &+& 4x_{2} &=&-10 \\\\2x_1 &-& 3x_{2} &=&-3\\end{array} by eliminating one of the variables. $(x_1,x_2)=$ [ANS]\nHelp: If there is a solution $(a,b)$, enter your answer as a point (a,b). If there is a free variable in the solution, use s1 to denote the variable $s_1$. If there is no solution, type no solution.", "answer_v3": [ "(-3,-1)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0003", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [ "linear equations", "system", "systems", "numerical", "applications" ], "problem_v1": "Determine if the system\n\\begin{array}{rcrcr}-x_1 &-& 6x_{2} &=& 5 \\\\-3x_1 &-& 4x_{2} &=&-1\\end{array} is diagonally dominant. If the system is already diagonally dominant, then copy exactly the equations as given. If not, then rewrite the system so that it is diagonally dominant. If it is not possible, then also copy exactly the equations as given. Use x1 and x2 to enter the variables $x_1$ and $x_2$. Diagonally dominant? [ANS] Rewrite to be diagonally dominant if possible, otherwise copy exactly the given equations: [ANS]=[ANS] [ANS]=[ANS]", "answer_v1": [ "no", "-x1+(-6)*x2", "5", "-3*x1+(-4)*x2", "-1" ], "answer_type_v1": [ "TF", "EX", "NV", "EX", "NV" ], "options_v1": [ [ "yes", "no" ], [], [], [], [] ], "problem_v2": "Determine if the system\n\\begin{array}{rcrcr}-2x_1 &+& 5x_{2} &=& 6 \\\\-3x_1 &-& 7x_{2} &=&-6\\end{array} is diagonally dominant. If the system is already diagonally dominant, then copy exactly the equations as given. If not, then rewrite the system so that it is diagonally dominant. If it is not possible, then also copy exactly the equations as given. Use x1 and x2 to enter the variables $x_1$ and $x_2$. Diagonally dominant? [ANS] Rewrite to be diagonally dominant if possible, otherwise copy exactly the given equations: [ANS]=[ANS] [ANS]=[ANS]", "answer_v2": [ "no", "-2*x1+5*x2", "6", "-3*x1+(-7)*x2", "-6" ], "answer_type_v2": [ "TF", "EX", "NV", "EX", "NV" ], "options_v2": [ [ "yes", "no" ], [], [], [], [] ], "problem_v3": "Determine if the system\n\\begin{array}{rcrcr}-x_1 &-& 7x_{2} &=& 8 \\\\-2x_1 &+& 6x_{2} &=& 7\\end{array} is diagonally dominant. If the system is already diagonally dominant, then copy exactly the equations as given. If not, then rewrite the system so that it is diagonally dominant. If it is not possible, then also copy exactly the equations as given. Use x1 and x2 to enter the variables $x_1$ and $x_2$. Diagonally dominant? [ANS] Rewrite to be diagonally dominant if possible, otherwise copy exactly the given equations: [ANS]=[ANS] [ANS]=[ANS]", "answer_v3": [ "no", "-x1+(-7)*x2", "8", "-2*x1+6*x2", "7" ], "answer_type_v3": [ "TF", "EX", "NV", "EX", "NV" ], "options_v3": [ [ "yes", "no" ], [], [], [], [] ] }, { "id": "Linear_algebra_0004", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "For what value(s) of $k$ is the linear system consistent?\n\\begin{array}{rcrcr} 7x_1 &+& 5x_2 &=&-2\\\\ 11x_1 &+& k x_2 &=&-1 \\end{array} $k$ [ANS] [ANS]", "answer_v1": [ "≠", "55/7" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "equals", "is not equal to" ], [] ], "problem_v2": "For what value(s) of $k$ is the linear system consistent?\n\\begin{array}{rcrcr} 4x_1 &-& 7x_2 &=&-2\\\\ 8x_1 &+& k x_2 &=&-2 \\end{array} $k$ [ANS] [ANS]", "answer_v2": [ "≠", "-14" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "equals", "is not equal to" ], [] ], "problem_v3": "For what value(s) of $k$ is the linear system consistent?\n\\begin{array}{rcrcr} 5x_1 &-& 6x_2 &=&-1\\\\ 10x_1 &+& k x_2 &=& 2 \\end{array} $k$ [ANS] [ANS]", "answer_v3": [ "≠", "-12" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "equals", "is not equal to" ], [] ] }, { "id": "Linear_algebra_0005", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [ "linear equations", "system", "systems", "numerical", "applications" ], "problem_v1": "First solve the system\n\\begin{array}{rcrcr}-7x_1 &-& 982x_{2} &=&-37 \\\\-50x_1 &-& 88x_{2} &=&-13\\end{array} with Gaussian elimination with three significant digits of accuracy. Then solve the system again with three significant digits of accuracy, incorporating partial pivoting. To enter scientific notation, use x and ^. For example, type 2.12x 10^(-3) to enter $2.12 \\times 10^{-3}$. Note that you must enter all answers in scientific notation, even if the power of $10$ is zero. Gaussian Elimination: $(x_1,x_2)=\\bigg($ [ANS], [ANS] $\\bigg)$ Partial Pivoting: $(x_1,x_2)=\\bigg($ [ANS], [ANS] $\\bigg)$", "answer_v1": [ "2.07 x 10^-1", "3.62 x 10^-2", "1.96 x 10^-1", "3.63 x 10^-2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "First solve the system\n\\begin{array}{rcrcr}3x_1 &-& 957x_{2} &=&-37 \\\\-58x_1 &-& 84x_{2} &=& 5\\end{array} with Gaussian elimination with three significant digits of accuracy. Then solve the system again with three significant digits of accuracy, incorporating partial pivoting. To enter scientific notation, use x and ^. For example, type 2.12x 10^(-3) to enter $2.12 \\times 10^{-3}$. Note that you must enter all answers in scientific notation, even if the power of $10$ is zero. Gaussian Elimination: $(x_1,x_2)=\\bigg($ [ANS], [ANS] $\\bigg)$ Partial Pivoting: $(x_1,x_2)=\\bigg($ [ANS], [ANS] $\\bigg)$", "answer_v2": [ "-1.48 x 10^-1", "3.82 x 10^-2", "-1.42 x 10^-1", "3.82 x 10^-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "First solve the system\n\\begin{array}{rcrcr}-14x_1 &-& 964x_{2} &=&-38 \\\\41x_1 &+& 99x_{2} &=&-8\\end{array} with Gaussian elimination with three significant digits of accuracy. Then solve the system again with three significant digits of accuracy, incorporating partial pivoting. To enter scientific notation, use x and ^. For example, type 2.12x 10^(-3) to enter $2.12 \\times 10^{-3}$. Note that you must enter all answers in scientific notation, even if the power of $10$ is zero. Gaussian Elimination: $(x_1,x_2)=\\bigg($ [ANS], [ANS] $\\bigg)$ Partial Pivoting: $(x_1,x_2)=\\bigg($ [ANS], [ANS] $\\bigg)$", "answer_v3": [ "-2.95 x 10^-1", "4.37 x 10^-2", "-3.01 x 10^-1", "4.38 x 10^-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0006", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [ "linear equations", "system", "systems", "numerical", "applications" ], "problem_v1": "Compute the first three Jacobi iterations of the system\n\\begin{array}{rcrcr}-3x_1 &-& x_{2} &=& 1 \\\\3x_1 &+& 4x_{2} &=&-13\\end{array} using 0 as the initial value for each variable. Then find the exact solution and compare. Jacobi Iteration: $\\begin{array}{ccc}\\hline n & x_1 & x_2 \\\\ \\hline 0 & 0 & 0 \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nActual solution: $(x_1,x_2)=$ [ANS]\nHelp: To enter a solution $(a,b)$, type your answer as a point (a,b).", "answer_v1": [ "-0.333333", "-3.25", "0.75", "-3", "0.666667", "-3.8125", "(1,-4)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "OL" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Compute the first three Jacobi iterations of the system\n\\begin{array}{rcrcr}5x_1 &+& x_{2} &=&-3 \\\\5x_1 &+& 4x_{2} &=&-12\\end{array} using 0 as the initial value for each variable. Then find the exact solution and compare. Jacobi Iteration: $\\begin{array}{ccc}\\hline n & x_1 & x_2 \\\\ \\hline 0 & 0 & 0 \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nActual solution: $(x_1,x_2)=$ [ANS]\nHelp: To enter a solution $(a,b)$, type your answer as a point (a,b).", "answer_v2": [ "-0.6", "-3", "0", "-2.25", "-0.15", "-3", "(0,-3)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "OL" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Compute the first three Jacobi iterations of the system\n\\begin{array}{rcrcr}5x_1 &-& x_{2} &=& 5 \\\\-3x_1 &+& 4x_{2} &=& 14\\end{array} using 0 as the initial value for each variable. Then find the exact solution and compare. Jacobi Iteration: $\\begin{array}{ccc}\\hline n & x_1 & x_2 \\\\ \\hline 0 & 0 & 0 \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nActual solution: $(x_1,x_2)=$ [ANS]\nHelp: To enter a solution $(a,b)$, type your answer as a point (a,b).", "answer_v3": [ "1", "3.5", "1.7", "4.25", "1.85", "4.775", "(2,5)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "OL" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0007", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Determine which of the points $\\left(5,-3\\right)$, $\\left(2,2\\right)$, and $\\left(-1,1\\right)$ lie on both the lines $-5x_1-3x_2=-16$ and $2x_1+3x_2=1$.\nAnswer: [ANS]", "answer_v1": [ "(5,-3)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Determine which of the points $\\left(5,-2\\right)$, $\\left(-2,2\\right)$, and $\\left(-1,4\\right)$ lie on both the lines $x_1+x_2=3$ and $-2x_1+x_2=6$.\nAnswer: [ANS]", "answer_v2": [ "(-1,4)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Determine which of the points $\\left(-2,2\\right)$, $\\left(3,5\\right)$, and $\\left(-3,1\\right)$ lie on both the lines $-x_1+x_2=4$ and $-3x_1+5x_2=16$.\nAnswer: [ANS]", "answer_v3": [ "(-2,2)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0008", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear equations", "system", "systems", "numerical", "applications" ], "problem_v1": "Use partial pivoting with Gaussian elimination to find the solutions to the linear system\n\\begin{array}{rcrcr} x_1 &+& 2x_{2} &=& 7 \\\\5x_1 &-& 2x_{2} &=&-1\\end{array} $(x_1,x_2)=$ [ANS]\nHelp: To enter a solution $(a,b)$, type your answer as a point (a,b).", "answer_v1": [ "(1,3)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Use partial pivoting with Gaussian elimination to find the solutions to the linear system\n\\begin{array}{rcrcr}3x_1 &+& 5x_{2} &=& 9 \\\\4x_1 &+& 3x_{2} &=& 1\\end{array} $(x_1,x_2)=$ [ANS]\nHelp: To enter a solution $(a,b)$, type your answer as a point (a,b).", "answer_v2": [ "(-2,3)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Use partial pivoting with Gaussian elimination to find the solutions to the linear system\n\\begin{array}{rcrcr}-5x_1 &-& x_{2} &=& 2 \\\\8x_1 &+& 3x_{2} &=& 1\\end{array} $(x_1,x_2)=$ [ANS]\nHelp: To enter a solution $(a,b)$, type your answer as a point (a,b).", "answer_v3": [ "(-1,3)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0009", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [ "financial mathematics", "algebra" ], "problem_v1": "Use the elimination method to find all solutions of the system \\begin{array}{l} 5x+2y=12, \\\\ 7x+3y=17. \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "2", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use the elimination method to find all solutions of the system \\begin{array}{l} 5x+2y=-12, \\\\ 7x+3y=-16. \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "-4", "4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use the elimination method to find all solutions of the system \\begin{array}{l} 5x+2y=-8, \\\\ 7x+3y=-11. \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "-2", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0010", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [ "financial mathematics", "algebra" ], "problem_v1": "Use the substitution method to solve the system \\begin{array}{l}-x+y=-1, \\\\ 4x-3y=5. \\\\ \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "2", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use the substitution method to solve the system \\begin{array}{l}-x+y=8, \\\\ 4x-3y=-28. \\\\ \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "-4", "4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use the substitution method to solve the system \\begin{array}{l}-x+y=3, \\\\ 4x-3y=-11. \\\\ \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "-2", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0011", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear", "equation", "solve", "system", "fraction" ], "problem_v1": "Solve the following system of equations, using whichever method you wish.\n$\\left\\{\\begin{aligned} {-{\\textstyle \\frac{1}{3} }x+{\\textstyle \\frac{1}{4} }y} &={-{\\textstyle \\frac{67}{672} }} \\\\ {-{\\textstyle \\frac{1}{3} }x-{\\textstyle \\frac{1}{4} }y} &={-{\\textstyle \\frac{445}{672} }} \\end{aligned}\\right.$\nIf there is one solution, enter it as an ordered pair. If there is no solution, enter no solution. If there is an infinite number of solutions, enter infinite number of solutions. [ANS]", "answer_v1": [ "(8/7,9/8)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Solve the following system of equations, using whichever method you wish.\n$\\left\\{\\begin{aligned} {-{\\textstyle \\frac{1}{5} }x-{\\textstyle \\frac{1}{2} }y} &={-{\\textstyle \\frac{181}{440} }} \\\\ {-{\\textstyle \\frac{1}{4} }x-{\\textstyle \\frac{1}{4} }y} &={-{\\textstyle \\frac{41}{176} }} \\end{aligned}\\right.$\nIf there is one solution, enter it as an ordered pair. If there is no solution, enter no solution. If there is an infinite number of solutions, enter infinite number of solutions. [ANS]", "answer_v2": [ "(2/11,3/4)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Solve the following system of equations, using whichever method you wish.\n$\\left\\{\\begin{aligned} {-{\\textstyle \\frac{1}{2} }x+{\\textstyle \\frac{1}{5} }y} &={-{\\textstyle \\frac{1}{18} }} \\\\ {-{\\textstyle \\frac{1}{5} }x-{\\textstyle \\frac{1}{3} }y} &={-{\\textstyle \\frac{11}{30} }} \\end{aligned}\\right.$\nIf there is one solution, enter it as an ordered pair. If there is no solution, enter no solution. If there is an infinite number of solutions, enter infinite number of solutions. [ANS]", "answer_v3": [ "(4/9,5/6)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0012", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "algebra", "pair of lines", "inconsistent", "dependent" ], "problem_v1": "Solve the system \\begin{array}{l} x+2y=4, \\\\ 5x-y=9. \\\\ \\end{array} If the system has infinitely many solutions, express your answer in the form $x=x$ and $y$ as a function of $x$ Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "2", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system \\begin{array}{l} x+2y=3, \\\\ 5x-y=-18. \\\\ \\end{array} If the system has infinitely many solutions, express your answer in the form $x=x$ and $y$ as a function of $x$ Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "-3", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system \\begin{array}{l} x+2y=1, \\\\ 5x-y=-6. \\\\ \\end{array} If the system has infinitely many solutions, express your answer in the form $x=x$ and $y$ as a function of $x$ Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "-1", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0013", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [], "problem_v1": "Use the method of elimination to solve the system \\begin{array}{r@{\\,}c@{\\,}r} x+2 y &=& 4, \\cr-2x+2 y &=&-2. \\cr \\end{array} Answer: [ANS]\nNote: If there is more than one point, type the points as a comma separated list (e.g.: (1,2),(3,4)). If the system has no solutions, enter None.", "answer_v1": [ "(2,1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Use the method of elimination to solve the system \\begin{array}{r@{\\,}c@{\\,}r}-3x-y &=& 6, \\cr 4x+2 y &=&-6. \\cr \\end{array} Answer: [ANS]\nNote: If there is more than one point, type the points as a comma separated list (e.g.: (1,2),(3,4)). If the system has no solutions, enter None.", "answer_v2": [ "(-3,3)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Use the method of elimination to solve the system \\begin{array}{r@{\\,}c@{\\,}r}-2x-3 y &=&-1, \\cr-x-3 y &=&-2. \\cr \\end{array} Answer: [ANS]\nNote: If there is more than one point, type the points as a comma separated list (e.g.: (1,2),(3,4)). If the system has no solutions, enter None.", "answer_v3": [ "(-1,1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0014", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "algebra", "system of linear equations", "graphical method" ], "problem_v1": "Solve the system \\begin{array}{l} 3x+2y=8, \\\\ x-2y=0. \\\\ \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "2", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system \\begin{array}{l} 3x+2y=-3, \\\\ x-2y=-9. \\\\ \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "-3", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system \\begin{array}{l} 3x+2y=-1, \\\\ x-2y=-3. \\\\ \\end{array} Your answer is $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "-1", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0015", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [], "problem_v1": "Solve the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r} x &+y &=& 7 \\\\ 4x &-3 y &=& 14 \\\\ 7x &-7 y &=& 21 \\end{array}\\right. If there is no solution, enter NONE in both answer blanks. $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "5", "2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r} x &+y &=& 2 \\\\ 2x &-4 y &=&-62 \\\\ 6x &-6 y &=&-120 \\end{array}\\right. If there is no solution, enter NONE in both answer blanks. $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "-9", "11" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r} x &+y &=&-2 \\\\ 3x &-3 y &=&-18 \\\\ 5x &-7 y &=&-34 \\end{array}\\right. If there is no solution, enter NONE in both answer blanks. $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "-4", "2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0016", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [], "problem_v1": "Solve the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r} 2x_1 &+x_2 &=& 7 \\\\-8x_1 &-4x_2 &=&-28 \\end{array} \\right. $ \\left\\lbrack \\begin{array}{c} x_1 \\\\ x_2 \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+s$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "3.5", "0", "-0.5", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Solve the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r}-4x_1 &+x_2 &=&-9 \\\\-12x_1 &+3x_2 &=&-27 \\end{array} \\right. $ \\left\\lbrack \\begin{array}{c} x_1 \\\\ x_2 \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+s$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "2.25", "0", "0.25", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Solve the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r}-2x_1 &+x_2 &=&-7 \\\\ 6x_1 &-3x_2 &=& 21 \\end{array} \\right. $ \\left\\lbrack \\begin{array}{c} x_1 \\\\ x_2 \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+s$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "3.5", "0", "0.5", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0017", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [], "problem_v1": "Solve the system using row operations (or elementary matrices). \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r} 4x &+7 y &=&-13 \\\\-7x &+5 y &=&-29 \\end{array}\\right. $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "2", "-3" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system using row operations (or elementary matrices). \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r}-8x &-9 y &=& 23 \\\\ 5x &-5 y &=&-25 \\end{array}\\right. $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "-4", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system using row operations (or elementary matrices). \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r}-4x &-7 y &=&-92 \\\\-7x &+5 y &=&-23 \\end{array}\\right. $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "9", "8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0018", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [], "problem_v1": "For each system, determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions.\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r}-3x &+6 y &=& 0 \\cr-6x &+6 y &=& 0 \\end{array} \\right.$ [ANS] A\\. No solutions B\\. Unique solution: $x=0,\\ y=0$ C\\. Unique solution: $x=-6,\\ y=-3$ D\\. Infinitely many solutions E\\. Unique solution: $x=3,\\ y=0$ F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r} 9x &+2 y &=& 42 \\cr-8x &+5 y &=&-78 \\end{array} \\right.$ [ANS] A\\. Infinitely many solutions B\\. Unique solution: $x=-6,\\ y=6$ C\\. Unique solution: $x=0,\\ y=0$ D\\. No solutions E\\. Unique solution: $x=6,\\ y=-6$ F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r} 4x &+5 y &=&-5 \\cr-8x &-10 y &=& 11 \\end{array} \\right.$ [ANS] A\\. Unique solution: $x=-5,\\ y=11$ B\\. Infinitely many solutions C\\. No solutions D\\. Unique solution: $x=0,\\ y=0$ E\\. Unique solution: $x=11,\\ y=-5$ F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r}-2x &+3 y &=& 9 \\cr 4x &-6 y &=&-18 \\end{array} \\right.$ [ANS] A\\. Unique solution: $x=0,\\ y=0$ B\\. Infinitely many solutions C\\. Unique solution: $x=-4.5,\\ y=0$ D\\. No solutions E\\. Unique solution: $x=9,\\ y=-18$ F\\. None of the above", "answer_v1": [ "B", "E", "C", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "For each system, determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions.\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r} 4x &-3 y &=& 0 \\cr-3x &-6 y &=& 0 \\end{array} \\right.$ [ANS] A\\. Infinitely many solutions B\\. Unique solution: $x=6,\\ y=4$ C\\. No solutions D\\. Unique solution: $x=0,\\ y=0$ E\\. Unique solution: $x=1,\\ y=-9$ F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r} 3x &-8 y &=& 41 \\cr 5x &-5 y &=& 35 \\end{array} \\right.$ [ANS] A\\. Unique solution: $x=0,\\ y=0$ B\\. Unique solution: $x=-4,\\ y=3$ C\\. No solutions D\\. Unique solution: $x=3,\\ y=-4$ E\\. Infinitely many solutions F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r} 2x &+5 y &=&-9 \\cr-4x &-10 y &=& 19 \\end{array} \\right.$ [ANS] A\\. No solutions B\\. Unique solution: $x=19,\\ y=-9$ C\\. Unique solution: $x=0,\\ y=0$ D\\. Infinitely many solutions E\\. Unique solution: $x=-9,\\ y=19$ F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r}-2x &+2 y &=& 8 \\cr 4x &-4 y &=&-16 \\end{array} \\right.$ [ANS] A\\. Unique solution: $x=0,\\ y=0$ B\\. Infinitely many solutions C\\. No solutions D\\. Unique solution: $x=8,\\ y=-16$ E\\. Unique solution: $x=-4,\\ y=0$ F\\. None of the above", "answer_v2": [ "D", "D", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "For each system, determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions.\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r}-2x &+5 y &=& 9 \\cr-6x &+15 y &=& 27 \\end{array} \\right.$ [ANS] A\\. Unique solution: $x=9,\\ y=27$ B\\. Infinitely many solutions C\\. Unique solution: $x=0,\\ y=0$ D\\. Unique solution: $x=-4.5,\\ y=0$ E\\. No solutions F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r} 3x &+3 y &=&-12 \\cr-6x &-6 y &=& 25 \\end{array} \\right.$ [ANS] A\\. Unique solution: $x=0,\\ y=0$ B\\. No solutions C\\. Unique solution: $x=25,\\ y=-12$ D\\. Unique solution: $x=-12,\\ y=25$ E\\. Infinitely many solutions F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r} 4x &+8 y &=& 0 \\cr 5x &+8 y &=& 0 \\end{array} \\right.$ [ANS] A\\. Unique solution: $x=-8,\\ y=4$ B\\. Unique solution: $x=0,\\ y=0$ C\\. Unique solution: $x=12,\\ y=13$ D\\. Infinitely many solutions E\\. No solutions F\\. None of the above\n$ \\left\\{\\begin{array}{r@{}r@{}r@{}r} 5x &-4 y &=& 53 \\cr-6x &+5 y &=&-64 \\end{array} \\right.$ [ANS] A\\. Unique solution: $x=0,\\ y=0$ B\\. No solutions C\\. Unique solution: $x=9,\\ y=-2$ D\\. Infinitely many solutions E\\. Unique solution: $x=-2,\\ y=9$ F\\. None of the above", "answer_v3": [ "B", "B", "B", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0019", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "systems of equations", "linear system of equations" ], "problem_v1": "Solve the following system of equations. Your answer must be a point. If there is no solution, type None and if there are infinitely many solutions, type x for $x$, and an expression in terms of $x$ for the $y$-coordinate. \\begin{array}{rl} 2x+y &=-2\\\\ x+y &=-2 \\end{array} Answer: [ANS]", "answer_v1": [ "(0,-2)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Solve the following system of equations. Your answer must be a point. If there is no solution, type None and if there are infinitely many solutions, type x for $x$, and an expression in terms of $x$ for the $y$-coordinate. \\begin{array}{rl}-4x+4 y &=-2\\\\-2x+y &=-3 \\end{array} Answer: [ANS]", "answer_v2": [ "(2.5,2)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Solve the following system of equations. Your answer must be a point. If there is no solution, type None and if there are infinitely many solutions, type x for $x$, and an expression in terms of $x$ for the $y$-coordinate. \\begin{array}{rl}-2x+y &=-1\\\\-x-y &=3 \\end{array} Answer: [ANS]", "answer_v3": [ "(-0.666667,-2.33333)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0020", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear system of equations" ], "problem_v1": "Solve the following system of equations. \\begin{array}{rl} 4x+y &=-10\\\\ 4x+2 y &=-12 \\end{array} Answer: [ANS]", "answer_v1": [ "(-2,-2)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Solve the following system of equations. \\begin{array}{rl} x+5 y &=-5\\\\ x-2 y &=9 \\end{array} Answer: [ANS]", "answer_v2": [ "(5,-2)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Solve the following system of equations. \\begin{array}{rl} 2x+y &=11\\\\ 2x-2 y &=-4 \\end{array} Answer: [ANS]", "answer_v3": [ "(3,5)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0021", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Solve the linear system below by substitution. \\left\\lbrace \\begin{array}{rcr} 3x+y &=& 1 \\\\ 9x+3y &=& 3 \\end{array} \\right. If the answer is a line such as $y=x+1$, enter $x=x$ and $y=x+1$. If there is no solution, enter DNE for both answers.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "x", "(1-3*x)/1" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Solve the linear system below by substitution. \\left\\lbrace \\begin{array}{rcr} 5y-5x &=&-4 \\\\ 25x-25y &=& 20 \\end{array} \\right. If the answer is a line such as $y=x+1$, enter $x=x$ and $y=x+1$. If there is no solution, enter DNE for both answers.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "x", "(-4--5*x)/5" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Solve the linear system below by substitution. \\left\\lbrace \\begin{array}{rcr} y-2x &=&-2 \\\\ 2y-4x &=&-4 \\end{array} \\right. If the answer is a line such as $y=x+1$, enter $x=x$ and $y=x+1$. If there is no solution, enter DNE for both answers.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "x", "(-2--2*x)/1" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0022", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "algebra" ], "problem_v1": "A linear system may have a unique solution, no solution, or infinitely many solutions. Indicate the type of the system for the following examples by U, N, or I, respectively. [ANS] 1. $\\begin{array}{rrcl}2x+3y&=&5\\\\ x+6y &=& 7\\\\ \\end{array}$ [ANS] 2. $\\begin{array}{rrcl}2x+3y&=&0\\\\ 4x+6y &=& 0\\\\ \\end{array}$ [ANS] 3. $\\begin{array}{rrcl}x-y&=&15\\\\ y-x &=& 15\\\\ \\end{array}$ [ANS] 4. $\\begin{array}{rrcl}x+y&=&5\\\\ x+2y &=& 10\\\\ \\end{array}$ [ANS] 5. $\\begin{array}{rrcl}2x+3y&=&0\\\\ 2x+4y &=& 0\\\\ \\end{array}$", "answer_v1": [ "U", "I", "N", "U", "U" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ] ], "problem_v2": "A linear system may have a unique solution, no solution, or infinitely many solutions. Indicate the type of the system for the following MCSamples by U, N, or I, respectively. [ANS] 1. $\\begin{array}{rrcl}-2x+y&=&5\\\\ 2x-y &=&-5\\\\ \\end{array}$ [ANS] 2. $\\begin{array}{rrcl}7x+3y&=&\\pi\\\\ 4x-6y &=& \\pi^2\\\\ \\end{array}$ [ANS] 3. $\\begin{array}{rrcl}2x-3y&=&5\\\\ 4x-6y &=& 10\\\\ \\end{array}$ [ANS] 4. $\\begin{array}{rrcl}x-y&=&15\\\\ y-x &=& 15\\\\ \\end{array}$ [ANS] 5. $\\begin{array}{rrcl}x+y&=&5\\\\ x+2y &=& 10\\\\ \\end{array}$", "answer_v2": [ "I", "U", "I", "N", "U" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ] ], "problem_v3": "A linear system may have a unique solution, no solution, or infinitely many solutions. Indicate the type of the system for the following MCSamples by U, N, or I, respectively. [ANS] 1. $\\begin{array}{rrcl}x-3y&=&5\\\\ x+3y &=& 5\\\\ \\end{array}$ [ANS] 2. $\\begin{array}{rrcl}2x+3y&=&1\\\\ 4x+6y &=& 1\\\\ \\end{array}$ [ANS] 3. $\\begin{array}{rrcl}2x-3y&=&5\\\\ 4x-6y &=& 10\\\\ \\end{array}$ [ANS] 4. $\\begin{array}{rrcl}2x+3y&=&0\\\\ 4x+6y &=& 0\\\\ \\end{array}$ [ANS] 5. $\\begin{array}{rrcl}2x+3y&=&0\\\\ 2x+4y &=& 0\\\\ \\end{array}$", "answer_v3": [ "U", "N", "I", "I", "U" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ], [ "U", "N", "I" ] ] }, { "id": "Linear_algebra_0023", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Solve the system $\\left\\{\\begin{array}{rrrrr} 8x &+& 6 y &=& 80 \\\\ 6x &+& 36 y &=& 186 \\end{array}\\right.$ $x=$ [ANS] $y=$ [ANS]", "answer_v1": [ "7", "4" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system $\\left\\{\\begin{array}{rrrrr} 2x &+& 9 y &=& 89 \\\\ 3x &+& 27 y &=& 255 \\end{array}\\right.$ $x=$ [ANS] $y=$ [ANS]", "answer_v2": [ "4", "9" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system $\\left\\{\\begin{array}{rrrrr} 4x &+& 6 y &=& 42 \\\\ 4x &+& 24 y &=& 96 \\end{array}\\right.$ $x=$ [ANS] $y=$ [ANS]", "answer_v3": [ "6", "3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0024", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [ "algebra" ], "problem_v1": "The following system has infinitely many solutions. Express $x$ in terms of $y$ (we know $y$ can be any real number). $\\left\\{\\begin{array}{rrrrr} 56x &+& 42 y &=& 42 \\\\ 32x &+& 24 y &=& 24 \\end{array}\\right.$ $x=$ [ANS]", "answer_v1": [ "(6-6*y)/8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The following system has infinitely many solutions. Express $x$ in terms of $y$ (we know $y$ can be any real number). $\\left\\{\\begin{array}{rrrrr} 8x &+& 36 y &=& 12 \\\\ 18x &+& 81 y &=& 27 \\end{array}\\right.$ $x=$ [ANS]", "answer_v2": [ "(3-9*y)/2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The following system has infinitely many solutions. Express $x$ in terms of $y$ (we know $y$ can be any real number). $\\left\\{\\begin{array}{rrrrr} 24x &+& 36 y &=& 24 \\\\ 12x &+& 18 y &=& 12 \\end{array}\\right.$ $x=$ [ANS]", "answer_v3": [ "(4-6*y)/4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0025", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "systems of linear equations" ], "problem_v1": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 6r+5s &=& 3 \\\\ 5r+6s &=& 9 \\end{array} \\right.$ $r$=[ANS]\n$s$=[ANS]", "answer_v1": [ "-27/11", "39/11" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 4r+3s &=& 5 \\\\ 3r+4s &=& 7 \\end{array} \\right.$ $r$=[ANS]\n$s$=[ANS]", "answer_v2": [ "-1/7", "13/7" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 4r+3s &=& 3 \\\\ 3r+4s &=& 7 \\end{array} \\right.$ $r$=[ANS]\n$s$=[ANS]", "answer_v3": [ "-9/7", "19/7" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0026", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "systems of linear equations" ], "problem_v1": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 3a+b &=& 28 \\\\ 2b-3a &=& 1 \\end{array} \\right.$ $a$=[ANS]\n$b$=[ANS]", "answer_v1": [ "55/9", "87/9" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 3a+b &=& 20 \\\\ 2b-3a &=& 1 \\end{array} \\right.$ $a$=[ANS]\n$b$=[ANS]", "answer_v2": [ "39/9", "63/9" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 3a+b &=& 23 \\\\ 2b-3a &=& 1 \\end{array} \\right.$ $a$=[ANS]\n$b$=[ANS]", "answer_v3": [ "45/9", "72/9" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0027", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "systems of linear equations" ], "problem_v1": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 3(e+f) &=& 5e+f+8 \\\\ 4(f-e) &=& e+2f-3 \\end{array} \\right.$ $e$=[ANS]\n$f$=[ANS]", "answer_v1": [ "22/6", "46/6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 3(e+f) &=& 5e+f+2 \\\\ 4(f-e) &=& e+2f-1 \\end{array} \\right.$ $e$=[ANS]\n$f$=[ANS]", "answer_v2": [ "6/6", "12/6" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system of equations.\n$ \\left\\lbrace \\begin{array}{rcl} 3(e+f) &=& 5e+f+4 \\\\ 4(f-e) &=& e+2f-3 \\end{array} \\right.$ $e$=[ANS]\n$f$=[ANS]", "answer_v3": [ "14/6", "26/6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0028", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [], "problem_v1": "Solve the linear system \\left\\lbrace\\begin{array}{r@{\\}c@{\\}rl} 0.2x+0.2 y &=&-1 &\\\\ 0.2x+0.7 y &=&-2.5 & \\vphantom{9^{9^9}} \\end{array} \\right.\nIf there are infinitely many solutions, enter x for $x$ and write $y$ as a function of $x$ in the answer blank for $y$. If there are no solutions, enter None for $x$.\nIf there is only one solution, just enter the values for $x$ and $y$.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "-2", "-3" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the linear system \\left\\lbrace\\begin{array}{r@{\\}c@{\\}rl} 0.2x+0.8 y &=&-1.4 &\\\\-0.7x+0.4 y &=&-4.7 & \\vphantom{9^{9^9}} \\end{array} \\right.\nIf there are infinitely many solutions, enter x for $x$ and write $y$ as a function of $x$ in the answer blank for $y$. If there are no solutions, enter None for $x$.\nIf there is only one solution, just enter the values for $x$ and $y$.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "5", "-3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the linear system \\left\\lbrace\\begin{array}{r@{\\}c@{\\}rl} 0.2x+0.2 y &=&-1.2 &\\\\-0.4x+0.5 y &=&-0.3 & \\vphantom{9^{9^9}} \\end{array} \\right.\nIf there are infinitely many solutions, enter x for $x$ and write $y$ as a function of $x$ in the answer blank for $y$. If there are no solutions, enter None for $x$.\nIf there is only one solution, just enter the values for $x$ and $y$.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "-3", "-3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0029", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Solve the system using matrices (row operations) \\left\\{\\begin{array}{rl} x+2 y &=4, \\\\-2x+2 y &=-2. \\\\ \\end{array}\\right. How many solutions are there to this system? [ANS] A\\. None B\\. Exactly 1 C\\. Exactly 2 D\\. Exactly 3 E\\. Infinitely many F\\. None of the above\nIf there is one solution, give its coordinates in the answer spaces below. If there are infinitely many solutions, enter x in the answer blank for $x$ and enter a formula for $y$ in terms of $x$ in the answer blank for $y$. If there are no solutions, leave the answer blanks for $x$ and $y$ empty.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "B", "2", "1" ], "answer_type_v1": [ "MCS", "NV", "NV" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [], [] ], "problem_v2": "Solve the system using matrices (row operations) \\left\\{\\begin{array}{rl}-3x-y &=6, \\\\ 4x+2 y &=-6. \\\\ \\end{array}\\right. How many solutions are there to this system? [ANS] A\\. None B\\. Exactly 1 C\\. Exactly 2 D\\. Exactly 3 E\\. Infinitely many F\\. None of the above\nIf there is one solution, give its coordinates in the answer spaces below. If there are infinitely many solutions, enter x in the answer blank for $x$ and enter a formula for $y$ in terms of $x$ in the answer blank for $y$. If there are no solutions, leave the answer blanks for $x$ and $y$ empty.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "B", "-3", "3" ], "answer_type_v2": [ "MCS", "NV", "NV" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [], [] ], "problem_v3": "Solve the system using matrices (row operations) \\left\\{\\begin{array}{rl}-2x-3 y &=-1, \\\\-x-3 y &=-2. \\\\ \\end{array}\\right. How many solutions are there to this system? [ANS] A\\. None B\\. Exactly 1 C\\. Exactly 2 D\\. Exactly 3 E\\. Infinitely many F\\. None of the above\nIf there is one solution, give its coordinates in the answer spaces below. If there are infinitely many solutions, enter x in the answer blank for $x$ and enter a formula for $y$ in terms of $x$ in the answer blank for $y$. If there are no solutions, leave the answer blanks for $x$ and $y$ empty.\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "B", "-1", "1" ], "answer_type_v3": [ "MCS", "NV", "NV" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [], [] ] }, { "id": "Linear_algebra_0030", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [], "problem_v1": "Linear System-Two Variables Solve the following system of equations using elimination or substitution method. If there are no solutions, type \"N\" for both $x$ and $y$. If there are infinitely many solutions, type \"x\" for $x$, and an expression in terms of $x$ for $y$. \\begin{aligned} 5x+2 y &=-4\\\\ 2x+4 y &=-3 \\end{aligned} $x=$ [ANS] $y=$ [ANS]", "answer_v1": [ "-0.625", "-0.4375" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Linear System-Two Variables Solve the following system of equations using elimination or substitution method. If there are no solutions, type \"N\" for both $x$ and $y$. If there are infinitely many solutions, type \"x\" for $x$, and an expression in terms of $x$ for $y$. \\begin{aligned}-8x+8 y &=8\\\\-7x-3 y &=-3 \\end{aligned} $x=$ [ANS] $y=$ [ANS]", "answer_v2": [ "0", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Linear System-Two Variables Solve the following system of equations using elimination or substitution method. If there are no solutions, type \"N\" for both $x$ and $y$. If there are infinitely many solutions, type \"x\" for $x$, and an expression in terms of $x$ for $y$. \\begin{aligned}-4x+2 y &=-6\\\\-4x+y &=-3 \\end{aligned} $x=$ [ANS] $y=$ [ANS]", "answer_v3": [ "0", "-3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0031", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "equations", "linear system of equations" ], "problem_v1": "Solve the following system of equations:\n\\left\\lbrace \\begin{array}{rcrcr} 0.4x &-& 0.3 y &=& 0.9\\\\ 0.3x &+& 0.3 y &=& 1.2 \\end{array}\\right. Answer: [ANS]", "answer_v1": [ "(3,1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Solve the following system of equations:\n\\left\\lbrace \\begin{array}{rcrcr}-0.3x &+& 0.3 y &=& 3\\\\-0.3x &+& 0.2 y &=& 2.5 \\end{array}\\right. Answer: [ANS]", "answer_v2": [ "(-5,5)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Solve the following system of equations:\n\\left\\lbrace \\begin{array}{rcrcr}-0.4x &-& 0.3 y &=& 0.5\\\\ 0.5x &+& 0.2 y &=&-0.8 \\end{array}\\right. Answer: [ANS]", "answer_v3": [ "(-2,1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0032", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [ "linear", "system", "consistent" ], "problem_v1": "$\\left\\lbrace\\begin{array}{rrrrrr}& 10 \\,x&+& 8 \\, y&=&-4\\\\& 30 \\,x&+& 24 \\, y&=&k\\\\ \\end{array}\\right.$\nFor the above system of equations to be consistent, $k$ must equal [ANS]", "answer_v1": [ "-12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$\\left\\lbrace\\begin{array}{rrrrrr}& 4 \\,x&-&4 \\,y&=&10\\\\& 10 \\,x&-&10 \\,y&=&k\\\\ \\end{array}\\right.$\nFor the above system of equations to be consistent, $k$ must equal [ANS]", "answer_v2": [ "25" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$\\left\\lbrace\\begin{array}{rrrrrr}& 9 \\,x&+& 9 \\, y&=&-9\\\\& 21 \\,x&+& 21 \\, y&=&k\\\\ \\end{array}\\right.$\nFor the above system of equations to be consistent, $k$ must equal [ANS]", "answer_v3": [ "-21" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0033", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "3", "keywords": [ "linear", "system", "infinite", "solutions" ], "problem_v1": "If there are an infinite number of solutions to the system\n$\\begin{array}{rrrrrr}& 7 \\,x&+& 6 \\, y&=&h\\\\&-4 \\, x&+& k \\, y&=&6\\\\ \\end{array}$\nthen $k=$ [ANS], $h=$ [ANS]", "answer_v1": [ "-3.42857142857143", "-10.5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If there are an infinite number of solutions to the system\n$\\begin{array}{rrrrrr}& \\,x&-&2 \\,y&=&h\\\\&-8 \\, x&+& k \\, y&=&-2\\\\ \\end{array}$\nthen $k=$ [ANS], $h=$ [ANS]", "answer_v2": [ "16", "0.25" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If there are an infinite number of solutions to the system\n$\\begin{array}{rrrrrr}& 3 \\,x&+& 3 \\, y&=&h\\\\&-2 \\, x&+& k \\, y&=&9\\\\ \\end{array}$\nthen $k=$ [ANS], $h=$ [ANS]", "answer_v3": [ "-2", "-13.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0034", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear equations' 'system" ], "problem_v1": "Give a geometric description of the following systems of equations.\n[ANS] 1. $\\left\\lbrace\\begin{array}{rrrrrr}& 8 \\,x&+& 8 \\, y&=&-4\\\\&-20 \\, x&-&20 \\,y&=&10\\\\ \\end{array}\\right.$ [ANS] 2. $\\left\\lbrace\\begin{array}{rrrrrr}& 8 \\,x&+& 8 \\, y&=&-4\\\\&-20 \\, x&-&20 \\,y&=&12\\\\ \\end{array}\\right.$ [ANS] 3. $\\left\\lbrace\\begin{array}{rrrrrr}& 3 \\,x&-&5 \\,y&=&-2\\\\& 4 \\,x&-&2 \\,y&=&0\\\\ \\end{array}\\right.$", "answer_v1": [ "TWO LINES THAT ARE THE SAME", "TWO PARALLEL LINES", "TWO LINES INTERSECTING IN A POINT" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ], [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ], [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ] ], "problem_v2": "Give a geometric description of the following systems of equations.\n[ANS] 1. $\\left\\lbrace\\begin{array}{rrrrrr}&-2 \\, x&+& 2 \\, y&=&-10\\\\& 4 \\,x&-&4 \\,y&=&21\\\\ \\end{array}\\right.$ [ANS] 2. $\\left\\lbrace\\begin{array}{rrrrrr}&-2 \\, x&+& 2 \\, y&=&-10\\\\& 4 \\,x&-&4 \\,y&=&20\\\\ \\end{array}\\right.$ [ANS] 3. $\\left\\lbrace\\begin{array}{rrrrrr}& 5 \\,x&-& \\,y&=&1\\\\& 4 \\,x&-&2 \\,y&=&-7\\\\ \\end{array}\\right.$", "answer_v2": [ "TWO PARALLEL LINES", "TWO LINES THAT ARE THE SAME", "TWO LINES INTERSECTING IN A POINT" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ], [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ], [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ] ], "problem_v3": "Give a geometric description of the following systems of equations.\n[ANS] 1. $\\left\\lbrace\\begin{array}{rrrrrr}& 4 \\,x&+& 4 \\, y&=&-6\\\\& 10 \\,x&+& 10 \\, y&=&-15\\\\ \\end{array}\\right.$ [ANS] 2. $\\left\\lbrace\\begin{array}{rrrrrr}&-3 \\, x&+& 5 \\, y&=&3\\\\& 8 \\,x&-& \\,y&=&4\\\\ \\end{array}\\right.$ [ANS] 3. $\\left\\lbrace\\begin{array}{rrrrrr}& 4 \\,x&+& 4 \\, y&=&-6\\\\& 10 \\,x&+& 10 \\, y&=&-16\\\\ \\end{array}\\right.$", "answer_v3": [ "TWO LINES THAT ARE THE SAME", "TWO LINES INTERSECTING IN A POINT", "TWO PARALLEL LINES" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ], [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ], [ "Two parallel lines", "Two lines that are the same", "Two lines intersecting in a point" ] ] }, { "id": "Linear_algebra_0035", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 2 variables", "level": "2", "keywords": [ "linear equations' 'system" ], "problem_v1": "Solve the system:\n$\\left\\lbrace\\begin{array}{rrrrrr}& 4 \\,x&+& 5 \\, y&=&a\\\\& 5 \\,x&+& 6 \\, y&=&b\\\\ \\end{array}\\right.$\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "(6*a-5*b)*-1", "(4*b-5*a)*-1" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system:\n$\\left\\lbrace\\begin{array}{rrrrrr}& 4 \\,x&+& 3 \\, y&=&a\\\\&-3 \\, x&-&2 \\,y&=&b\\\\ \\end{array}\\right.$\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "(-2*a-3*b)*1", "(4*b--3*a)*1" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system:\n$\\left\\lbrace\\begin{array}{rrrrrr}& 4 \\,x&+& 3 \\, y&=&a\\\\& 3 \\,x&+& 2 \\, y&=&b\\\\ \\end{array}\\right.$\n$x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "(2*a-3*b)*-1", "(4*b-3*a)*-1" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0036", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Find the set of solutions for the linear system\n\\begin{array}{rcrcrcr}3x_1 &+& 6x_{2} &-& 3x_{3} &=&-8 \\\\ & & 5x_{2} &-& 8x_{3} &=& 9\\end{array} Use s1, s2, etc. for the free variables if necessary. $(x_1, x_2, x_3)=\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v1": [ "-(94/15)-11/5*s1", "9/5+8/5*s1", "s1" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the set of solutions for the linear system\n\\begin{array}{rcrcrcr}-3x_1 &-& 5x_{2} &+& 3x_{3} &=& 9 \\\\ &-& 4x_{2} &+& 7x_{3} &=&-7\\end{array} Use s1, s2, etc. for the free variables if necessary. $(x_1, x_2, x_3)=\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v2": [ "-(71/12)-23/12*s1", "7/4+7/4*s1", "s1" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the set of solutions for the linear system\n\\begin{array}{rcrcrcr}-3x_1 &-& 5x_{2} &-& 3x_{3} &=&-9 \\\\ & & 6x_{2} &+& 7x_{3} &=& 7\\end{array} Use s1, s2, etc. for the free variables if necessary. $(x_1, x_2, x_3)=\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v3": [ "19/18+17/18*s1", "7/6-7/6*s1", "s1" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0037", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "3", "keywords": [ "linear equations", "system", "systems", "numerical", "applications" ], "problem_v1": "The values of the first three Gauss-Seidel iterations are given for an unknown system. Find the values of the fourth iteration.\n$\\begin{array}{cccc}\\hline n & x_1 & x_2 & x_3 \\\\ \\hline 0 & 0 & 0 & 0 \\\\ \\hline 1 &-3 &-4 & 12 \\\\ \\hline 2 &-7 & 24 &-76 \\\\ \\hline 3 & 25 &-176 & 556 \\\\ \\hline 4 & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "-207", "1256", "-3972" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The values of the first three Gauss-Seidel iterations are given for an unknown system. Find the values of the fourth iteration.\n$\\begin{array}{cccc}\\hline n & x_1 & x_2 & x_3 \\\\ \\hline 0 & 0 & 0 & 0 \\\\ \\hline 1 & 3 &-4 & 12 \\\\ \\hline 2 & 7 & 24 &-76 \\\\ \\hline 3 &-25 &-176 & 556 \\\\ \\hline 4 & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "207", "1256", "-3972" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The values of the first three Gauss-Seidel iterations are given for an unknown system. Find the values of the fourth iteration.\n$\\begin{array}{cccc}\\hline n & x_1 & x_2 & x_3 \\\\ \\hline 0 & 0 & 0 & 0 \\\\ \\hline 1 & 3 &-4 &-12 \\\\ \\hline 2 & 7 & 24 & 76 \\\\ \\hline 3 &-25 &-176 &-556 \\\\ \\hline 4 & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "207", "1256", "3972" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0038", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Determine which of A-D form a solution to the given system for any choice of the free parameter $s_1$. List all letters that apply. If there is more than one answer, type them as a comma separated list.\n\\begin{array}{rcrcrcr}2x_1 &-& x_{2} &+& 3x_{3} &=& 14 \\\\-x_1 &+& x_{2} &-& x_{3} &=&-9\\end{array} HINT: All of the parameters of a solution must cancel completely when substituted into each equation. A. $\\left(13+2s_1,s_1,-\\left(4+s_1\\right)\\right)$ B. $\\left(5-2s_1,-\\left(4+s_1\\right),s_1\\right)$ C. $\\left( \\frac{6-3s_1}{4} ,s_1,- \\frac{2+s_1}{6} \\right)$ D. $\\left(5+s_1,3+2s_1,s_1\\right)$ Answer(s): [ANS]", "answer_v1": [ "AB" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Determine which of A-D form a solution to the given system for any choice of the free parameter $s_1$. List all letters that apply. If there is more than one answer, type them as a comma separated list.\n\\begin{array}{rcrcrcr}-2x_1 &+& 3x_{2} &+& 35x_{3} &=& 15 \\\\-x_1 &+& x_{2} &+& 15x_{3} &=& 8\\end{array} HINT: All of the parameters of a solution must cancel completely when substituted into each equation. A. $\\left(- \\frac{3s_1-6}{4} ,s_1,- \\frac{5-s_1}{70} \\right)$ B. $\\left(2-3s_1,4-6s_1,s_1\\right)$ C. $\\left(10s_1-9,-\\left(1+5s_1\\right),s_1\\right)$ D. $\\left(-\\left(11+2s_1\\right),s_1,- \\frac{1+s_1}{5} \\right)$ Answer(s): [ANS]", "answer_v2": [ "CD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Determine which of A-D form a solution to the given system for any choice of the free parameter $s_1$. List all letters that apply. If there is more than one answer, type them as a comma separated list.\n\\begin{array}{rcrcrcr}3x_1 &-& 2x_{2} &+& x_{3} &=&-8 \\\\-x_1 &+& x_{2} & & &=& 2\\end{array} HINT: All of the parameters of a solution must cancel completely when substituted into each equation. A. $\\left( \\frac{2-s_1}{6} ,s_1,- \\frac{2-s_1}{3} \\right)$ B. $\\left(s_1-2,s_1,-\\left(2+s_1\\right)\\right)$ C. $\\left(-\\left(4+s_1\\right),-\\left(2+s_1\\right),s_1\\right)$ D. $\\left(7+6s_1,7-4s_1,s_1\\right)$ Answer(s): [ANS]", "answer_v3": [ "BC" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Linear_algebra_0039", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "The linear system\n\\begin{array}{rcrcrcr} & & & & 4x_{3} &=& 8 \\\\ &-& x_{2} &+& x_{3} &=& 14 \\\\2x_1 &+& 2x_{2} &+& 6x_{3} &=& 0\\end{array} is not in echelon form. Begin by choosing which of the following statements are correct. If there is more than one reason why the system is not in echelon form, type the letters as a comma separated list. A. The system is not in echelon form because a variable is the leading variable of two or more equations. B. The system is not in echelon form because the system is not organized in a descending \"stair step\" pattern so that the index of the leading variables increases from the top to bottom. C. The system is not in echelon form because not every equation has a leading variable. Correct Letter(s): [ANS]\nNow write the system in echelon form. Equation 1: [ANS]\nEquation 2: [ANS]\nEquation 3: [ANS]\nFinally, solve the system. Use x1, x2, and x3 to enter the variables $x_1$, $x_2$, and $x_3$. If necessary, use s1, s2, etc. to enter the free variables $s_1$, $s_2$, etc. $(x_1, x_2, x_3)=\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v1": [ "B", "2*x1+2*x2+6*x3 = 0", "x3-x2 = 14", "4*x3 = 8", "6", "-12", "2" ], "answer_type_v1": [ "EX", "EQ", "EQ", "EQ", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "The linear system\n\\begin{array}{rcrcrcr} & & &-& 2x_{3} &=&-14 \\\\ &-& x_{2} &+& 2x_{3} &=& 8 \\\\2x_1 &+& 3x_{2} &+& 2x_{3} &=& 0\\end{array} is not in echelon form. Begin by choosing which of the following statements are correct. If there is more than one reason why the system is not in echelon form, type the letters as a comma separated list. A. The system is not in echelon form because a variable is the leading variable of two or more equations. B. The system is not in echelon form because the system is not organized in a descending \"stair step\" pattern so that the index of the leading variables increases from the top to bottom. C. The system is not in echelon form because not every equation has a leading variable. Correct Letter(s): [ANS]\nNow write the system in echelon form. Equation 1: [ANS]\nEquation 2: [ANS]\nEquation 3: [ANS]\nFinally, solve the system. Use x1, x2, and x3 to enter the variables $x_1$, $x_2$, and $x_3$. If necessary, use s1, s2, etc. to enter the free variables $s_1$, $s_2$, etc. $(x_1, x_2, x_3)=\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v2": [ "B", "2*x1+3*x2+2*x3 = 0", "2*x3-x2 = 8", "2*x3 = 14", "-16", "6", "7" ], "answer_type_v2": [ "EX", "EQ", "EQ", "EQ", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "The linear system\n\\begin{array}{rcrcrcr} & & & & 3x_{3} &=& 3 \\\\ &-& x_{2} &+& 4x_{3} &=&-6 \\\\6x_1 &+& 3x_{2} &+& 6x_{3} &=& 0\\end{array} is not in echelon form. Begin by choosing which of the following statements are correct. If there is more than one reason why the system is not in echelon form, type the letters as a comma separated list. A. The system is not in echelon form because a variable is the leading variable of two or more equations. B. The system is not in echelon form because the system is not organized in a descending \"stair step\" pattern so that the index of the leading variables increases from the top to bottom. C. The system is not in echelon form because not every equation has a leading variable. Correct Letter(s): [ANS]\nNow write the system in echelon form. Equation 1: [ANS]\nEquation 2: [ANS]\nEquation 3: [ANS]\nFinally, solve the system. Use x1, x2, and x3 to enter the variables $x_1$, $x_2$, and $x_3$. If necessary, use s1, s2, etc. to enter the free variables $s_1$, $s_2$, etc. $(x_1, x_2, x_3)=\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v3": [ "B", "6*x1+3*x2+6*x3 = 0", "4*x3-x2 = -6", "3*x3 = 3", "-6", "10", "1" ], "answer_type_v3": [ "EX", "EQ", "EQ", "EQ", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0040", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Determine which of the points $\\left(-4,-2,6\\right)$, $\\left(4,6,-1\\right)$, and $\\left(-6,-6,-2\\right)$ satisfy the linear system\n\\begin{array}{rcrcrcr}-3x_1 &+& 7x_{2} &+& 5x_{3} &=& 25 \\\\-9x_1 &+& 3x_{2} &+& 5x_{3} &=&-23\\end{array} Answer: [ANS]", "answer_v1": [ "(4,6,-1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Determine which of the points $\\left(2,-3,1\\right)$, $\\left(-2,-5,3\\right)$, and $\\left(1,-3,3\\right)$ satisfy the linear system\n\\begin{array}{rcrcrcr}-x_1 &+& x_{2} &-& 7x_{3} &=&-25 \\\\4x_1 &+& x_{2} &-& x_{3} &=&-2\\end{array} Answer: [ANS]", "answer_v2": [ "(1,-3,3)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Determine which of the points $\\left(6,1,-5\\right)$, $\\left(3,-1,-6\\right)$, and $\\left(-6,-3,-5\\right)$ satisfy the linear system\n\\begin{array}{rcrcrcr}2x_1 &+& 2x_{2} &+& 5x_{3} &=&-43 \\\\-2x_1 &-& 9x_{2} &-& 8x_{3} &=& 79\\end{array} Answer: [ANS]", "answer_v3": [ "(-6,-3,-5)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0041", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [ "system", "linear equations", "linear equations' 'system" ], "problem_v1": "Solve the system using elimination.\n\\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} 6x &+4 y &-3 z &=&-28 \\\\ 5x &-4 y &-4 z &=&-16 \\\\-4x &-3 y &-6 z &=& 19 \\end{array}\\right. $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v1": [ "-4", "-1", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the system using elimination.\n\\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} 2x &-2 y &-5 z &=&-7 \\\\-3x &-4 y &-4 z &=& 13 \\\\-6x &-3 y &+2 z &=& 26 \\end{array}\\right. $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v2": [ "-3", "-2", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the system using elimination.\n\\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} 2x &+3 y &-3 z &=&-20 \\\\ 5x &-6 y &-2 z &=&-12 \\\\ 2x &+5 y &+2 z &=&-14 \\end{array}\\right. $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v3": [ "-4", "-2", "2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0042", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [], "problem_v1": "Solve the equation 8x+7 y-5 z=3. $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+s$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.375", "0", "0", "-0.875", "1", "0", "0.625", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Solve the equation 2x-3 y-9 z=-12. $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+s$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-6", "0", "0", "1.5", "1", "0", "4.5", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Solve the equation 2x+5 y-3 z=12. $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+s$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "6", "0", "0", "-2.5", "1", "0", "1.5", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0043", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [ "algebra", "Gaussian elimination", "solving system of equations" ], "problem_v1": "The system of equations \\begin{array}{l} x-2y+z=1, \\\\ y+2z=3, \\\\ x+y+3z=6 \\\\ \\end{array} has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method. $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v1": [ "2", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The system of equations \\begin{array}{l} x-2y+z=-11, \\\\ y+2z=-1, \\\\ x+y+3z=-6 \\\\ \\end{array} has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method. $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v2": [ "-3", "3", "-2" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The system of equations \\begin{array}{l} x-2y+z=-5, \\\\ y+2z=-3, \\\\ x+y+3z=-6 \\\\ \\end{array} has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method. $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v3": [ "-1", "1", "-2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0044", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "3", "keywords": [ "System of Equations", "Matrix", "Augmented" ], "problem_v1": "Solve the system associated with the augmented matrix below. If the system is inconsistent, type \"No Solution\" in each blank. If the system is dependent, use the variable \"z\" as your free variable. \\left[\\begin{array}{ccc|c} 8 &3 &4 &3 \\cr-22 &-17 &24 &4 \\cr 2 &2 &-4 &-1 \\cr \\end{array}\\right] The solution to the system is: (x,y,z)=([ANS], [ANS], [ANS])", "answer_v1": [ "(((-1-3*2/8) - (-4-4*2/8)*z)/(2-3*2/8))*(-3/8) + 3/8 - 4/8*z", "((-1-3*2/8) - (-4-4*2/8)*z)/(2-3*2/8)", "z" ], "answer_type_v1": [ "EX", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the system associated with the augmented matrix below. If the system is inconsistent, type \"No Solution\" in each blank. If the system is dependent, use the variable \"z\" as your free variable. \\left[\\begin{array}{ccc|c}-13 &13 &-11 &-8 \\cr 138 &-48 &54 &52 \\cr-10 &-5 &2 &-1 \\cr \\end{array}\\right] The solution to the system is: (x,y,z)=([ANS], [ANS], [ANS])", "answer_v2": [ "NO SOLUTION", "NO SOLUTION", "NO SOLUTION" ], "answer_type_v2": [ "OE", "OE", "OE" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the system associated with the augmented matrix below. If the system is inconsistent, type \"No Solution\" in each blank. If the system is dependent, use the variable \"z\" as your free variable. \\left[\\begin{array}{ccc|c}-6 &3 &-7 &-6 \\cr 2 &-9 &-5 &-4 \\cr 10 &13 &12 &-5 \\cr \\end{array}\\right] The solution to the system is: (x,y,z)=([ANS], [ANS], [ANS])", "answer_v3": [ "-1.48969072164948", "-0.865979381443299", "1.76288659793814" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0045", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "3", "keywords": [ "algebra", "Gaussian elimination", "inconsistent", "dependent" ], "problem_v1": "Given the system of equations \\begin{array}{l} 2x-3y-9z=-8, \\\\ x+3z=5, \\\\-3x+y-4z=-9, \\\\ \\end{array}\n(a) determine whether the system is inconsistent or dependent; Your answer is (input inconsistent or dependent) [ANS]\n(b) if your answer is dependent in (a), find the complete solution. Write $x$ and $y$ as functions of $z$. $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "DEPENDENT", "5-3*z", "-9+3*(5-3*z)+4*z" ], "answer_type_v1": [ "MCS", "EX", "EX" ], "options_v1": [ [ "inconsistent", "dependent" ], [], [] ], "problem_v2": "Given the system of equations \\begin{array}{l} 2x-3y-9z=3, \\\\ x+3z=-9, \\\\-3x+y-4z=20, \\\\ \\end{array}\n(a) determine whether the system is inconsistent or dependent; Your answer is (input inconsistent or dependent) [ANS]\n(b) if your answer is dependent in (a), find the complete solution. Write $x$ and $y$ as functions of $z$. $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "DEPENDENT", "-9-3*z", "20+3*(-9-3*z)+4*z" ], "answer_type_v2": [ "MCS", "EX", "EX" ], "options_v2": [ [ "inconsistent", "dependent" ], [], [] ], "problem_v3": "Given the system of equations \\begin{array}{l} 2x-3y-9z=13, \\\\ x+3z=-7, \\\\-3x+y-4z=12, \\\\ \\end{array}\n(a) determine whether the system is inconsistent or dependent; Your answer is (input inconsistent or dependent) [ANS]\n(b) if your answer is dependent in (a), find the complete solution. Write $x$ and $y$ as functions of $z$. $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "DEPENDENT", "-7-3*z", "12+3*(-7-3*z)+4*z" ], "answer_type_v3": [ "MCS", "EX", "EX" ], "options_v3": [ [ "inconsistent", "dependent" ], [], [] ] }, { "id": "Linear_algebra_0046", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Solve the system of equations. Your answers must be fractions (decimals are not allowed). \\left\\lbrace \\begin{array}{rcr} 2x_{1}+x_{2}+x_{3} &=& 2 \\\\-x_{1}-x_{2}+x_{3} &=&-1 \\\\ x_{1}-2x_{2}-x_{3} &=&-1 \\end{array} \\right. $\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "5/9", "2/3", "2/9" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the system of equations. Your answers must be fractions (decimals are not allowed). \\left\\lbrace \\begin{array}{rcr}-2x_{1}-2x_{2}+x_{3} &=&-2 \\\\-x_{1}-3x_{2}-3x_{3} &=&-3 \\\\-x_{1}-3x_{2}+3x_{3} &=&-1 \\end{array} \\right. $\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "3/4", "5/12", "1/3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the system of equations. Your answers must be fractions (decimals are not allowed). \\left\\lbrace \\begin{array}{rcr}-x_{1}+x_{2}-2x_{3} &=&-2 \\\\-x_{1}+2x_{2}+3x_{3} &=& 3 \\\\-2x_{1}-x_{2}-2x_{3} &=&-3 \\end{array} \\right. $\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "7/17", "5/17", "16/17" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0047", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Solve the linear system below by substitution. \\left\\lbrace \\begin{array}{rcr} x-2y-z &=& 1 \\\\ x+y+z &=&-1 \\\\-2y-z &=& 2 \\end{array} \\right. Note: your answers must be fractions (decimals are not allowed).\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v1": [ "-1", "-2", "2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the linear system below by substitution. \\left\\lbrace \\begin{array}{rcr} x-5y-z &=&-4 \\\\ 5x+4y+z &=&-2 \\\\-5y-z &=& 5 \\end{array} \\right. Note: your answers must be fractions (decimals are not allowed).\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v2": [ "-9", "-48", "235" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the linear system below by substitution. \\left\\lbrace \\begin{array}{rcr} x-2y-z &=&-2 \\\\ x+y+z &=& 1 \\\\-2y-z &=&-3 \\end{array} \\right. Note: your answers must be fractions (decimals are not allowed).\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v3": [ "1", "3", "-3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0048", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "3", "keywords": [ "algebra" ], "problem_v1": "Solve the following system: \\begin{array}{rcrcrcl} x &+& y &+& z &=& 5 \\\\ 2x &-& y &+& 3z &=& 8 \\\\ 3x &+& y &-& 4z &=& 6 \\\\ \\end{array} Keep track of your calculations, because in the next problem you will have the same system, except that the right hand side is different. The solution is $x=$ [ANS], $y=$ [ANS], $z=$ [ANS].", "answer_v1": [ "3", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the following system: \\begin{array}{rcrcrcl} x &+& y &+& z &=&-4 \\\\ 2x &-& y &+& 3z &=&-31 \\\\ 3x &+& y &-& 4z &=& 11 \\\\ \\end{array} Keep track of your calculations, because in the next problem you will have the same system, except that the right hand side is different. The solution is $x=$ [ANS], $y=$ [ANS], $z=$ [ANS].", "answer_v2": [ "-5", "6", "-5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the following system: \\begin{array}{rcrcrcl} x &+& y &+& z &=&-5 \\\\ 2x &-& y &+& 3z &=&-16 \\\\ 3x &+& y &-& 4z &=& 4 \\\\ \\end{array} Keep track of your calculations, because in the next problem you will have the same system, except that the right hand side is different. The solution is $x=$ [ANS], $y=$ [ANS], $z=$ [ANS].", "answer_v3": [ "-3", "1", "-3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0049", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "3", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Solve the system by finding the reduced row-echelon form of the augmented matrix. \\left\\{\\begin{array}{ll}-2x-y-5 z &=-4 \\\\ x-y+z &=-1 \\\\ x-2 y &=-5 \\end{array}\\right. Fill in the blanks for the first 3 columns of the reduced row-echelon form of the augmented matrix:\n$\\begin{array}{ccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right|******\\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******& [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right|******\\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******&***& \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******& [ANS] & [ANS] & [ANS] &***& [ANS] & [ANS] & [ANS] &***\\\\ \\hline [ANS] & [ANS] & [ANS] &***\\\\ \\hline [ANS] & [ANS] & [ANS] &***\\\\ \\hline \\end{array}$ How many solutions are there to this system? [ANS] A\\. None B\\. Exactly 1 C\\. Exactly 2 D\\. Exactly 3 E\\. Infinitely many F\\. None of the above\nIf there is one solution, give its coordinates in the answer spaces below. If there are infinitely many solutions, enter z in the answer blank for $z$, enter a formula for $y$ in terms of $z$ in the answer blank for $y$ and enter a formula for $x$ in terms of $z$ in the answer blank for $x$. If there are no solutions, leave the answer blanks for $x$, $y$ and $z$ empty.\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v1": [ "1", "0", "2", "0", "1", "1", "0", "0", "0", "A", "No correct answer specified", "No correct answer specified", "No correct answer specified" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "OE", "OE", "OE" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F" ], [], [], [] ], "problem_v2": "Solve the system by finding the reduced row-echelon form of the augmented matrix. \\left\\{\\begin{array}{ll} 5x-y-17 z &=26 \\\\-2x-y+11 z &=-9 \\\\-3x+y+9 z &=-11 \\end{array}\\right. Fill in the blanks for the first 3 columns of the reduced row-echelon form of the augmented matrix:\n$\\begin{array}{ccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right|******\\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******& [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right|******\\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******&***& \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******& [ANS] & [ANS] & [ANS] &***& [ANS] & [ANS] & [ANS] &***\\\\ \\hline [ANS] & [ANS] & [ANS] &***\\\\ \\hline [ANS] & [ANS] & [ANS] &***\\\\ \\hline \\end{array}$ How many solutions are there to this system? [ANS] A\\. None B\\. Exactly 1 C\\. Exactly 2 D\\. Exactly 3 E\\. Infinitely many F\\. None of the above\nIf there is one solution, give its coordinates in the answer spaces below. If there are infinitely many solutions, enter z in the answer blank for $z$, enter a formula for $y$ in terms of $z$ in the answer blank for $y$ and enter a formula for $x$ in terms of $z$ in the answer blank for $x$. If there are no solutions, leave the answer blanks for $x$, $y$ and $z$ empty.\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v2": [ "1", "0", "-4", "0", "1", "-3", "0", "0", "0", "A", "No correct answer specified", "No correct answer specified", "No correct answer specified" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "OE", "OE", "OE" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F" ], [], [], [] ], "problem_v3": "Solve the system by finding the reduced row-echelon form of the augmented matrix. \\left\\{\\begin{array}{ll}-2x+2 y &=-8 \\\\ 3x+3 y-12 z &=-6 \\\\-2x-y+6 z &=-1 \\end{array}\\right. Fill in the blanks for the first 3 columns of the reduced row-echelon form of the augmented matrix:\n$\\begin{array}{ccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right|******\\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******& [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right|******\\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******&***& \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS]******[ANS] [ANS] [ANS]******& [ANS] & [ANS] & [ANS] &***& [ANS] & [ANS] & [ANS] &***\\\\ \\hline [ANS] & [ANS] & [ANS] &***\\\\ \\hline [ANS] & [ANS] & [ANS] &***\\\\ \\hline \\end{array}$ How many solutions are there to this system? [ANS] A\\. None B\\. Exactly 1 C\\. Exactly 2 D\\. Exactly 3 E\\. Infinitely many F\\. None of the above\nIf there is one solution, give its coordinates in the answer spaces below. If there are infinitely many solutions, enter z in the answer blank for $z$, enter a formula for $y$ in terms of $z$ in the answer blank for $y$ and enter a formula for $x$ in terms of $z$ in the answer blank for $x$. If there are no solutions, leave the answer blanks for $x$, $y$ and $z$ empty.\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v3": [ "1", "0", "-2", "0", "1", "-2", "0", "0", "0", "A", "No correct answer specified", "No correct answer specified", "No correct answer specified" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "OE", "OE", "OE" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F" ], [], [], [] ] }, { "id": "Linear_algebra_0050", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "2", "keywords": [], "problem_v1": "Use the Gauss-Jordan reduction to solve the following linear system:\n\\left\\{\\begin{array}{rcrcrcr} x_1 &-& x_2 &+& 4x_3 &=& 2 \\cr 4x_1 &-& 3x_2 &+& 7x_3 &=& 7 \\cr-2x_1 & & &+& 10x_3 &=&-2 \\end{array} \\right.\n$\\left[\\begin{array}{c} x_1 \\\\ x_2 \\\\ x_3 \\end{array} \\right]$ $=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\+\\ s\\ $ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "1", "-1", "0", "5", "9", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Use the Gauss-Jordan reduction to solve the following linear system:\n\\left\\{\\begin{array}{rcrcrcr} x_1 &-& x_2 &-& 5x_3 &=&-6 \\cr 4x_1 &-& 5x_2 &-& 4x_3 &=&-5 \\cr 3x_1 & & &-& 63x_3 &=&-75 \\end{array} \\right.\n$\\left[\\begin{array}{c} x_1 \\\\ x_2 \\\\ x_3 \\end{array} \\right]$ $=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\+\\ s\\ $ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-25", "-19", "0", "21", "16", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Use the Gauss-Jordan reduction to solve the following linear system:\n\\left\\{\\begin{array}{rcrcrcr} x_1 &-& x_2 &-& 4x_3 &=& 2 \\cr 4x_1 &-& 3x_2 &-& 4x_3 &=& 9 \\cr-3x_1 & & &-& 24x_3 &=&-9 \\end{array} \\right.\n$\\left[\\begin{array}{c} x_1 \\\\ x_2 \\\\ x_3 \\end{array} \\right]$ $=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\+\\ s\\ $ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "3", "1", "0", "-8", "-12", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0051", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "3", "keywords": [ "linear", "system", "no solution" ], "problem_v1": "Determine all values of $h$ and $k$ for which the linear system \\begin{array}{rrrrrrrr}& 6 \\,x&+& 6 \\, y&-&4 \\,z&=&5\\\\&-4 \\, x&-&7 \\,y&-&6 \\,z&=&-5\\\\&-10 \\, x&-&4 \\,y&+& h \\, z&=&k\\\\ \\end{array} has no solution.\nThe linear system has no solution if $k \\neq$ [ANS] and $h=$ [ANS].", "answer_v1": [ "-5", "24" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Determine all values of $h$ and $k$ for which the linear system \\begin{array}{rrrrrrrr}& 3 \\,x&-&3 \\,y&-&7 \\,z&=&-3\\\\&-6 \\, x&-&7 \\,y&-&8 \\,z&=&-3\\\\&-21 \\, x&-&5 \\,y&+& h \\, z&=&k\\\\ \\end{array} has no solution.\nThe linear system has no solution if $k \\neq$ [ANS] and $h=$ [ANS].", "answer_v2": [ "3", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Determine all values of $h$ and $k$ for which the linear system \\begin{array}{rrrrrrrr}& 4 \\,x&+& 4 \\, y&-&4 \\,z&=&7\\\\&-8 \\, x&-&4 \\,y&+& 2 \\, z&=&9\\\\&-28 \\, x&-&20 \\,y&+& h \\, z&=&k\\\\ \\end{array} has no solution.\nThe linear system has no solution if $k \\neq$ [ANS] and $h=$ [ANS].", "answer_v3": [ "-3", "16" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0052", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 3 variables", "level": "3", "keywords": [ "linear equations' 'system" ], "problem_v1": "Determine whether the following system has no solution, an infinite number of solutions or a unique solution.\n[ANS] 1. $\\left\\lbrace\\begin{array}{rrrrrrrr}&-8 \\, x&-&4 \\,y&+& 10 \\, z&=&-4\\\\& 16 \\,x&+& 8 \\, y&-&20 \\,z&=&8\\\\&-28 \\, x&-&14 \\,y&+& 35 \\, z&=&-14\\\\ \\end{array}\\right.$ [ANS] 2. $\\left\\lbrace\\begin{array}{rrrrrrrr}& 7 \\,x&+& 5 \\, y&-&3 \\,z&=&5\\\\&-2 \\, x&-&4 \\,y&-&3 \\,z&=&-2\\\\&-17 \\, x&-&7 \\,y&+& 15 \\, z&=&-11\\\\ \\end{array}\\right.$ [ANS] 3. $\\left\\lbrace\\begin{array}{rrrrrrrr}& 7 \\,x&+& 5 \\, y&-&3 \\,z&=&5\\\\&-2 \\, x&-&4 \\,y&-&3 \\,z&=&-2\\\\&-17 \\, x&-&7 \\,y&+& 15 \\, z&=&-10\\\\ \\end{array}\\right.$ [ANS] 4. $\\left\\lbrace\\begin{array}{rrrrrrrr}&-3 \\, x&-&6 \\,y&-&13 \\,z&=&5\\\\& 2 \\,x&+& 3 \\, y&+& 7 \\, z&=&5\\\\& \\,x&+& 2 \\, y&+& 4 \\, z&=&1\\\\ \\end{array}\\right.$", "answer_v1": [ "INFINITE SOLUTIONS", "INFINITE SOLUTIONS", "NO SOLUTION", "Unique Solution" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ] ], "problem_v2": "Determine whether the following system has no solution, an infinite number of solutions or a unique solution.\n[ANS] 1. $\\left\\lbrace\\begin{array}{rrrrrrrr}& \\,x&-& \\,y&-&7 \\,z&=&-1\\\\&-3 \\, x&-&4 \\,y&-&5 \\,z&=&-1\\\\&-9 \\, x&-&5 \\,y&+& 11 \\, z&=&1\\\\ \\end{array}\\right.$ [ANS] 2. $\\left\\lbrace\\begin{array}{rrrrrrrr}&-3 \\, x&-&6 \\,y&-&16 \\,z&=&2\\\\& 4 \\,x&+& 7 \\, y&+& 19 \\, z&=&0\\\\& 3 \\,x&+& 6 \\, y&+& 15 \\, z&=&10\\\\ \\end{array}\\right.$ [ANS] 3. $\\left\\lbrace\\begin{array}{rrrrrrrr}&-3 \\, x&-&6 \\,y&+& 3 \\, z&=&-6\\\\&-4 \\, x&-&8 \\,y&+& 4 \\, z&=&-8\\\\& 7 \\,x&+& 14 \\, y&-&7 \\,z&=&14\\\\ \\end{array}\\right.$ [ANS] 4. $\\left\\lbrace\\begin{array}{rrrrrrrr}& \\,x&-& \\,y&-&7 \\,z&=&-1\\\\&-3 \\, x&-&4 \\,y&-&5 \\,z&=&-1\\\\&-9 \\, x&-&5 \\,y&+& 11 \\, z&=&-4\\\\ \\end{array}\\right.$", "answer_v2": [ "INFINITE SOLUTIONS", "UNIQUE SOLUTION", "INFINITE SOLUTIONS", "No Solution" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ] ], "problem_v3": "Determine whether the following system has no solution, an infinite number of solutions or a unique solution.\n[ANS] 1. $\\left\\lbrace\\begin{array}{rrrrrrrr}& 3 \\,x&+& 3 \\, y&-& \\,z&=&7\\\\&-5 \\, x&-&2 \\,y&+& \\, z&=&5\\\\&-19 \\, x&-&13 \\,y&+& 5 \\, z&=&-8\\\\ \\end{array}\\right.$ [ANS] 2. $\\left\\lbrace\\begin{array}{rrrrrrrr}&-4 \\, x&-&8 \\,y&-&17 \\,z&=&8\\\\& \\,x&+& \\, y&+& 3 \\, z&=&1\\\\& 3 \\,x&+& 6 \\, y&+& 12 \\, z&=&3\\\\ \\end{array}\\right.$ [ANS] 3. $\\left\\lbrace\\begin{array}{rrrrrrrr}& 3 \\,x&+& 3 \\, y&-& \\,z&=&7\\\\&-5 \\, x&-&2 \\,y&+& \\, z&=&5\\\\&-19 \\, x&-&13 \\,y&+& 5 \\, z&=&-11\\\\ \\end{array}\\right.$ [ANS] 4. $\\left\\lbrace\\begin{array}{rrrrrrrr}& 9 \\,x&-&15 \\,y&-&15 \\,z&=&-9\\\\&-15 \\, x&+& 25 \\, y&+& 25 \\, z&=&15\\\\& 21 \\,x&-&35 \\,y&-&35 \\,z&=&-21\\\\ \\end{array}\\right.$", "answer_v3": [ "NO SOLUTION", "UNIQUE SOLUTION", "INFINITE SOLUTIONS", "Infinite Solutions" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ], [ "Unique Solution", "Infinite Solutions", "No Solution" ] ] }, { "id": "Linear_algebra_0053", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 4 or more variables", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Suppose that a system of six equations with twelve unknowns is in echelon form. How many leading variables are there? Number of leading variables: [ANS]", "answer_v1": [ "6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that a system of two equations with ten unknowns is in echelon form. How many leading variables are there? Number of leading variables: [ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that a system of three equations with nine unknowns is in echelon form. How many leading variables are there? Number of leading variables: [ANS]", "answer_v3": [ "3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0054", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 4 or more variables", "level": "3", "keywords": [ "Matrix", "System of Equations" ], "problem_v1": "Given the augmented matrix below, solve the associated system of equations. For your variables, use $x1$, $x2$, $x3$, $x4$, $x5$, $x6$, $x7$, and $x8$.\n\\left[\\begin{array}{rrrrrrrr|r} 1 &4 &-4 &-3 &1 &1 &-2 &0 &-1 \\cr 0 &0 &0 &0 &1 &3 &-4 &0 &0 \\cr 0 &0 &0 &0 &0 &1 &-1 &-2 &-7 \\cr 0 &0 &0 &0 &0 &0 &0 &1 &-6 \\cr \\end{array}\\right] The solution is ([ANS], [ANS], [ANS], [ANS], [ANS], [ANS], [ANS], [ANS])", "answer_v1": [ "-1-4*(x2)--4*(x3)--3*(x4)-1*(0-3*(-7--1*(x7)--2*(-6))--4*(x7)-0*(-6))-1*(-7--1*(x7)--2*(-6))--2*(x7)-0*(-6)", "x2", "x3", "x4", "0-3*(-7--1*(x7)--2*(-6))--4*(x7)-0*(-6)", "-7--1*(x7)--2*(-6)", "x7", "-6" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Given the augmented matrix below, solve the associated system of equations. For your variables, use $x1$, $x2$, $x3$, $x4$, $x5$, $x6$, $x7$, and $x8$.\n\\left[\\begin{array}{rrrrrrrr|r} 1 &-3 &8 &-3 &-6 &-3 &1 &-8 &-3 \\cr 0 &0 &1 &3 &-1 &6 &-6 &-6 &1 \\cr 0 &0 &0 &0 &0 &1 &-4 &1 &-7 \\cr 0 &0 &0 &0 &0 &0 &1 &-6 &-7 \\cr \\end{array}\\right] The solution is ([ANS], [ANS], [ANS], [ANS], [ANS], [ANS], [ANS], [ANS])", "answer_v2": [ "-3--3*(x2)-8*(1-3*(x4)--1*(x5)-6*(-7--4*(-7--6*(x8))-1*(x8))--6*(-7--6*(x8))--6*(x8))--3*(x4)--6*(x5)--3*(-7--4*(-7--6*(x8))-1*(x8))-1*(-7--6*(x8))--8*(x8)", "x2", "1-3*(x4)--1*(x5)-6*(-7--4*(-7--6*(x8))-1*(x8))--6*(-7--6*(x8))--6*(x8)", "x4", "x5", "-7--4*(-7--6*(x8))-1*(x8)", "-7--6*(x8)", "x8" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Given the augmented matrix below, solve the associated system of equations. For your variables, use $x1$, $x2$, $x3$, $x4$, $x5$, $x6$, $x7$, and $x8$.\n\\left[\\begin{array}{rrrrrrrr|r} 1 &1 &-6 &-3 &6 &8 &7 &-6 &4 \\cr 0 &0 &1 &-4 &-5 &-9 &1 &9 &0 \\cr 0 &0 &0 &0 &1 &6 &2 &-7 &-2 \\cr 0 &0 &0 &0 &0 &1 &-4 &3 &8 \\cr \\end{array}\\right] The solution is ([ANS], [ANS], [ANS], [ANS], [ANS], [ANS], [ANS], [ANS])", "answer_v3": [ "4-1*(x2)--6*(0--4*(x4)--5*(-2-6*(8--4*(x7)-3*(x8))-2*(x7)--7*(x8))--9*(8--4*(x7)-3*(x8))-1*(x7)-9*(x8))--3*(x4)-6*(-2-6*(8--4*(x7)-3*(x8))-2*(x7)--7*(x8))-8*(8--4*(x7)-3*(x8))-7*(x7)--6*(x8)", "x2", "0--4*(x4)--5*(-2-6*(8--4*(x7)-3*(x8))-2*(x7)--7*(x8))--9*(8--4*(x7)-3*(x8))-1*(x7)-9*(x8)", "x4", "-2-6*(8--4*(x7)-3*(x8))-2*(x7)--7*(x8)", "8--4*(x7)-3*(x8)", "x7", "x8" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0055", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 4 or more variables", "level": "3", "keywords": [ "System of Equations", "Matrix", "Independant", "Inconsistent" ], "problem_v1": "Given the augmented matrix below, determine if the associated system of equations is independent, dependent, or inconsistent. \\left[\\begin{array}{cccc} 5 &2 &2 &4 \\cr 27 &10 &7 &24 \\cr-2 &0 &3 &-4 \\cr 0 &-1 &-2 &-1 \\cr \\end{array}\\right] \\left[\\begin{array}{c} 0 \\cr 6 \\cr-6 \\cr-2 \\cr \\end{array}\\right] The system is [ANS].", "answer_v1": [ "DEPENDENT" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "independent", "dependent", "inconsistent" ] ], "problem_v2": "Given the augmented matrix below, determine if the associated system of equations is independent, dependent, or inconsistent. \\left[\\begin{array}{cccc}-8 &8 &-7 &-3 \\cr 8 &-3 &-6 &-3 \\cr 1 &-8 &3 &-1 \\cr 6 &-6 &-6 &-4 \\cr \\end{array}\\right] \\left[\\begin{array}{c} 1 \\cr-6 \\cr-3 \\cr 1 \\cr \\end{array}\\right] The system is [ANS].", "answer_v2": [ "INDEPENDENT" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "independent", "dependent", "inconsistent" ] ], "problem_v3": "Given the augmented matrix below, determine if the associated system of equations is independent, dependent, or inconsistent. \\left[\\begin{array}{cccc}-4 &2 &-4 &1 \\cr 64 &-52 &-24 &-42 \\cr 7 &-6 &-4 &-5 \\cr-9 &1 &9 &6 \\cr \\end{array}\\right] \\left[\\begin{array}{c} 2 \\cr-28 \\cr-4 \\cr 3 \\cr \\end{array}\\right] The system is [ANS].", "answer_v3": [ "INCONSISTENT" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "independent", "dependent", "inconsistent" ] ] }, { "id": "Linear_algebra_0056", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 4 or more variables", "level": "3", "keywords": [ "algebra" ], "problem_v1": "The solution of the linear system \\begin{array}{rcrcrcrcl} r &+& s &+& t &+& u &=& 0 \\\\ r &+& 2s &+& 3t &+& 4u &=&-1 \\\\ r &-& s &+& 2t &-& u &=& 5 \\\\ r &+& 2s &-& 3t &+& 2u &=&-13 \\\\\n\\end{array} is $r=$ [ANS], $s=$ [ANS], $t=$ [ANS], $u=$ [ANS].", "answer_v1": [ "-2", "2", "3", "-3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The solution of the linear system \\begin{array}{rcrcrcrcl} r &+& s &+& t &+& u &=&-3 \\\\ r &+& 2s &+& 3t &+& 4u &=& 6 \\\\ r &-& s &+& 2t &-& u &=&-7 \\\\ r &+& 2s &-& 3t &+& 2u &=& 6 \\\\\n\\end{array} is $r=$ [ANS], $s=$ [ANS], $t=$ [ANS], $u=$ [ANS].", "answer_v2": [ "-2", "-5", "-2", "6" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The solution of the linear system \\begin{array}{rcrcrcrcl} r &+& s &+& t &+& u &=&-8 \\\\ r &+& 2s &+& 3t &+& 4u &=&-21 \\\\ r &-& s &+& 2t &-& u &=& 7 \\\\ r &+& 2s &-& 3t &+& 2u &=&-19 \\\\\n\\end{array} is $r=$ [ANS], $s=$ [ANS], $t=$ [ANS], $u=$ [ANS].", "answer_v3": [ "-2", "-3", "1", "-4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0057", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 4 or more variables", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Solve: \\left(\\begin{array}{cc} 8x & y+6 \\cr z-6 & w \\end{array}\\right)=\\left(\\begin{array}{cc} 7 & 4 \\cr 4 & 6 \\end{array}\\right) $x=$ [ANS] $y=$ [ANS] $z=$ [ANS] $w=$ [ANS]", "answer_v1": [ "0.875", "-2", "10", "6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Solve: \\left(\\begin{array}{cc} 2x & y+9 \\cr z-3 & w \\end{array}\\right)=\\left(\\begin{array}{cc} 4 & 9 \\cr 4 & 3 \\end{array}\\right) $x=$ [ANS] $y=$ [ANS] $z=$ [ANS] $w=$ [ANS]", "answer_v2": [ "2", "0", "7", "3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Solve: \\left(\\begin{array}{cc} 4x & y+6 \\cr z-4 & w \\end{array}\\right)=\\left(\\begin{array}{cc} 6 & 3 \\cr 4 & 8 \\end{array}\\right) $x=$ [ANS] $y=$ [ANS] $z=$ [ANS] $w=$ [ANS]", "answer_v3": [ "1.5", "-3", "8", "8" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0058", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Systems with 4 or more variables", "level": "4", "keywords": [ "algebra" ], "problem_v1": "The following system has an infinite number of solutions. Write the solution in terms of $y$ and $z$. \\begin{array}{rcrcrcrcr} 2 w &-& x &+& 4 y &-& z &=& 3 \\\\ w &+& x &-& y &+& 3 z &=& 3 \\\\ 3 w & & &+& 3 y &+& 2 z &=& 6 \\\\ 3 w &-& 3x &+& 9 y &-& 5 z &=& 3 \\end{array} $w$=[ANS]\n$x$=[ANS]", "answer_v1": [ "(3+3+(1-4)*y+(1-3)*z)/3", "(-3+2*3+(2+4)*y-(1+2*3)*z)/3" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "The following system has an infinite number of solutions. Write the solution in terms of $y$ and $z$. \\begin{array}{rcrcrcrcr} 2 w &-& x &+& 1 y &-& z &=& 4 \\\\ w &+& x &-& y &+& 2 z &=& 1 \\\\ 3 w & & &+& 0 y &+& 1 z &=& 5 \\\\ 3 w &-& 3x &+& 3 y &-& 4 z &=& 7 \\end{array} $w$=[ANS]\n$x$=[ANS]", "answer_v2": [ "(4+1+(1-1)*y+(1-2)*z)/3", "(-4+2*1+(2+1)*y-(1+2*2)*z)/3" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "The following system has an infinite number of solutions. Write the solution in terms of $y$ and $z$. \\begin{array}{rcrcrcrcr} 2 w &-& x &+& 2 y &-& z &=& 3 \\\\ w &+& x &-& y &+& 3 z &=& 2 \\\\ 3 w & & &+& 1 y &+& 2 z &=& 5 \\\\ 3 w &-& 3x &+& 5 y &-& 5 z &=& 4 \\end{array} $w$=[ANS]\n$x$=[ANS]", "answer_v3": [ "(3+2+(1-2)*y+(1-3)*z)/3", "(-3+2*2+(2+2)*y-(1+2*3)*z)/3" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0059", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Matrix-vector forms", "level": "2", "keywords": [ "linear", "system", "matrix" ], "problem_v1": "Write the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} & 9 y &+10 z &=& 11 \\\\-5x &-4 y & &=&-3 \\\\-6x &+3 y &+4 z &=&-7 \\end{array} \\right. in matrix form.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "0", "9", "10", "-5", "-4", "0", "-6", "3", "4", "11", "-3", "-7" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Write the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} & 4 y &-5 z &=&-2 \\\\ 3x &-3 y & &=& 9 \\\\ 11x &-6 y &+2 z &=& 10 \\end{array} \\right. in matrix form.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "0", "4", "-5", "3", "-3", "0", "11", "-6", "2", "-2", "9", "10" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Write the system \\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} & 10 y &-5 z &=& 9 \\\\-6x &-2 y & &=& 4 \\\\ 2x &-7 y &-3 z &=& 11 \\end{array} \\right. in matrix form.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "0", "10", "-5", "-6", "-2", "0", "2", "-7", "-3", "9", "4", "11" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0060", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Matrix-vector forms", "level": "2", "keywords": [ "Matrix", "System of Equations", "algebra", "matrix operation", "matrix" ], "problem_v1": "Write the system of equations \\begin{array}{r} 5x+4 y-3 z=-4\\\\ 4x-3 y-4 z=-1\\\\-4x-3 y-4 z=0\\\\ \\end{array} as a matrix equation, that is, rewrite it in the form \\left[\\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23}\\\\ a_{31} & a_{32} & a_{33}\\\\ \\end{array}\\right] \\left[\\begin{array}{l} x \\\\ y\\\\ z\\\\ \\end{array}\\right]=\\left[\\begin{array}{l} b_1 \\\\ b_2\\\\ b_3\\\\ \\end{array}\\right] Input your answer below: $a_{11}=$ [ANS]\n$a_{12}=$ [ANS]\n$a_{13}=$ [ANS]\n$a_{21}=$ [ANS]\n$a_{22}=$ [ANS]\n$a_{23}=$ [ANS]\n$a_{31}=$ [ANS]\n$a_{32}=$ [ANS]\n$a_{33}=$ [ANS]\n$b_1=$ [ANS]\n$b_2=$ [ANS]\n$b_3=$ [ANS]", "answer_v1": [ "5", "4", "-3", "4", "-3", "-4", "-4", "-3", "-4", "-4", "-1", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Write the system of equations \\begin{array}{r} 2x-2 y-5 z=-2\\\\-2x-4 y-4 z=1\\\\-5x-2 y-4 z=-4\\\\ \\end{array} as a matrix equation, that is, rewrite it in the form \\left[\\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23}\\\\ a_{31} & a_{32} & a_{33}\\\\ \\end{array}\\right] \\left[\\begin{array}{l} x \\\\ y\\\\ z\\\\ \\end{array}\\right]=\\left[\\begin{array}{l} b_1 \\\\ b_2\\\\ b_3\\\\ \\end{array}\\right] Input your answer below: $a_{11}=$ [ANS]\n$a_{12}=$ [ANS]\n$a_{13}=$ [ANS]\n$a_{21}=$ [ANS]\n$a_{22}=$ [ANS]\n$a_{23}=$ [ANS]\n$a_{31}=$ [ANS]\n$a_{32}=$ [ANS]\n$a_{33}=$ [ANS]\n$b_1=$ [ANS]\n$b_2=$ [ANS]\n$b_3=$ [ANS]", "answer_v2": [ "2", "-2", "-5", "-2", "-4", "-4", "-5", "-2", "-4", "-2", "1", "-4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Write the system of equations \\begin{array}{r} 3x+3 y-2 z=-2\\\\ 5x-5 y-3 z=2\\\\ 2x+5 y-4 z=2\\\\ \\end{array} as a matrix equation, that is, rewrite it in the form \\left[\\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23}\\\\ a_{31} & a_{32} & a_{33}\\\\ \\end{array}\\right] \\left[\\begin{array}{l} x \\\\ y\\\\ z\\\\ \\end{array}\\right]=\\left[\\begin{array}{l} b_1 \\\\ b_2\\\\ b_3\\\\ \\end{array}\\right] Input your answer below: $a_{11}=$ [ANS]\n$a_{12}=$ [ANS]\n$a_{13}=$ [ANS]\n$a_{21}=$ [ANS]\n$a_{22}=$ [ANS]\n$a_{23}=$ [ANS]\n$a_{31}=$ [ANS]\n$a_{32}=$ [ANS]\n$a_{33}=$ [ANS]\n$b_1=$ [ANS]\n$b_2=$ [ANS]\n$b_3=$ [ANS]", "answer_v3": [ "3", "3", "-2", "5", "-5", "-3", "2", "5", "-4", "-2", "2", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0061", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Matrix-vector forms", "level": "2", "keywords": [ "linear algebra", "matrix equation" ], "problem_v1": "Solve the matrix equation $Ax=b$, where\nA=\\left[\\begin{array}{cc} 5 &2\\cr 2 &2 \\end{array}\\right], \\quad x=\\left[\\begin{array}{cc} x_1 \\\\ x_2 \\end{array} \\right] \\text{and} \\quad b=\\left[\\begin{array}{c}-14\\cr-8 \\end{array}\\right]. The solution is: $x_1=$ [ANS]\n$x_2=$ [ANS]", "answer_v1": [ "-2", "-2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the matrix equation $Ax=b$, where\nA=\\left[\\begin{array}{cc}-8 &8\\cr-7 &-1 \\end{array}\\right], \\quad x=\\left[\\begin{array}{cc} x_1 \\\\ x_2 \\end{array} \\right] \\text{and} \\quad b=\\left[\\begin{array}{c}-56\\cr-33 \\end{array}\\right]. The solution is: $x_1=$ [ANS]\n$x_2=$ [ANS]", "answer_v2": [ "5", "-2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the matrix equation $Ax=b$, where\nA=\\left[\\begin{array}{cc}-4 &2\\cr-4 &0 \\end{array}\\right], \\quad x=\\left[\\begin{array}{cc} x_1 \\\\ x_2 \\end{array} \\right] \\text{and} \\quad b=\\left[\\begin{array}{c} 8\\cr 12 \\end{array}\\right]. The solution is: $x_1=$ [ANS]\n$x_2=$ [ANS]", "answer_v3": [ "-3", "-2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0062", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Matrix-vector forms", "level": "3", "keywords": [ "algebra" ], "problem_v1": "Solve the division problem \\left[\\begin{array}{ccc} 3 &0 &0\\cr-4 &2 &0\\cr-4 &-2 &-2 \\end{array}\\right] \\left[\\begin{aligned} x \\\\ y \\\\ z\\end{aligned}\\right]=\\left[\\begin{array}{c}-3\\cr 6\\cr 4 \\end{array}\\right] $ \\left[\\begin{aligned} x \\\\ y \\\\ z\\end{aligned}\\right]=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-1", "1", "-1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the division problem \\left[\\begin{array}{ccc}-5 &0 &0\\cr-4 &-2 &0\\cr-4 &4 &-2 \\end{array}\\right] \\left[\\begin{aligned} x \\\\ y \\\\ z\\end{aligned}\\right]=\\left[\\begin{array}{c} 10\\cr 10\\cr 8 \\end{array}\\right] $ \\left[\\begin{aligned} x \\\\ y \\\\ z\\end{aligned}\\right]=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-2", "-1", "-2" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the division problem \\left[\\begin{array}{ccc}-2 &0 &0\\cr-4 &1 &0\\cr-4 &-3 &-2 \\end{array}\\right] \\left[\\begin{aligned} x \\\\ y \\\\ z\\end{aligned}\\right]=\\left[\\begin{array}{c}-4\\cr-6\\cr-18 \\end{array}\\right] $ \\left[\\begin{aligned} x \\\\ y \\\\ z\\end{aligned}\\right]=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "2", "2", "2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0063", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Matrix-vector forms", "level": "4", "keywords": [ "matrix' 'equation" ], "problem_v1": "The vector $\\left[\\begin{array}{c} 9\\cr 3\\cr-9 \\end{array}\\right]$ is a linear combination of the vectors $\\left[\\begin{array}{c} 3\\cr-2\\cr-1 \\end{array}\\right]$ and $\\left[\\begin{array}{c} 8\\cr 7\\cr 8 \\end{array}\\right]$ if and only if the matrix equation $A\\vec{x}=\\vec{b}$ has a solution $\\vec{x}$, where\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} and $\\vec{b}$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "3", "8", "-2", "7", "-1", "8", "9", "3", "-9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "The vector $\\left[\\begin{array}{c}-7\\cr-7\\cr-2 \\end{array}\\right]$ is a linear combination of the vectors $\\left[\\begin{array}{c}-5\\cr 5\\cr 1 \\end{array}\\right]$ and $\\left[\\begin{array}{c}-10\\cr-7\\cr 6 \\end{array}\\right]$ if and only if the matrix equation $A\\vec{x}=\\vec{b}$ has a solution $\\vec{x}$, where\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} and $\\vec{b}$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-5", "-10", "5", "-7", "1", "6", "-7", "-7", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "The vector $\\left[\\begin{array}{c} 2\\cr 17\\cr-10 \\end{array}\\right]$ is a linear combination of the vectors $\\left[\\begin{array}{c}-2\\cr-3\\cr 4 \\end{array}\\right]$ and $\\left[\\begin{array}{c}-9\\cr 7\\cr-7 \\end{array}\\right]$ if and only if the matrix equation $A\\vec{x}=\\vec{b}$ has a solution $\\vec{x}$, where\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} and $\\vec{b}$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "-2", "-9", "-3", "7", "4", "-7", "2", "17", "-10" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0064", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Matrix-vector forms", "level": "4", "keywords": [ "matrix' 'equation" ], "problem_v1": "Let $A$ be a $3 \\times 2$ matrix with linearly independent columns. Suppose we know that $\\vec{u}=\\left[\\begin{array}{c} 3\\cr 1 \\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{c} 1\\cr 2 \\end{array}\\right]$ satisfy the equations $A\\vec{u}=\\vec{a}$ and $A\\vec{v}=\\vec{b}$. Find a solution $\\vec{x}$ to $A\\vec{x}=-3 \\vec{a}+4 \\vec{b}$.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "-5", "5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $A$ be a $3 \\times 2$ matrix with linearly independent columns. Suppose we know that $\\vec{u}=\\left[\\begin{array}{c}-5\\cr-4 \\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{c} 5\\cr-2 \\end{array}\\right]$ satisfy the equations $A\\vec{u}=\\vec{a}$ and $A\\vec{v}=\\vec{b}$. Find a solution $\\vec{x}$ to $A\\vec{x}=3 \\vec{a}-3 \\vec{b}$.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-30", "-6" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $A$ be a $3 \\times 2$ matrix with linearly independent columns. Suppose we know that $\\vec{u}=\\left[\\begin{array}{c}-2\\cr-2 \\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{c} 1\\cr 1 \\end{array}\\right]$ satisfy the equations $A\\vec{u}=\\vec{a}$ and $A\\vec{v}=\\vec{b}$. Find a solution $\\vec{x}$ to $A\\vec{x}=-3 \\vec{a}+5 \\vec{b}$.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "11", "11" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0065", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Matrix-vector forms", "level": "2", "keywords": [ "matrix", "solution' 'nontrivial" ], "problem_v1": "Given A=\\left[\\begin{array}{cc} 2 &-6\\cr 1 &-3\\cr 1 &-3 \\end{array}\\right], find one nontrivial solution of $A\\vec{x}=\\vec{0}$ by inspection.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-3", "-1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Given A=\\left[\\begin{array}{cc} 3 &15\\cr-3 &-15\\cr 2 &10 \\end{array}\\right], find one nontrivial solution of $A\\vec{x}=\\vec{0}$ by inspection.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-5", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Given A=\\left[\\begin{array}{cc} 1 &-4\\cr-1 &4\\cr 2 &-8 \\end{array}\\right], find one nontrivial solution of $A\\vec{x}=\\vec{0}$ by inspection.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "4", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0066", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Matrix-vector forms", "level": "3", "keywords": [ "matrix", "solutions' 'free variables" ], "problem_v1": "Let A=\\left[\\begin{array}{cccccc} 1 &3 &1 &1 &2 &-2\\cr 0 &0 &1 &-2 &0 &1\\cr 0 &0 &0 &0 &1 &1\\cr 0 &0 &0 &0 &0 &0 \\end{array}\\right]. Describe all solutions of $A\\vec{x}=\\vec{0}$.\n$\\vec{x}=x_2$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+x_4$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+x_6$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "-3", "1", "0", "0", "0", "0", "-3", "0", "2", "1", "0", "0", "5", "0", "-1", "0", "-1", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{cccccc} 1 &-5 &5 &-4 &-2 &5\\cr 0 &0 &1 &-2 &0 &-3\\cr 0 &0 &0 &0 &1 &-2\\cr 0 &0 &0 &0 &0 &0 \\end{array}\\right]. Describe all solutions of $A\\vec{x}=\\vec{0}$.\n$\\vec{x}=x_2$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+x_4$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+x_6$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "5", "1", "0", "0", "0", "0", "-6", "0", "2", "1", "0", "0", "-16", "0", "3", "0", "2", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{cccccc} 1 &-2 &1 &-2 &1 &-3\\cr 0 &0 &1 &-2 &0 &3\\cr 0 &0 &0 &0 &1 &5\\cr 0 &0 &0 &0 &0 &0 \\end{array}\\right]. Describe all solutions of $A\\vec{x}=\\vec{0}$.\n$\\vec{x}=x_2$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+x_4$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+x_6$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "2", "1", "0", "0", "0", "0", "0", "0", "2", "1", "0", "0", "11", "0", "-3", "0", "-5", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0068", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Vector equations", "level": "2", "keywords": [ "linear algebra", "vectors" ], "problem_v1": "Solve for the unknowns in the vector equation below. $6 \\left[\\begin{matrix} a \\cr 2 \\cr \\end{matrix}\\right]+3 \\left[\\begin{matrix} 4 \\cr b \\cr \\end{matrix}\\right]=\\left[\\begin{matrix} 6 \\cr-2 \\cr \\end{matrix}\\right]$\n$a=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "-1", "-4.66666666666667" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve for the unknowns in the vector equation below. $6 \\left[\\begin{matrix} a \\cr 8 \\cr \\end{matrix}\\right]+4 \\left[\\begin{matrix}-3 \\cr b \\cr \\end{matrix}\\right]=\\left[\\begin{matrix}-2 \\cr 1 \\cr \\end{matrix}\\right]$\n$a=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "1.66666666666667", "-11.75" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve for the unknowns in the vector equation below. $3 \\left[\\begin{matrix} a \\cr 2 \\cr \\end{matrix}\\right]-3 \\left[\\begin{matrix} 1 \\cr b \\cr \\end{matrix}\\right]=\\left[\\begin{matrix} 8 \\cr 7 \\cr \\end{matrix}\\right]$\n$a=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "3.66666666666667", "-0.333333333333333" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0069", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Vector equations", "level": "3", "keywords": [ "linear algebra", "vectors" ], "problem_v1": "Consider the following vector equation.\nx_1 \\left[{\\begin{matrix} 7 \\cr 4 \\cr 6 \\cr \\end{matrix}}\\right]+x_2 \\left[{\\begin{matrix} 2 \\cr 3 \\cr-2 \\cr \\end{matrix}}\\right]=\\left[{\\begin{matrix} 2 \\cr-3 \\cr 5 \\cr \\end{matrix}}\\right]. Express the vector equation as a system of linear equations. (Order your equations from the top.) The first equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS]. The second equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS]. The third equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS].", "answer_v1": [ "7", "2", "2", "4", "3", "-3", "6", "-2", "5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the following vector equation.\nx_1 \\left[{\\begin{matrix} 1 \\cr-3 \\cr-2 \\cr \\end{matrix}}\\right]+x_2 \\left[{\\begin{matrix} 8 \\cr 9 \\cr 1 \\cr \\end{matrix}}\\right]=\\left[{\\begin{matrix}-7 \\cr-3 \\cr 1 \\cr \\end{matrix}}\\right]. Express the vector equation as a system of linear equations. (Order your equations from the top.) The first equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS]. The second equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS]. The third equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS].", "answer_v2": [ "1", "8", "-7", "-3", "9", "-3", "-2", "1", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the following vector equation.\nx_1 \\left[{\\begin{matrix} 3 \\cr 1 \\cr 8 \\cr \\end{matrix}}\\right]+x_2 \\left[{\\begin{matrix} 2 \\cr 2 \\cr 7 \\cr \\end{matrix}}\\right]=\\left[{\\begin{matrix}-4 \\cr-3 \\cr 2 \\cr \\end{matrix}}\\right]. Express the vector equation as a system of linear equations. (Order your equations from the top.) The first equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS]. The second equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS]. The third equation is [ANS] $x_1+$ [ANS] $x_2$ [ANS].", "answer_v3": [ "3", "2", "-4", "1", "2", "-3", "8", "7", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0070", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Vector equations", "level": "2", "keywords": [ "linear algebra", "linear system" ], "problem_v1": "Find $a$ and $b$ such that \\left\\lbrack \\begin{array}{r} 5 \\\\ 2 \\\\ 7 \\end{array} \\right\\rbrack=a \\left\\lbrack \\begin{array}{r} 1 \\\\ 1 \\\\ 2 \\end{array} \\right\\rbrack+b \\left\\lbrack \\begin{array}{r} 2 \\\\-1 \\\\ 1 \\end{array} \\right\\rbrack. $a=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "3", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find $a$ and $b$ such that \\left\\lbrack \\begin{array}{r} 20 \\\\-15 \\\\ 20 \\end{array} \\right\\rbrack=a \\left\\lbrack \\begin{array}{r} 1 \\\\-3 \\\\-1 \\end{array} \\right\\rbrack+b \\left\\lbrack \\begin{array}{r} 5 \\\\-6 \\\\ 3 \\end{array} \\right\\rbrack. $a=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "-5", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find $a$ and $b$ such that \\left\\lbrack \\begin{array}{r} 5 \\\\ 6 \\\\ 7 \\end{array} \\right\\rbrack=a \\left\\lbrack \\begin{array}{r} 1 \\\\-2 \\\\-2 \\end{array} \\right\\rbrack+b \\left\\lbrack \\begin{array}{r} 7 \\\\ 2 \\\\ 3 \\end{array} \\right\\rbrack. $a=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "-2", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0071", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Vector equations", "level": "2", "keywords": [ "linear", "system", "vector' 'equation" ], "problem_v1": "Write a vector equation\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $x+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $y+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $z=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nthat is equivalent to the system of equations:\n\\left\\lbrace \\begin{array}{rcl} 5x+2y+2z &=& 4, \\\\-4x-3y+z &=& 1, \\\\ 3y-2x-4z &=&-1. \\end{array} \\right.", "answer_v1": [ "5", "-4", "-2", "2", "-3", "3", "2", "1", "-4", "4", "1", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Write a vector equation\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $x+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $y+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $z=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nthat is equivalent to the system of equations:\n\\left\\lbrace \\begin{array}{rcl} 8y-8x-7z &=&-3, \\\\ 8x-3y-6z &=&-3, \\\\ x-8y+3z &=&-1. \\end{array} \\right.", "answer_v2": [ "-8", "8", "1", "8", "-3", "-8", "-7", "-6", "3", "-3", "-3", "-1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Write a vector equation\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $x+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $y+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $z=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nthat is equivalent to the system of equations:\n\\left\\lbrace \\begin{array}{rcl} 2y-4x-4z &=& 1, \\\\-6x-3y+6z &=& 8, \\\\ 7x-6y-4z &=&-5. \\end{array} \\right.", "answer_v3": [ "-4", "-6", "7", "2", "-3", "-6", "-4", "6", "-4", "1", "8", "-5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0072", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Vector equations", "level": "2", "keywords": [ "vector' 'line" ], "problem_v1": "Suppose the solution set of a certain system of equations can be described as $x_1=3-3t$, $x_2=1-2t$, $x_3=2+t$, $x_4=3+t$, where $t$ is a free variable. Use vectors to describe this solution set as a line in $\\mathbb{R}^4$.\n$L(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "3", "1", "2", "3", "-3", "-2", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose the solution set of a certain system of equations can be described as $x_1=6t-5$, $x_2=6-2t$, $x_3=-5-4t$, $x_4=-2-2t$, where $t$ is a free variable. Use vectors to describe this solution set as a line in $\\mathbb{R}^4$.\n$L(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-5", "6", "-5", "-2", "6", "-2", "-4", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose the solution set of a certain system of equations can be described as $x_1=-2-4t$, $x_2=1-2t$, $x_3=4t-3$, $x_4=1+5t$, where $t$ is a free variable. Use vectors to describe this solution set as a line in $\\mathbb{R}^4$.\n$L(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "-2", "1", "-3", "1", "-4", "-2", "4", "5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0074", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Augmented matrices", "level": "2", "keywords": [ "linear equations", "system", "systems", "matrices" ], "problem_v1": "Convert the augmented matrix\n\\left[\\begin{array}{rrr} 3 & 1 & 1 \\\\ 2 &-2 & 0 \\\\ 1 & 1 &-1\\end{array}\\right] to the equivalent linear system. Use x1 and x2 to enter the variables $x_1$ and $x_2$. [ANS] $=$ [ANS] [ANS] $=$ [ANS] [ANS] $=$ [ANS]", "answer_v1": [ "3*x1+x2", "1", "2*x1-2*x2", "0", "x1+x2", "-1" ], "answer_type_v1": [ "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Convert the augmented matrix\n\\left[\\begin{array}{rrr}-5 & 5 & 0 \\\\-2 & 5 &-2 \\\\-3 &-2 & 1\\end{array}\\right] to the equivalent linear system. Use x1 and x2 to enter the variables $x_1$ and $x_2$. [ANS] $=$ [ANS] [ANS] $=$ [ANS] [ANS] $=$ [ANS]", "answer_v2": [ "5*x2-5*x1", "0", "5*x2-2*x1", "-2", "-(3*x1+2*x2)", "1" ], "answer_type_v2": [ "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Convert the augmented matrix\n\\left[\\begin{array}{rrr}-2 & 1 & 0 \\\\ 1 &-3 &-2 \\\\ 3 & 5 & 4\\end{array}\\right] to the equivalent linear system. Use x1 and x2 to enter the variables $x_1$ and $x_2$. [ANS] $=$ [ANS] [ANS] $=$ [ANS] [ANS] $=$ [ANS]", "answer_v3": [ "x2-2*x1", "0", "x1-3*x2", "-2", "3*x1+5*x2", "4" ], "answer_type_v3": [ "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0075", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Augmented matrices", "level": "3", "keywords": [], "problem_v1": "The reduced row-echelon forms of the augmented matrices of four systems are given below. How many solutions does each system have?\n$\\left\\lbrack \\begin{array}{rrr|r} 1 & 0 &-8 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\end{array} \\right\\rbrack$ [ANS] A\\. No solutions B\\. Infinitely many solutions C\\. Unique solution D\\. None of the above\n$\\left\\lbrack \\begin{array}{rr|r} 1 & 0 & 10 \\\\ 0 & 1 & 6 \\\\ 0 & 0 & 0 \\end{array} \\right\\rbrack$ [ANS] A\\. Unique solution B\\. Infinitely many solutions C\\. No solutions D\\. None of the above\n$\\left\\lbrack \\begin{array}{rrr|r} 1 & 0 & 0 & 16 \\\\ 0 & 0 & 1 &-2 \\end{array} \\right\\rbrack$ [ANS] A\\. Unique solution B\\. Infinitely many solutions C\\. No solutions D\\. None of the above\n$\\left\\lbrack \\begin{array}{rrr|r} 0 & 1 & 0 &-5 \\\\ 0 & 0 & 1 & 9 \\end{array} \\right\\rbrack$ [ANS] A\\. Infinitely many solutions B\\. Unique solution C\\. No solutions D\\. None of the above", "answer_v1": [ "A", "A", "B", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "The reduced row-echelon forms of the augmented matrices of four systems are given below. How many solutions does each system have?\n$\\left\\lbrack \\begin{array}{rr|r} 1 & 0 & 18 \\\\ 0 & 1 &-3 \\end{array} \\right\\rbrack$ [ANS] A\\. Unique solution B\\. No solutions C\\. Infinitely many solutions D\\. None of the above\n$\\left\\lbrack \\begin{array}{rrr|r} 1 & 0 & 7 & 0 \\\\ 0 & 1 & 19 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right\\rbrack$ [ANS] A\\. Infinitely many solutions B\\. Unique solution C\\. No solutions D\\. None of the above\n$\\left\\lbrack \\begin{array}{rrr|r} 0 & 1 & 0 &-6 \\\\ 0 & 0 & 1 & 4 \\end{array} \\right\\rbrack$ [ANS] A\\. Infinitely many solutions B\\. No solutions C\\. Unique solution D\\. None of the above\n$\\left\\lbrack \\begin{array}{rr|r} 1 & 0 & 9 \\\\ 0 & 1 & 12 \\\\ 0 & 0 & 0 \\end{array} \\right\\rbrack$ [ANS] A\\. No solutions B\\. Infinitely many solutions C\\. Unique solution D\\. None of the above", "answer_v2": [ "A", "C", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The reduced row-echelon forms of the augmented matrices of four systems are given below. How many solutions does each system have?\n$\\left\\lbrack \\begin{array}{rrr|r} 1 & 0 &-2 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\end{array} \\right\\rbrack$ [ANS] A\\. Infinitely many solutions B\\. Unique solution C\\. No solutions D\\. None of the above\n$\\left\\lbrack \\begin{array}{rrr|r} 1 & 0 & 11 & 0 \\\\ 0 & 1 & 3 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right\\rbrack$ [ANS] A\\. No solutions B\\. Infinitely many solutions C\\. Unique solution D\\. None of the above\n$\\left\\lbrack \\begin{array}{rrr|r} 1 & 0 & 0 & 4 \\\\ 0 & 0 & 1 &-7 \\end{array} \\right\\rbrack$ [ANS] A\\. Infinitely many solutions B\\. No solutions C\\. Unique solution D\\. None of the above\n$\\left\\lbrack \\begin{array}{rr|r} 1 & 0 & 18 \\\\ 0 & 1 & 16 \\\\ 0 & 0 & 0 \\end{array} \\right\\rbrack$ [ANS] A\\. Unique solution B\\. No solutions C\\. Infinitely many solutions D\\. None of the above", "answer_v3": [ "C", "A", "A", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Linear_algebra_0076", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Augmented matrices", "level": "3", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Determine the value of $h$ such that the matrix is the augmented matrix of a consistent linear system.\n\\left[\\begin{array} {rr} 7 &-4 \\cr-21 & 12 \\end{array} \\right| \\left. \\begin{array}{r} h \\cr 7 \\end{array} \\right] $h=$ [ANS]", "answer_v1": [ "-7/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the value of $h$ such that the matrix is the augmented matrix of a consistent linear system.\n\\left[\\begin{array} {rr} 3 &-7 \\cr-12 & 28 \\end{array} \\right| \\left. \\begin{array}{r} h \\cr 4 \\end{array} \\right] $h=$ [ANS]", "answer_v2": [ "-4/4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the value of $h$ such that the matrix is the augmented matrix of a consistent linear system.\n\\left[\\begin{array} {rr} 4 &-7 \\cr-12 & 21 \\end{array} \\right| \\left. \\begin{array}{r} h \\cr 6 \\end{array} \\right] $h=$ [ANS]", "answer_v3": [ "-6/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0077", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Augmented matrices", "level": "", "keywords": [ "linear", "system", "augmented", "matrix" ], "problem_v1": "Write the augmented matrix of the system\n\\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} &-70 y&-6 z&=&1 \\\\-58x&&+50 z&=&3 \\\\ 4x&-5 y&-z&=&-1 \\\\ \\end{array} \\right.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0", "-70", "-6", "1", "-58", "0", "50", "3", "4", "-5", "-1", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Write the augmented matrix of the system\n\\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} &-7 y&-6 z&=&-6 \\\\ 34x&&+8 z&=&3 \\\\ 5x&-3 y&-z&=&-4 \\\\ \\end{array} \\right.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "0", "-7", "-6", "-6", "34", "0", "8", "3", "5", "-3", "-1", "-4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Write the augmented matrix of the system\n\\left\\lbrace \\begin{array}{r@{}r@{}r@{}r@{}r} &-79 y&-2 z&=&6 \\\\ 91x&&+21 z&=&-4 \\\\-4x&+6 y&-z&=&6 \\\\ \\end{array} \\right.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "0", "-79", "-2", "6", "91", "0", "21", "-4", "-4", "6", "-1", "6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0078", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Augmented matrices", "level": "2", "keywords": [], "problem_v1": "Consider a linear system whose augmented matrix is\n\\left[\\begin{array}{rrr|r} 1 & 1 & 4 &-1 \\cr 1 & 2 &-3 &-1 \\cr 6 & 13 & k &-5 \\end{array} \\right] For what value of $k$ will the system have no solutions? $k=$ [ANS]", "answer_v1": [ "-25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider a linear system whose augmented matrix is\n\\left[\\begin{array}{rrr|r} 1 & 1 & 2 & 3 \\cr 1 & 2 &-4 &-1 \\cr 6 & 16 & k &-21 \\end{array} \\right] For what value of $k$ will the system have no solutions? $k=$ [ANS]", "answer_v2": [ "-48" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider a linear system whose augmented matrix is\n\\left[\\begin{array}{rrr|r} 1 & 1 & 3 &-2 \\cr 1 & 2 &-3 &-1 \\cr 5 & 13 & k &-1 \\end{array} \\right] For what value of $k$ will the system have no solutions? $k=$ [ANS]", "answer_v3": [ "-33" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0080", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Augmented matrices", "level": "3", "keywords": [ "matrix' 'augmented' 'determined variables" ], "problem_v1": "How many determined (basic) variables does each augmented matrix have?\n[ANS] $\\left\\lbrack \\begin{array}{rrr|r} 1 & 0 & 0 &-4 \\\\ 0 & 1 & 0 & 7 \\\\ 0 & 0 & 1 &-5 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rrr|r} 1 &-7 &-6 & 5 \\\\ 0 & 0 & 0 & 0 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rrrr|r} 1 & 0 & 0 & 6 &-10 \\\\ 0 & 1 &-2 & 0 &-6 \\\\ 0 & 0 & 0 & 0 & 0 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rr|r} 1 & 0 & 8 \\\\ 0 & 1 & 7 \\\\ 0 & 0 & 0 \\end{array} \\right\\rbrack$", "answer_v1": [ "Three", "One", "Two", "Two" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ] ], "problem_v2": "How many determined (basic) variables does each augmented matrix have?\n[ANS] $\\left\\lbrack \\begin{array}{rrr|r} 1 &-7 &-9 & 6 \\\\ 0 & 0 & 0 & 0 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rrrr|r} 1 & 0 & 0 &-7 & 4 \\\\ 0 & 1 & 0 & 0 &-3 \\\\ 0 & 0 & 1 & 0 &-2 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rrrr|r} 1 & 0 & 0 &-6 &-10 \\\\ 0 & 1 & 5 & 0 &-10 \\\\ 0 & 0 & 0 & 0 & 0 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rrr|r} 1 & 0 & 0 &-10 \\\\ 0 & 1 & 0 &-3 \\\\ 0 & 0 & 1 &-7 \\end{array} \\right\\rbrack$", "answer_v2": [ "One", "Three", "Two", "Three" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ] ], "problem_v3": "How many determined (basic) variables does each augmented matrix have?\n[ANS] $\\left\\lbrack \\begin{array}{rrr|r} 1 &-4 & 2 & 10 \\\\ 0 & 0 & 0 & 0 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rr|r} 1 & 0 & 4 \\\\ 0 & 1 & 4 \\\\ 0 & 0 & 0 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rrrr|r} 1 & 0 & 0 &-7 & 4 \\\\ 0 & 1 & 0 & 0 & 8 \\\\ 0 & 0 & 1 & 0 & 5 \\end{array} \\right\\rbrack$\n[ANS] $\\left\\lbrack \\begin{array}{rr|r} 1 & 3 & 10 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right\\rbrack$", "answer_v3": [ "One", "Two", "Three", "One" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ], [ "Zero", "One", "Two", "Three", "Four" ] ] }, { "id": "Linear_algebra_0081", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "The points $(-2,-1,3)$, $(-1,-2,14)$, and $(3,-3,4)$ lie on a unique plane. Use linear algebra to find the equation of the plane and then determine where the line crosses the $z$-axis. Equation of plane (use $x$, $y$, and $z$ as the variables): [ANS]\nCrosses the $z$-axis at the point: $\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v1": [ "7*x+18*y+z = -29", "0", "0", "-29" ], "answer_type_v1": [ "EQ", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The points $(-4,-1,5)$, $(-1,2,-13)$, and $(-6,1,-3)$ lie on a unique plane. Use linear algebra to find the equation of the plane and then determine where the line crosses the $x$-axis. Equation of plane (use $x$, $y$, and $z$ as the variables): [ANS]\nCrosses the $x$-axis at the point: $\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v2": [ "x+5*y+z = -4", "-4", "0", "0" ], "answer_type_v2": [ "EQ", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The points $(-5,1,-1)$, $(1,3,13)$, and $(5,4,3)$ lie on a unique plane. Use linear algebra to find the equation of the plane and then determine where the line crosses the $z$-axis. Equation of plane (use $x$, $y$, and $z$ as the variables): [ANS]\nCrosses the $z$-axis at the point: $\\bigg($ [ANS], [ANS], [ANS] $\\bigg)$", "answer_v3": [ "17*x-58*y+z = -144", "0", "0", "-144" ], "answer_type_v3": [ "EQ", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0082", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear algebra", "vectors" ], "problem_v1": "One $8.3$ ounce can of Red Bull contains energy in two forms: $27$ grams of sugar and $80$ milligrams of caffeine. One $23.5$ ounce can of Jolt Cola contains $94$ grams of sugar and $280$ milligrams of caffeine. Determine the number of cans of each drink that when combined will contain $296$ grams sugar and $880$ milligrams caffeine. We need [ANS] cans of Red Bull, and [ANS] cans of Jolt.", "answer_v1": [ "4", "2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "One $8.3$ ounce can of Red Bull contains energy in two forms: $27$ grams of sugar and $80$ milligrams of caffeine. One $23.5$ ounce can of Jolt Cola contains $94$ grams of sugar and $280$ milligrams of caffeine. Determine the number of cans of each drink that when combined will contain $524$ grams sugar and $1560$ milligrams caffeine. We need [ANS] cans of Red Bull, and [ANS] cans of Jolt.", "answer_v2": [ "2", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "One $8.3$ ounce can of Red Bull contains energy in two forms: $27$ grams of sugar and $80$ milligrams of caffeine. One $23.5$ ounce can of Jolt Cola contains $94$ grams of sugar and $280$ milligrams of caffeine. Determine the number of cans of each drink that when combined will contain $269$ grams sugar and $800$ milligrams caffeine. We need [ANS] cans of Red Bull, and [ANS] cans of Jolt.", "answer_v3": [ "3", "2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0083", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Find the values of the coefficients $a$, $b$ and $c$ so that the conditions\nf(0)=4, \\quad f'(0)=3, \\quad \\mbox{and} \\quad \\ f''(0)=5 hold for the function\nf(x)=a e^{-4x}+b e^{x}+c x e^{x}. $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v1": [ "3/25", "97/25", "-2/5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the values of the coefficients $a$, $b$ and $c$ so that the conditions\nf(0)=5, \\quad f'(0)=-2, \\quad \\mbox{and} \\quad \\ f''(0)=-3 hold for the function\nf(x)=a e^{2x}+b e^{-x}+c x e^{-x}. $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v2": [ "-2/9", "47/9", "11/3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the values of the coefficients $a$, $b$ and $c$ so that the conditions\nf(0)=3, \\quad f'(0)=5, \\quad \\mbox{and} \\quad \\ f''(0)=4 hold for the function\nf(x)=a e^{2x}+b e^{-x}+c x e^{-x}. $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v3": [ "17/9", "10/9", "7/3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0084", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear algebra", "vectors" ], "problem_v1": "One serving of Lucky Charms contains $10\\%$ of the percent daily values (PDV) for calcium, $25\\%$ of the PDV for iron, and $25\\%$ of the PDV for zinc. One serving of Raisin Bran contains $2\\%$ of the PDV for calcium, $25\\%$ of the PDV for iron, and $10\\%$ of the PDV for zinc. Determine the number of servings of each cereal required to get $44 \\%$ of the PDV for calcium, $150 \\%$ of the PDV for iron, and $120 \\%$ of the PDV for zinc.\nWe need [ANS] servings of Lucky Charms, and [ANS] servings or Raisin Bran.", "answer_v1": [ "4", "2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "One serving of Lucky Charms contains $10\\%$ of the percent daily values (PDV) for calcium, $25\\%$ of the PDV for iron, and $25\\%$ of the PDV for zinc. One serving of Raisin Bran contains $2\\%$ of the PDV for calcium, $25\\%$ of the PDV for iron, and $10\\%$ of the PDV for zinc. Determine the number of servings of each cereal required to get $30 \\%$ of the PDV for calcium, $175 \\%$ of the PDV for iron, and $100 \\%$ of the PDV for zinc.\nWe need [ANS] servings of Lucky Charms, and [ANS] servings or Raisin Bran.", "answer_v2": [ "2", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "One serving of Lucky Charms contains $10\\%$ of the percent daily values (PDV) for calcium, $25\\%$ of the PDV for iron, and $25\\%$ of the PDV for zinc. One serving of Raisin Bran contains $2\\%$ of the PDV for calcium, $25\\%$ of the PDV for iron, and $10\\%$ of the PDV for zinc. Determine the number of servings of each cereal required to get $34 \\%$ of the PDV for calcium, $125 \\%$ of the PDV for iron, and $95 \\%$ of the PDV for zinc.\nWe need [ANS] servings of Lucky Charms, and [ANS] servings or Raisin Bran.", "answer_v3": [ "3", "2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0085", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "For tax and accounting purposes, corporations depreciate the value of equipment each year. One method used is called \"linear depreciation,\" where the value decreases over time in a linear manner. Suppose that four years after purchase, an industrial milling machine is worth $\\$840{,}000.00$, and nine years after purchase, the machine is worth $\\$470{,}000.00$. Find a formula, $V(t)$, representing the machine value in thousands of dollars at time $t\\geq 0$ after purchase. $V(t)=$ [ANS]", "answer_v1": [ "1136-74*t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "For tax and accounting purposes, corporations depreciate the value of equipment each year. One method used is called \"linear depreciation,\" where the value decreases over time in a linear manner. Suppose that three years after purchase, an industrial milling machine is worth $\\$840{,}000.00$, and five years after purchase, the machine is worth $\\$560{,}000.00$. Find a formula, $V(t)$, representing the machine value in thousands of dollars at time $t\\geq 0$ after purchase. $V(t)=$ [ANS]", "answer_v2": [ "1260-140*t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "For tax and accounting purposes, corporations depreciate the value of equipment each year. One method used is called \"linear depreciation,\" where the value decreases over time in a linear manner. Suppose that two years after purchase, an industrial milling machine is worth $\\$730{,}000.00$, and seven years after purchase, the machine is worth $\\$410{,}000.00$. Find a formula, $V(t)$, representing the machine value in thousands of dollars at time $t\\geq 0$ after purchase. $V(t)=$ [ANS]", "answer_v3": [ "858-64*t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0086", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "In another universe, temperature is measured differently than here on Earth. One scale is measured in degrees Frankenheit (F) and the other is measured in degrees Crazius (C). Suppose that the two scales are related by a linear equation $C=a F+b$. Use the fact that pure water freezes at-40 ${}^{\\circ}$ F and 60 ${}^{\\circ}$ C, and boils at 80 ${}^{\\circ}$ F and 120 ${}^{\\circ}$ C to find $a$ and $b$. $a=$ [ANS] and $b=$ [ANS]", "answer_v1": [ "1/2", "80" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "In another universe, temperature is measured differently than here on Earth. One scale is measured in degrees Frankenheit (F) and the other is measured in degrees Crazius (C). Suppose that the two scales are related by a linear equation $C=a F+b$. Use the fact that pure water freezes at-60 ${}^{\\circ}$ F and-20 ${}^{\\circ}$ C, and boils at 90 ${}^{\\circ}$ F and 40 ${}^{\\circ}$ C to find $a$ and $b$. $a=$ [ANS] and $b=$ [ANS]", "answer_v2": [ "2/5", "4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "In another universe, temperature is measured differently than here on Earth. One scale is measured in degrees Frankenheit (F) and the other is measured in degrees Crazius (C). Suppose that the two scales are related by a linear equation $C=a F+b$. Use the fact that pure water freezes at-30 ${}^{\\circ}$ F and 100 ${}^{\\circ}$ C, and boils at 220 ${}^{\\circ}$ F and 150 ${}^{\\circ}$ C to find $a$ and $b$. $a=$ [ANS] and $b=$ [ANS]", "answer_v3": [ "1/5", "106" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0087", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "A new \"LAI\" (for Linear Algebra Index) formula has been used to rank the eight college football teams shown below. Determine the formula for the LAI.\n\\begin{array}{clcccc} \\mbox{Rank} & \\mbox{Team} & \\mbox{USA} & \\mbox{Harris} & \\mbox{Computer} & \\mbox{LAI} \\\\\\hline 1 & \\mbox{Oklahoma} & 1482 & 2699 & 100 & 0.9928438 \\\\ 2 & \\mbox{Florida} & 1481 & 2776 & 89 & 0.9917167 \\\\ 3 & \\mbox{Texas} & 1408 & 2616 & 94 & 0.9493332 \\\\ 4 & \\mbox{Alabama} & 1309 & 2442 & 81 & 0.8773579 \\\\ 5 & \\mbox{Southern Cal} & 1309 & 2413 & 75 & 0.8671931 \\\\ 6 & \\mbox{Penn State} & 1193 & 2186 & 66 & 0.7861117 \\end{array} LAI=[ANS] (USA)+[ANS] (Harris)+[ANS] (Computer) Help: The use of a computer algebra system is advised. Be sure to use all 8 digits of the LAI in your computations, or else suffer huge rounding errors!", "answer_v1": [ "0.0003505", "0.0001372", "0.001031" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A new \"LAI\" (for Linear Algebra Index) formula has been used to rank the eight college football teams shown below. Determine the formula for the LAI.\n\\begin{array}{clcccc} \\mbox{Rank} & \\mbox{Team} & \\mbox{USA} & \\mbox{Harris} & \\mbox{Computer} & \\mbox{LAI} \\\\\\hline 1 & \\mbox{Oklahoma} & 1482 & 2699 & 100 & 0.7214719 \\\\ 2 & \\mbox{Florida} & 1481 & 2776 & 89 & 0.7175712 \\\\ 3 & \\mbox{Texas} & 1408 & 2616 & 94 & 0.6895524 \\\\ 4 & \\mbox{Alabama} & 1309 & 2442 & 81 & 0.6354842 \\\\ 5 & \\mbox{Southern Cal} & 1309 & 2413 & 75 & 0.6264353 \\\\ 6 & \\mbox{Penn State} & 1193 & 2186 & 66 & 0.5671948 \\end{array} LAI=[ANS] (USA)+[ANS] (Harris)+[ANS] (Computer) Help: The use of a computer algebra system is advised. Be sure to use all 8 digits of the LAI in your computations, or else suffer huge rounding errors!", "answer_v2": [ "0.0002366", "9.93E-05", "0.0010282" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A new \"LAI\" (for Linear Algebra Index) formula has been used to rank the eight college football teams shown below. Determine the formula for the LAI.\n\\begin{array}{clcccc} \\mbox{Rank} & \\mbox{Team} & \\mbox{USA} & \\mbox{Harris} & \\mbox{Computer} & \\mbox{LAI} \\\\\\hline 1 & \\mbox{Oklahoma} & 1482 & 2699 & 100 & 0.8707406 \\\\ 2 & \\mbox{Florida} & 1481 & 2776 & 89 & 0.8701655 \\\\ 3 & \\mbox{Texas} & 1408 & 2616 & 94 & 0.8335304 \\\\ 4 & \\mbox{Alabama} & 1309 & 2442 & 81 & 0.7698419 \\\\ 5 & \\mbox{Southern Cal} & 1309 & 2413 & 75 & 0.7596747 \\\\ 6 & \\mbox{Penn State} & 1193 & 2186 & 66 & 0.6881139 \\end{array} LAI=[ANS] (USA)+[ANS] (Harris)+[ANS] (Computer) Help: The use of a computer algebra system is advised. Be sure to use all 8 digits of the LAI in your computations, or else suffer huge rounding errors!", "answer_v3": [ "0.0002627", "0.0001408", "0.001014" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0088", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "The BCS ranking system was more complicated in 2001 than in 2008. The table below gives the BCS rankings at the end of the regular season. (A lower BCS Index gave a higher rank.)\nTable headings: AP and USA gives the rank of each team in two opinion polls of writers and coaches, respectively. AP and USA gives the rank of each team in two opinion polls of writers and coaches, respectively. SS stands for strength of schedule ranking, with 1 being the most challenging. SS stands for strength of schedule ranking, with 1 being the most challenging. L is the number of losses during the season. L is the number of losses during the season. CA (Computer Average) is the average of computer rankings from various sources. CA (Computer Average) is the average of computer rankings from various sources. QW (Quality Wins) gives a measurement of the number of victories over highly ranked teams. QW (Quality Wins) gives a measurement of the number of victories over highly ranked teams.\n\\begin{array}{clccccccc} \\mbox{Rank} & \\mbox{Team} & \\mbox{AP} & \\mbox{USA} & \\mbox{SS} & \\mbox{L} & \\mbox{CA} & \\mbox{QW} & \\mbox{BCS Index} \\\\\\hline 1 & \\mbox{Miami} & 1 & 1 & 17 & 0 & 0.98 & 0.1 & 2.53 \\\\ 2 & \\mbox{Nebraska} & 4 & 4 & 14 & 1 & 2.13 & 0.6 & 7.07 \\\\ 3 & \\mbox{Colorado} & 3 & 3 & 2 & 2 & 4.5 & 2.2 & 7.28 \\\\ 4 & \\mbox{Oregon} & 2 & 2 & 29 & 1 & 4.73 & 0.5 & 8.68 \\\\ 5 & \\mbox{Florida} & 5 & 5 & 19 & 2 & 5.73 & 0.7 & 13.14 \\\\ 6 & \\mbox{Tennessee} & 8 & 8 & 3 & 2 & 6.12 & 1.7 & 13.97 \\\\ 7 & \\mbox{Texas} & 7 & 7 & 34 & 2 & 6.7 & 1.1 & 16.03 \\\\ 8 & \\mbox{Illinois} & 7 & 7 & 38 & 1 & 9.93 & 0 & 19.18 \\\\ 9 & \\mbox{Stanford} & 11 & 11 & 23 & 2 & 7.74 & 1.1 & 19.98 \\\\ 10 & \\mbox{Oklahoma} & 10 & 10 & 36 & 2 & 8.99 & 0.9 & 21.22 \\\\ 11 & \\mbox{Maryland} & 6 & 6 & 75 & 1 & 11.13 & 0 & 21.30 \\\\ 12 & \\mbox{Washington St} & 13 & 13 & 47 & 2 & 10.96 & 0.8 & 26.43 \\\\ 13 & \\mbox{LSU} & 12 & 12 & 10 & 3 & 13 & 1 & 27.01 \\\\ 14 & \\mbox{South Carolina} & 14 & 14 & 41 & 3 & 19 & 0 & 37.43 \\\\ 15 & \\mbox{Washington} & 22 & 21 & 20 & 3 & 14.7 & 1 & 38.11 \\end{array} Find the 2001 BCS ranking formula. (Hint: To avoid a aystem with infinitely many solutions, include Washington among the schools used to develop the formula.) BCS Index=[ANS] (AP) $+$ [ANS] (USA) $+$ [ANS] (SS) $+$ [ANS] (L) $+$ [ANS] (CA) $+$ [ANS] (QW)", "answer_v1": [ "1.05957", "-0.179576", "0.0502484", "1.6459", "0.953084", "-1.38234" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The BCS ranking system was more complicated in 2001 than in 2008. The table below gives the BCS rankings at the end of the regular season. (A lower BCS Index gave a higher rank.)\nTable headings: AP and USA gives the rank of each team in two opinion polls of writers and coaches, respectively. AP and USA gives the rank of each team in two opinion polls of writers and coaches, respectively. SS stands for strength of schedule ranking, with 1 being the most challenging. SS stands for strength of schedule ranking, with 1 being the most challenging. L is the number of losses during the season. L is the number of losses during the season. CA (Computer Average) is the average of computer rankings from various sources. CA (Computer Average) is the average of computer rankings from various sources. QW (Quality Wins) gives a measurement of the number of victories over highly ranked teams. QW (Quality Wins) gives a measurement of the number of victories over highly ranked teams.\n\\begin{array}{clccccccc} \\mbox{Rank} & \\mbox{Team} & \\mbox{AP} & \\mbox{USA} & \\mbox{SS} & \\mbox{L} & \\mbox{CA} & \\mbox{QW} & \\mbox{BCS Index} \\\\\\hline 1 & \\mbox{Miami} & 1 & 1 & 16 & 0 & 0.94 & 0.2 & 2.58 \\\\ 2 & \\mbox{Nebraska} & 4 & 4 & 13 & 1 & 2.08 & 0.8 & 7.02 \\\\ 3 & \\mbox{Colorado} & 3 & 3 & 2 & 2 & 4.51 & 2.3 & 7.3 \\\\ 4 & \\mbox{Oregon} & 2 & 2 & 31 & 1 & 4.74 & 0.6 & 8.74 \\\\ 5 & \\mbox{Florida} & 5 & 5 & 19 & 2 & 5.76 & 0.8 & 13.04 \\\\ 6 & \\mbox{Tennessee} & 8 & 8 & 3 & 2 & 6.17 & 1.4 & 15.25 \\\\ 7 & \\mbox{Texas} & 7 & 7 & 32 & 2 & 6.62 & 1.1 & 16.34 \\\\ 8 & \\mbox{Illinois} & 7 & 7 & 37 & 1 & 9.73 & 0 & 20.09 \\\\ 9 & \\mbox{Stanford} & 11 & 11 & 23 & 2 & 7.7 & 1.2 & 21.15 \\\\ 10 & \\mbox{Maryland} & 6 & 6 & 71 & 1 & 11.13 & 0 & 22.10 \\\\ 11 & \\mbox{Oklahoma} & 10 & 10 & 36 & 2 & 8.99 & 1 & 22.26 \\\\ 12 & \\mbox{Washington St} & 13 & 13 & 47 & 2 & 10.72 & 0.9 & 27.84 \\\\ 13 & \\mbox{LSU} & 12 & 12 & 10 & 3 & 13 & 1 & 28.17 \\\\ 14 & \\mbox{Washington} & 16 & 24 & 20 & 3 & 14.81 & 1 & 38.1 \\\\ 15 & \\mbox{South Carolina} & 14 & 14 & 41 & 3 & 19 & 0 & 39.00 \\end{array} Find the 2001 BCS ranking formula. (Hint: To avoid a aystem with infinitely many solutions, include Washington among the schools used to develop the formula.) BCS Index=[ANS] (AP) $+$ [ANS] (USA) $+$ [ANS] (SS) $+$ [ANS] (L) $+$ [ANS] (CA) $+$ [ANS] (QW)", "answer_v2": [ "0.630198", "0.421386", "0.0471169", "0.852779", "1.04156", "-1.0226" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The BCS ranking system was more complicated in 2001 than in 2008. The table below gives the BCS rankings at the end of the regular season. (A lower BCS Index gave a higher rank.)\nTable headings: AP and USA gives the rank of each team in two opinion polls of writers and coaches, respectively. AP and USA gives the rank of each team in two opinion polls of writers and coaches, respectively. SS stands for strength of schedule ranking, with 1 being the most challenging. SS stands for strength of schedule ranking, with 1 being the most challenging. L is the number of losses during the season. L is the number of losses during the season. CA (Computer Average) is the average of computer rankings from various sources. CA (Computer Average) is the average of computer rankings from various sources. QW (Quality Wins) gives a measurement of the number of victories over highly ranked teams. QW (Quality Wins) gives a measurement of the number of victories over highly ranked teams.\n\\begin{array}{clccccccc} \\mbox{Rank} & \\mbox{Team} & \\mbox{AP} & \\mbox{USA} & \\mbox{SS} & \\mbox{L} & \\mbox{CA} & \\mbox{QW} & \\mbox{BCS Index} \\\\\\hline 1 & \\mbox{Miami} & 1 & 1 & 16 & 0 & 1.09 & 0.2 & 2.59 \\\\ 2 & \\mbox{Nebraska} & 4 & 4 & 13 & 1 & 2.23 & 0.4 & 7.05 \\\\ 3 & \\mbox{Colorado} & 3 & 3 & 2 & 2 & 4.52 & 2.4 & 7.3 \\\\ 4 & \\mbox{Oregon} & 2 & 2 & 29 & 1 & 4.73 & 0.8 & 8.7 \\\\ 5 & \\mbox{Florida} & 5 & 5 & 19 & 2 & 5.76 & 0.7 & 13.03 \\\\ 6 & \\mbox{Tennessee} & 8 & 8 & 4 & 2 & 6.19 & 1.4 & 14.35 \\\\ 7 & \\mbox{Texas} & 7 & 7 & 35 & 2 & 6.74 & 1.2 & 15.94 \\\\ 8 & \\mbox{Stanford} & 11 & 11 & 24 & 2 & 7.73 & 1.2 & 19.66 \\\\ 9 & \\mbox{Illinois} & 7 & 7 & 38 & 1 & 9.85 & 0 & 19.84 \\\\ 10 & \\mbox{Oklahoma} & 10 & 10 & 36 & 2 & 9.05 & 0.8 & 21.44 \\\\ 11 & \\mbox{Maryland} & 6 & 6 & 72 & 1 & 11.36 & 0 & 22.45 \\\\ 12 & \\mbox{Washington St} & 13 & 13 & 44 & 2 & 10.95 & 0.7 & 26.47 \\\\ 13 & \\mbox{LSU} & 12 & 12 & 10 & 3 & 13 & 1 & 27.23 \\\\ 14 & \\mbox{Washington} & 18 & 21 & 21 & 3 & 14.98 & 1 & 38.03 \\\\ 15 & \\mbox{South Carolina} & 14 & 14 & 40 & 3 & 19 & 0 & 38.29 \\end{array} Find the 2001 BCS ranking formula. (Hint: To avoid a aystem with infinitely many solutions, include Washington among the schools used to develop the formula.) BCS Index=[ANS] (AP) $+$ [ANS] (USA) $+$ [ANS] (SS) $+$ [ANS] (L) $+$ [ANS] (CA) $+$ [ANS] (QW)", "answer_v3": [ "-0.320058", "1.09993", "0.0487026", "1.17799", "1.15229", "-1.12557" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0089", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear algebra", "vectors" ], "problem_v1": "Fertilizer is sold in 100 pound bags labelled with the amount of nitrogen (N), phosphoric acid (P2O5), and potash (K2O) present. The mixture of these nutrients varies from one type of fertilizer to the next. For example, a $100$ pound bag of Vigoro Ultra Turf fertilizer contains $29$ pounds of nitrogen, $3$ pounds of phosphoric acid, and $4$ pounds of potash. Another type of fertilizer, Parker’s Premium Starter, has $18$ pounds of nitrogen, $25$ pounds of phosphoric acid, and $6$ pounds of potash per $100$ pounds. Determine the total amount of nitrogen, phosphoric acid, and potash in the mixture of $400$ pounds of Vigoro and $300$ pounds of Parker’s. The mixture contains [ANS] pounds of nitrogen, [ANS] pounds of phosphoric acid, and [ANS] pounds of potash.", "answer_v1": [ "170", "87", "34" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Fertilizer is sold in 100 pound bags labelled with the amount of nitrogen (N), phosphoric acid (P2O5), and potash (K2O) present. The mixture of these nutrients varies from one type of fertilizer to the next. For example, a $100$ pound bag of Vigoro Ultra Turf fertilizer contains $29$ pounds of nitrogen, $3$ pounds of phosphoric acid, and $4$ pounds of potash. Another type of fertilizer, Parker’s Premium Starter, has $18$ pounds of nitrogen, $25$ pounds of phosphoric acid, and $6$ pounds of potash per $100$ pounds. Determine the total amount of nitrogen, phosphoric acid, and potash in the mixture of $100$ pounds of Vigoro and $500$ pounds of Parker’s. The mixture contains [ANS] pounds of nitrogen, [ANS] pounds of phosphoric acid, and [ANS] pounds of potash.", "answer_v2": [ "119", "128", "34" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Fertilizer is sold in 100 pound bags labelled with the amount of nitrogen (N), phosphoric acid (P2O5), and potash (K2O) present. The mixture of these nutrients varies from one type of fertilizer to the next. For example, a $100$ pound bag of Vigoro Ultra Turf fertilizer contains $29$ pounds of nitrogen, $3$ pounds of phosphoric acid, and $4$ pounds of potash. Another type of fertilizer, Parker’s Premium Starter, has $18$ pounds of nitrogen, $25$ pounds of phosphoric acid, and $6$ pounds of potash per $100$ pounds. Determine the total amount of nitrogen, phosphoric acid, and potash in the mixture of $200$ pounds of Vigoro and $400$ pounds of Parker’s. The mixture contains [ANS] pounds of nitrogen, [ANS] pounds of phosphoric acid, and [ANS] pounds of potash.", "answer_v3": [ "130", "106", "32" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0090", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems", "partial fractions" ], "problem_v1": "When using partial fractions to find antiderivatives in calculus we decompose complicated rational expressions into the sum of simpler expressions that can be integrated individually. The required decomposition is\n \\frac{30}{x(x^2+5)} = \\frac{A}{x} + \\frac{Bx+C}{x^2+5} Find the values of the missing constants. $A=$ [ANS]\n$B=$ [ANS]\n$C=$ [ANS]", "answer_v1": [ "6", "-6", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "When using partial fractions to find antiderivatives in calculus we decompose complicated rational expressions into the sum of simpler expressions that can be integrated individually. The required decomposition is\n \\frac{16}{x(x^2+8)} = \\frac{A}{x} + \\frac{Bx+C}{x^2+8} Find the values of the missing constants. $A=$ [ANS]\n$B=$ [ANS]\n$C=$ [ANS]", "answer_v2": [ "2", "-2", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "When using partial fractions to find antiderivatives in calculus we decompose complicated rational expressions into the sum of simpler expressions that can be integrated individually. The required decomposition is\n \\frac{6}{x(x^2+2)} = \\frac{A}{x} + \\frac{Bx+C}{x^2+2} Find the values of the missing constants. $A=$ [ANS]\n$B=$ [ANS]\n$C=$ [ANS]", "answer_v3": [ "3", "-3", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0091", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "The points $(-4,6)$ and $(-3,4)$ lie on a line. Use linear algebra to find the equation of the line and then determine where the line crosses the $x$-axis. Equation of line: $y=$ [ANS]\nCrosses $x$-axis at the point: $\\bigg($ [ANS], [ANS] $\\bigg)$", "answer_v1": [ "-2*x+(-2)", "-1", "0" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The points $(5,4)$ and $(6,8)$ lie on a line. Use linear algebra to find the equation of the line and then determine where the line crosses the $x$-axis. Equation of line: $y=$ [ANS]\nCrosses $x$-axis at the point: $\\bigg($ [ANS], [ANS] $\\bigg)$", "answer_v2": [ "4*x+(-16)", "4", "0" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The points $(-9,2)$ and $(-3,8)$ lie on a line. Use linear algebra to find the equation of the line and then determine where the line crosses the $x$-axis. Equation of line: $y=$ [ANS]\nCrosses $x$-axis at the point: $\\bigg($ [ANS], [ANS] $\\bigg)$", "answer_v3": [ "1*x+11", "-11", "0" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0092", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Balance the chemical equation\n\\mbox{Na}+{\\mbox{H}_{4}}{\\mbox{O}} \\longrightarrow {\\mbox{Na}_2}{\\mbox{O}_4}{\\mbox{H}_2}+\\mbox{H}_3 and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Na}+$ [ANS] $\\mbox{H}_{4} \\mbox{O} \\longrightarrow$ [ANS] $\\mbox{Na}_2 \\mbox{O}_4 \\mbox{H}_2+$ [ANS] $\\mbox{H}_3$", "answer_v1": [ "6", "12", "3", "14" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Balance the chemical equation\n\\mbox{Na}+{\\mbox{H}_{2}}{\\mbox{O}} \\longrightarrow {\\mbox{Na}_4}{\\mbox{O}_5}{\\mbox{H}_4}+\\mbox{H}_3 and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Na}+$ [ANS] $\\mbox{H}_{2} \\mbox{O} \\longrightarrow$ [ANS] $\\mbox{Na}_4 \\mbox{O}_5 \\mbox{H}_4+$ [ANS] $\\mbox{H}_3$", "answer_v2": [ "4", "5", "1", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Balance the chemical equation\n\\mbox{Na}+{\\mbox{H}_{4}}{\\mbox{O}} \\longrightarrow {\\mbox{Na}_4}{\\mbox{O}_4}{\\mbox{H}_4}+\\mbox{H}_3 and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Na}+$ [ANS] $\\mbox{H}_{4} \\mbox{O} \\longrightarrow$ [ANS] $\\mbox{Na}_4 \\mbox{O}_4 \\mbox{H}_4+$ [ANS] $\\mbox{H}_3$", "answer_v3": [ "4", "4", "1", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0093", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Find a polynomial of the form f(x)=a x^3+bx^2+cx+d such that $f(0)=-1$, $f(2)=-2$, $f(-3)=-1$, and $f(-5)=-5$. Answer: $f(x)=$ [ANS]", "answer_v1": [ "0.0428571*x^3+(-0.0571429)*x^2+(-0.557143)*x-1" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a polynomial of the form f(x)=a x^3+bx^2+cx+d such that $f(0)=-5$, $f(1)=-3$, $f(-3)=1$, and $f(-5)=-6$. Answer: $f(x)=$ [ANS]", "answer_v2": [ "0.35*x^3+1.7*x^2+(-0.05)*x+(-5)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a polynomial of the form f(x)=a x^3+bx^2+cx+d such that $f(0)=5$, $f(1)=6$, $f(3)=6$, and $f(-4)=-4$. Answer: $f(x)=$ [ANS]", "answer_v3": [ "-0.0119048*x^3+(-0.285714)*x^2+1.29762*x+5" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0094", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "The data given below provides the distance required for a particular type of car to stop when traveling at a variety of speeds. A reasonable model for braking distance is $d=as^k$, where $d$ is distance, $s$ is speed, and $a$ and $k$ are constants. Use the data in the table to find values for $a$ and $k$, and also type out the equation that models the breaking distance.\n\\begin{array}{c|c|c|c|c|} \\mbox{Speed (MPH)} & 10 & 20 & 30 & 40 \\\\\\hline \\mbox{Distance (Feed)} & 10.5 & 42 & 94.5 & 168 \\end{array} $a=$ [ANS]\n$k=$ [ANS]\nModel: [ANS]", "answer_v1": [ "0.105", "2", "d = 0.105*s^2" ], "answer_type_v1": [ "NV", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "The data given below provides the distance required for a particular type of car to stop when traveling at a variety of speeds. A reasonable model for braking distance is $d=as^k$, where $d$ is distance, $s$ is speed, and $a$ and $k$ are constants. Use the data in the table to find values for $a$ and $k$, and also type out the equation that models the breaking distance.\n\\begin{array}{c|c|c|c|c|} \\mbox{Speed (MPH)} & 10 & 20 & 30 & 40 \\\\\\hline \\mbox{Distance (Feed)} & 4.5 & 18 & 40.5 & 72 \\end{array} $a=$ [ANS]\n$k=$ [ANS]\nModel: [ANS]", "answer_v2": [ "0.045", "2", "d = 0.045*s^2" ], "answer_type_v2": [ "NV", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "The data given below provides the distance required for a particular type of car to stop when traveling at a variety of speeds. A reasonable model for braking distance is $d=as^k$, where $d$ is distance, $s$ is speed, and $a$ and $k$ are constants. Use the data in the table to find values for $a$ and $k$, and also type out the equation that models the breaking distance.\n\\begin{array}{c|c|c|c|c|} \\mbox{Speed (MPH)} & 10 & 20 & 30 & 40 \\\\\\hline \\mbox{Distance (Feed)} & 8.5 & 34 & 76.5 & 136 \\end{array} $a=$ [ANS]\n$k=$ [ANS]\nModel: [ANS]", "answer_v3": [ "0.085", "2", "d = 0.085*s^2" ], "answer_type_v3": [ "NV", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0095", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Balance the chemical equation\n\\mbox{Fe}_{2}+\\mbox{O}_{6} \\longrightarrow \\mbox{Fe}_{7}\\mbox{O}_{5} and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Fe}_{2}+$ [ANS] $\\mbox{O}_{6} \\longrightarrow$ [ANS] $\\mbox{Fe}_{7}\\mbox{O}_{5}$", "answer_v1": [ "21", "5", "6" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Balance the chemical equation\n\\mbox{Fe}+\\mbox{O}_{3} \\longrightarrow \\mbox{Fe}_{2}\\mbox{O}_{7} and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Fe}+$ [ANS] $\\mbox{O}_{3} \\longrightarrow$ [ANS] $\\mbox{Fe}_{2}\\mbox{O}_{7}$", "answer_v2": [ "6", "7", "3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Balance the chemical equation\n\\mbox{Fe}_{3}+\\mbox{O}_{5} \\longrightarrow \\mbox{Fe}_{5}\\mbox{O}_{2} and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Fe}_{3}+$ [ANS] $\\mbox{O}_{5} \\longrightarrow$ [ANS] $\\mbox{Fe}_{5}\\mbox{O}_{2}$", "answer_v3": [ "25", "6", "15" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0096", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "The table below consists of the planetary orbital distances and periods for all planets in our solar system. Find a model for planetary orbital period using the data for Jupiter and Saturn. Use $p$ to represent orbital period and $d$ the distance from the sun.\n\\begin{array}{lcc} \\mbox{Planet} \\quad & \\quad\\mbox{Distance from Sun} (\\times 10^6 \\mbox{km})\\quad & \\quad \\mbox{Orbital Period (days)} \\\\\\hline \\mbox{Mercury} & 57.9 & 88 \\\\ \\mbox{Venus} & 108.2 & 224.7 \\\\ \\mbox{Earth} & 149.6 & 365.2 \\\\ \\mbox{Mars} & 227.9 & 687 \\\\ \\mbox{Jupiter} & 778.6 & 4331 \\\\ \\mbox{Saturn} & 1433.5 & 10747 \\\\ \\mbox{Uranus} & 2872.5 & 30589 \\\\ \\mbox{Neptune} & 4495.1 & 59800 \\end{array} Model: [ANS]", "answer_v1": [ "p = 0.214551*d^1.48896" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The table below consists of the planetary orbital distances and periods for all planets in our solar system. Find a model for planetary orbital period using the data for Mercury and Neptune. Use $p$ to represent orbital period and $d$ the distance from the sun.\n\\begin{array}{lcc} \\mbox{Planet} \\quad & \\quad\\mbox{Distance from Sun} (\\times 10^6 \\mbox{km})\\quad & \\quad \\mbox{Orbital Period (days)} \\\\\\hline \\mbox{Mercury} & 57.9 & 88 \\\\ \\mbox{Venus} & 108.2 & 224.7 \\\\ \\mbox{Earth} & 149.6 & 365.2 \\\\ \\mbox{Mars} & 227.9 & 687 \\\\ \\mbox{Jupiter} & 778.6 & 4331 \\\\ \\mbox{Saturn} & 1433.5 & 10747 \\\\ \\mbox{Uranus} & 2872.5 & 30589 \\\\ \\mbox{Neptune} & 4495.1 & 59800 \\end{array} Model: [ANS]", "answer_v2": [ "p = 0.200976*d^1.49848" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The table below consists of the planetary orbital distances and periods for all planets in our solar system. Find a model for planetary orbital period using the data for Earth and Jupiter. Use $p$ to represent orbital period and $d$ the distance from the sun.\n\\begin{array}{lcc} \\mbox{Planet} \\quad & \\quad\\mbox{Distance from Sun} (\\times 10^6 \\mbox{km})\\quad & \\quad \\mbox{Orbital Period (days)} \\\\\\hline \\mbox{Mercury} & 57.9 & 88 \\\\ \\mbox{Venus} & 108.2 & 224.7 \\\\ \\mbox{Earth} & 149.6 & 365.2 \\\\ \\mbox{Mars} & 227.9 & 687 \\\\ \\mbox{Jupiter} & 778.6 & 4331 \\\\ \\mbox{Saturn} & 1433.5 & 10747 \\\\ \\mbox{Uranus} & 2872.5 & 30589 \\\\ \\mbox{Neptune} & 4495.1 & 59800 \\end{array} Model: [ANS]", "answer_v3": [ "p = 0.20031*d^1.49928" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0097", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "A $70$-gallon bathtub is to be filled with water that is exactly 105 $^{\\circ}$ F. Unfortunately, the two sources of water available are 140 $^{\\circ}$ F and 65 $^{\\circ}$ F. When mixed, the temperature will be a weighted average based on the amount of each water source in the mix. How much of each should be used to fill the tub as specified? Amount of 140 $^{\\circ}$ F water: [ANS] gallons Amount of 65 $^{\\circ}$ F water: [ANS] gallons", "answer_v1": [ "37.3333", "32.6667" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A $35$-gallon bathtub is to be filled with water that is exactly 125 $^{\\circ}$ F. Unfortunately, the two sources of water available are 140 $^{\\circ}$ F and 100 $^{\\circ}$ F. When mixed, the temperature will be a weighted average based on the amount of each water source in the mix. How much of each should be used to fill the tub as specified? Amount of 140 $^{\\circ}$ F water: [ANS] gallons Amount of 100 $^{\\circ}$ F water: [ANS] gallons", "answer_v2": [ "21.875", "13.125" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A $50$-gallon bathtub is to be filled with water that is exactly 105 $^{\\circ}$ F. Unfortunately, the two sources of water available are 125 $^{\\circ}$ F and 75 $^{\\circ}$ F. When mixed, the temperature will be a weighted average based on the amount of each water source in the mix. How much of each should be used to fill the tub as specified? Amount of 125 $^{\\circ}$ F water: [ANS] gallons Amount of 75 $^{\\circ}$ F water: [ANS] gallons", "answer_v3": [ "30", "20" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0098", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "The equation for a parabola has the form $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants and $a \\neq 0$. Find an equation for the parabola that passes through the points $(2,-5)$, $(-2,-17)$, and $(4,-11)$. Answer: $y=$ [ANS]", "answer_v1": [ "-x^2+3*x+(-7)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The equation for a parabola has the form $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants and $a \\neq 0$. Find an equation for the parabola that passes through the points $(3,-2)$, $(1,-6)$, and $(-4,19)$. Answer: $y=$ [ANS]", "answer_v2": [ "x^2+(-2)*x+(-5)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The equation for a parabola has the form $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants and $a \\neq 0$. Find an equation for the parabola that passes through the points $(1,-4)$, $(2,-19)$, and $(-6,-11)$. Answer: $y=$ [ANS]", "answer_v3": [ "-2*x^2+(-9)*x+7" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0099", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Balance the chemical equation\n\\mbox{K} \\mbox{O}_{4}+\\mbox{C} \\mbox{O}_{4} \\longrightarrow \\mbox{K}_2 \\mbox{C}_4 \\mbox{O}_2+\\mbox{O}_4 and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{K} \\mbox{O}_{4}+$ [ANS] $\\mbox{C} \\mbox{O}_{4} \\longrightarrow$ [ANS] $\\mbox{K}_2 \\mbox{C}_4 \\mbox{O}_2+$ [ANS] $\\mbox{O}_4$", "answer_v1": [ "4", "8", "2", "11" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Balance the chemical equation\n\\mbox{K} \\mbox{O}_{5}+\\mbox{C} \\mbox{O}_{2} \\longrightarrow \\mbox{K}_5 \\mbox{C}_4 \\mbox{O}_4+\\mbox{O}_2 and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{K} \\mbox{O}_{5}+$ [ANS] $\\mbox{C} \\mbox{O}_{2} \\longrightarrow$ [ANS] $\\mbox{K}_5 \\mbox{C}_4 \\mbox{O}_4+$ [ANS] $\\mbox{O}_2$", "answer_v2": [ "10", "8", "2", "29" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Balance the chemical equation\n\\mbox{K} \\mbox{O}_{4}+\\mbox{C} \\mbox{O}_{3} \\longrightarrow \\mbox{K}_4 \\mbox{C}_4 \\mbox{O}_3+\\mbox{O}_5 and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{K} \\mbox{O}_{4}+$ [ANS] $\\mbox{C} \\mbox{O}_{3} \\longrightarrow$ [ANS] $\\mbox{K}_4 \\mbox{C}_4 \\mbox{O}_3+$ [ANS] $\\mbox{O}_5$", "answer_v3": [ "4", "4", "1", "5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0100", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations", "system", "systems" ], "problem_v1": "Balance the chemical equation\n\\mbox{Mn}_{2} \\mbox{O}_{10}+\\mbox{H}_{2} \\mbox{Cl}_{2} \\longrightarrow \\mbox{Mn}_{2} \\mbox{Cl}_{10}+\\mbox{H}_{10} \\mbox{O}_{2}+\\mbox{Cl}_{10} and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Mn}_{2} \\mbox{O}_{10}+$ [ANS] $\\mbox{H}_{2} \\mbox{Cl}_{2} \\longrightarrow$ [ANS] $\\mbox{Mn}_{2} \\mbox{Cl}_{10}+$ [ANS] $\\mbox{H}_{10} \\mbox{O}_{2}+$ [ANS] $\\mbox{Cl}_{10}$", "answer_v1": [ "1", "25", "1", "5", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Balance the chemical equation\n\\mbox{Mn}_{3} \\mbox{O}_{6}+\\mbox{H}_{3} \\mbox{Cl}_{3} \\longrightarrow \\mbox{Mn}_{3} \\mbox{Cl}_{6}+\\mbox{H}_{6} \\mbox{O}_{3}+\\mbox{Cl}_{6} and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Mn}_{3} \\mbox{O}_{6}+$ [ANS] $\\mbox{H}_{3} \\mbox{Cl}_{3} \\longrightarrow$ [ANS] $\\mbox{Mn}_{3} \\mbox{Cl}_{6}+$ [ANS] $\\mbox{H}_{6} \\mbox{O}_{3}+$ [ANS] $\\mbox{Cl}_{6}$", "answer_v2": [ "1", "4", "1", "2", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Balance the chemical equation\n\\mbox{Mn}_{2} \\mbox{O}_{6}+\\mbox{H}_{2} \\mbox{Cl}_{2} \\longrightarrow \\mbox{Mn}_{2} \\mbox{Cl}_{6}+\\mbox{H}_{6} \\mbox{O}_{2}+\\mbox{Cl}_{6} and give your answer in lowest terms. That is, the coefficients should all be integers that do not all share a common factor. [ANS] $\\mbox{Mn}_{2} \\mbox{O}_{6}+$ [ANS] $\\mbox{H}_{2} \\mbox{Cl}_{2} \\longrightarrow$ [ANS] $\\mbox{Mn}_{2} \\mbox{Cl}_{6}+$ [ANS] $\\mbox{H}_{6} \\mbox{O}_{2}+$ [ANS] $\\mbox{Cl}_{6}$", "answer_v3": [ "1", "9", "1", "3", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Linear_algebra_0101", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "A test has $25$ problems, which are worth a total of $152$ points. There are two types of problems in the test. Each multiple-choice problem is worth $5$ points, and each short-answer problem is worth $8$ points. Write and solve a system equation to answer the following questions.\nThis test has [ANS] multiple-choice problems and [ANS] short-answer problems.", "answer_v1": [ "16", "9" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A test has $18$ problems, which are worth a total of $92$ points. There are two types of problems in the test. Each multiple-choice problem is worth $2$ points, and each short-answer problem is worth $10$ points. Write and solve a system equation to answer the following questions.\nThis test has [ANS] multiple-choice problems and [ANS] short-answer problems.", "answer_v2": [ "11", "7" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A test has $21$ problems, which are worth a total of $111$ points. There are two types of problems in the test. Each multiple-choice problem is worth $3$ points, and each short-answer problem is worth $9$ points. Write and solve a system equation to answer the following questions.\nThis test has [ANS] multiple-choice problems and [ANS] short-answer problems.", "answer_v3": [ "13", "8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0102", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "Eric invested a total of ${\\$9{,}000}$ in two accounts. One account pays $5\\%$ interest annually; the other pays $4\\%$ interest annually. At the end of the year, Eric earned a total of ${\\$400}$ in interest. How much money did Eric invest in each account? Write and solve a system of equations to answer the following questions.\nEric invested $[ANS] in the $5\\%$ account. Eric invested $[ANS] in the $4\\%$ account.", "answer_v1": [ "4000", "5000" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Hannah invested a total of ${\\$5{,}000}$ in two accounts. One account pays $7\\%$ interest annually; the other pays $2\\%$ interest annually. At the end of the year, Hannah earned a total of ${\\$250}$ in interest. How much money did Hannah invest in each account? Write and solve a system of equations to answer the following questions.\nHannah invested $[ANS] in the $7\\%$ account. Hannah invested $[ANS] in the $2\\%$ account.", "answer_v2": [ "3000", "2000" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Evan invested a total of ${\\$6{,}500}$ in two accounts. One account pays $5\\%$ interest annually; the other pays $2\\%$ interest annually. At the end of the year, Evan earned a total of ${\\$235}$ in interest. How much money did Evan invest in each account? Write and solve a system of equations to answer the following questions.\nEvan invested $[ANS] in the $5\\%$ account. Evan invested $[ANS] in the $2\\%$ account.", "answer_v3": [ "3500", "3000" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0103", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "You poured some $8\\%$ alcohol solution and some $12\\%$ alcohol solution into a mixing container. Now you have $640$ grams of $10.5 \\%$ alcohol solution. How many grams of $8\\%$ solution and how many grams of $12 \\%$ solution did you pour into the mixing container? Write and solve a system equation to answer the following questions.\nYou mixed [ANS] grams of $8\\%$ solution with [ANS] grams of $12\\%$ solution.", "answer_v1": [ "240", "400" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You poured some $10\\%$ alcohol solution and some $8\\%$ alcohol solution into a mixing container. Now you have $600$ grams of $8.8 \\%$ alcohol solution. How many grams of $10\\%$ solution and how many grams of $8 \\%$ solution did you pour into the mixing container? Write and solve a system equation to answer the following questions.\nYou mixed [ANS] grams of $10\\%$ solution with [ANS] grams of $8\\%$ solution.", "answer_v2": [ "240", "360" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You poured some $12\\%$ alcohol solution and some $6\\%$ alcohol solution into a mixing container. Now you have $800$ grams of $9 \\%$ alcohol solution. How many grams of $12\\%$ solution and how many grams of $6 \\%$ solution did you pour into the mixing container? Write and solve a system equation to answer the following questions.\nYou mixed [ANS] grams of $12\\%$ solution with [ANS] grams of $6\\%$ solution.", "answer_v3": [ "400", "400" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0104", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "Lily invested a total of ${\\$13{,}000}$ in two accounts. After a year, one account lost $7.6\\%$, while the other account gained $6.3\\%$. In total, Lily lost ${\\$362.50}$. How much money did Lily invest in each account? Write and solve a system of equations to answer the following questions.\nLily invested $[ANS] in the account with $7.6\\%$ loss. Lily invested $[ANS] in the account with $6.3\\%$ gain.", "answer_v1": [ "8500", "4500" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Eric invested a total of ${\\$10{,}000}$ in two accounts. After a year, one account lost $8.2\\%$, while the other account gained $4.9\\%$. In total, Eric lost ${\\$623.50}$. How much money did Eric invest in each account? Write and solve a system of equations to answer the following questions.\nEric invested $[ANS] in the account with $8.2\\%$ loss. Eric invested $[ANS] in the account with $4.9\\%$ gain.", "answer_v2": [ "8500", "1500" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Sydney invested a total of ${\\$11{,}000}$ in two accounts. After a year, one account lost $7.3\\%$, while the other account gained $4.2\\%$. In total, Sydney lost ${\\$515.50}$. How much money did Sydney invest in each account? Write and solve a system of equations to answer the following questions.\nSydney invested $[ANS] in the account with $7.3\\%$ loss. Sydney invested $[ANS] in the account with $4.2\\%$ gain.", "answer_v3": [ "8500", "2500" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0105", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "Company A’s revenue this fiscal year is ${\\$857{,}000}$, but its revenue is decreasing by ${\\$14{,}000}$ each year. Company B’s revenue this fiscal year is ${\\$317{,}000}$, and its revenue is increasing by ${\\$16{,}000}$ each year. Write and solve a system of equations to answer the following question.\nAfter [ANS] years, Company B will catch up with Company A in revenue.", "answer_v1": [ "18" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Company A’s revenue this fiscal year is ${\\$863{,}000}$, but its revenue is decreasing by ${\\$19{,}000}$ each year. Company B’s revenue this fiscal year is ${\\$603{,}000}$, and its revenue is increasing by ${\\$7{,}000}$ each year. Write and solve a system of equations to answer the following question.\nAfter [ANS] years, Company B will catch up with Company A in revenue.", "answer_v2": [ "10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Company A’s revenue this fiscal year is ${\\$904{,}000}$, but its revenue is decreasing by ${\\$14{,}000}$ each year. Company B’s revenue this fiscal year is ${\\$605{,}000}$, and its revenue is increasing by ${\\$9{,}000}$ each year. Write and solve a system of equations to answer the following question.\nAfter [ANS] years, Company B will catch up with Company A in revenue.", "answer_v3": [ "13" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0106", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "linear", "equation", "solve", "system", "integer", "application" ], "problem_v1": "A small fair charges different admission for adults and children; it charges ${\\$4}$ for adults, and ${\\$1.50}$ for children. On a certain day, the total revenue is ${\\$9{,}457.50}$ and the fair admits ${3800}$ people. How many adults and children were admitted?\nThere were [ANS] adults and [ANS] children at the fair.", "answer_v1": [ "1503", "2297" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A small fair charges different admission for adults and children; it charges ${\\$2.25}$ for adults, and ${\\$0.75}$ for children. On a certain day, the total revenue is ${\\$738}$ and the fair admits ${500}$ people. How many adults and children were admitted?\nThere were [ANS] adults and [ANS] children at the fair.", "answer_v2": [ "242", "258" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A small fair charges different admission for adults and children; it charges ${\\$2.75}$ for adults, and ${\\$1.25}$ for children. On a certain day, the total revenue is ${\\$2{,}963}$ and the fair admits ${1600}$ people. How many adults and children were admitted?\nThere were [ANS] adults and [ANS] children at the fair.", "answer_v3": [ "642", "958" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0107", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "Town A and Town B were located close to each other, and recently merged into one city. Town A had a population with $8\\%$ African Americans. Town B had a population with $12\\%$ African Americans. After the merge, the new city has a total of $4800$ residents, with $11\\%$ African Americans. How many residents did Town A and Town B used to have? Write and solve a system equation to answer the following questions.\nTown A used to have [ANS] residents, and Town B used to have [ANS] residents.", "answer_v1": [ "1200", "3600" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Town A and Town B were located close to each other, and recently merged into one city. Town A had a population with $6\\%$ Asians. Town B had a population with $8\\%$ Asians. After the merge, the new city has a total of $4000$ residents, with $7.4\\%$ Asians. How many residents did Town A and Town B used to have? Write and solve a system equation to answer the following questions.\nTown A used to have [ANS] residents, and Town B used to have [ANS] residents.", "answer_v2": [ "1200", "2800" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Town A and Town B were located close to each other, and recently merged into one city. Town A had a population with $12\\%$ native Americans. Town B had a population with $6\\%$ native Americans. After the merge, the new city has a total of $4800$ residents, with $7.75\\%$ native Americans. How many residents did Town A and Town B used to have? Write and solve a system equation to answer the following questions.\nTown A used to have [ANS] residents, and Town B used to have [ANS] residents.", "answer_v3": [ "1400", "3400" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0108", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "A school fund raising event sold a total of $178$ tickets and generated a total revenue of ${\\$776.25}$. There are two types of tickets: adult tickets and child tickets. Each adult ticket costs ${\\$5.75}$, and each child ticket costs ${\\$3.60}$. Write and solve a system of equations to answer the following questions. [ANS] adult tickets and [ANS] child tickets were sold.", "answer_v1": [ "63", "115" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A school fund raising event sold a total of $210$ tickets and generated a total revenue of ${\\$554.00}$. There are two types of tickets: adult tickets and child tickets. Each adult ticket costs ${\\$4.40}$, and each child ticket costs ${\\$1.90}$. Write and solve a system of equations to answer the following questions. [ANS] adult tickets and [ANS] child tickets were sold.", "answer_v2": [ "62", "148" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A school fund raising event sold a total of $174$ tickets and generated a total revenue of ${\\$467.20}$. There are two types of tickets: adult tickets and child tickets. Each adult ticket costs ${\\$4.55}$, and each child ticket costs ${\\$1.60}$. Write and solve a system of equations to answer the following questions. [ANS] adult tickets and [ANS] child tickets were sold.", "answer_v3": [ "64", "110" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0109", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "Phone Company A charges a monthly fee of ${\\$49.00}$, and ${\\$0.04}$ for each minute of talk time. Phone Company B charges a monthly fee of ${\\$35.00}$, and ${\\$0.09}$ for each minute of talk time. Write and solve a system equation to answer the following questions.\nThese two companies would charge the same amount on a monthly bill when the talk time was [ANS] minutes.", "answer_v1": [ "280" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Phone Company A charges a monthly fee of ${\\$27.00}$, and ${\\$0.05}$ for each minute of talk time. Phone Company B charges a monthly fee of ${\\$25.00}$, and ${\\$0.06}$ for each minute of talk time. Write and solve a system equation to answer the following questions.\nThese two companies would charge the same amount on a monthly bill when the talk time was [ANS] minutes.", "answer_v2": [ "200" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Phone Company A charges a monthly fee of ${\\$36.90}$, and ${\\$0.04}$ for each minute of talk time. Phone Company B charges a monthly fee of ${\\$30.00}$, and ${\\$0.07}$ for each minute of talk time. Write and solve a system equation to answer the following questions.\nThese two companies would charge the same amount on a monthly bill when the talk time was [ANS] minutes.", "answer_v3": [ "230" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0110", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "linear", "equation", "solve", "system", "integer", "application" ], "problem_v1": "The sum of two integers is ${122}$. The difference of the same numbers is ${8}$. Find the two numbers.\nThe numbers are [ANS] and [ANS].", "answer_v1": [ "65", "57" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The sum of two integers is ${-4}$. The difference of the same numbers is ${54}$. Find the two numbers.\nThe numbers are [ANS] and [ANS].", "answer_v2": [ "25", "-29" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The sum of two integers is ${75}$. The difference of the same numbers is ${3}$. Find the two numbers.\nThe numbers are [ANS] and [ANS].", "answer_v3": [ "39", "36" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0111", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "You will purchase some CDs and DVDs. If you purchase $8$ CDs and $11$ DVDs, it will cost you ${\\$133.05}$ ; if you purchase $11$ CDs and $8$ DVDs, it will cost you ${\\$120.60}$. Write and solve a system equation to answer the following questions.\nEach CD costs $[ANS] and each DVD costs $[ANS].", "answer_v1": [ "4.60", "8.75" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You will purchase some CDs and DVDs. If you purchase $15$ CDs and $8$ DVDs, it will cost you ${\\$79.70}$ ; if you purchase $8$ CDs and $15$ DVDs, it will cost you ${\\$111.20}$. Write and solve a system equation to answer the following questions.\nEach CD costs $[ANS] and each DVD costs $[ANS].", "answer_v2": [ "1.90", "6.40" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You will purchase some CDs and DVDs. If you purchase $7$ CDs and $8$ DVDs, it will cost you ${\\$78.60}$ ; if you purchase $8$ CDs and $7$ DVDs, it will cost you ${\\$73.65}$. Write and solve a system equation to answer the following questions.\nEach CD costs $[ANS] and each DVD costs $[ANS].", "answer_v3": [ "2.60", "7.55" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0112", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "solve", "system", "equation", "application" ], "problem_v1": "If a boat travels from Town A to Town B, it has to travel ${2210\\ {\\rm mi}}$ along a river. A boat traveled from Town A to Town B along the river’s current with its engine running at full speed. This trip took ${65\\ {\\rm hr}}$. Then the boat traveled back from Town B to Town A, again with the engine at full speed, but this time against the river’s current. This trip took ${110.5\\ {\\rm hr}}$. Write and solve a system of equations to answer the following questions.\nThe boat’s speed in still water with the engine running at full speed is [ANS]mi/hr. The river current’s speed was [ANS]mi/hr. Use mi for miles, and hr for hours.", "answer_v1": [ "27", "7" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If a boat travels from Town A to Town B, it has to travel ${437.5\\ {\\rm mi}}$ along a river. A boat traveled from Town A to Town B along the river’s current with its engine running at full speed. This trip took ${17.5\\ {\\rm hr}}$. Then the boat traveled back from Town B to Town A, again with the engine at full speed, but this time against the river’s current. This trip took ${62.5\\ {\\rm hr}}$. Write and solve a system of equations to answer the following questions.\nThe boat’s speed in still water with the engine running at full speed is [ANS]mi/hr. The river current’s speed was [ANS]mi/hr. Use mi for miles, and hr for hours.", "answer_v2": [ "16", "9" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If a boat travels from Town A to Town B, it has to travel ${1228.5\\ {\\rm mi}}$ along a river. A boat traveled from Town A to Town B along the river’s current with its engine running at full speed. This trip took ${45.5\\ {\\rm hr}}$. Then the boat traveled back from Town B to Town A, again with the engine at full speed, but this time against the river’s current. This trip took ${94.5\\ {\\rm hr}}$. Write and solve a system of equations to answer the following questions.\nThe boat’s speed in still water with the engine running at full speed is [ANS]mi/hr. The river current’s speed was [ANS]mi/hr. Use mi for miles, and hr for hours.", "answer_v3": [ "20", "7" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0113", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "4", "keywords": [ "polynomial' 'quartic", "logarithms", "exponentials", "exponential growth", "decay" ], "problem_v1": "Given the table below, find a quartic formula for $g(x)$.\n$\\begin{array}{cccccc}\\hline x &-4 &-2 & 1 & 2 & 4 \\\\ \\hline g(x) & 1085.99 & 58.17 & 8.64 & 97.73 & 1306.23 \\\\ \\hline \\end{array}$\n$g(x)=$ [ANS].", "answer_v1": [ "4.55*x^4+1.47*x^3+2.18*x^2+4.01*x+-3.57" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Given the table below, find a quartic formula for $g(x)$.\n$\\begin{array}{cccccc}\\hline x &-4 &-2 & 1 & 2 & 4 \\\\ \\hline g(x) &-2500.79 &-193.53 &-0.99 &-81.17 &-1530.15 \\\\ \\hline \\end{array}$\n$g(x)=$ [ANS].", "answer_v2": [ "-7.51*x^4+7.77*x^3+-6.31*x^2+-2.99*x+8.05" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Given the table below, find a quartic formula for $g(x)$.\n$\\begin{array}{cccccc}\\hline x &-4 &-2 & 1 & 2 & 4 \\\\ \\hline g(x) &-1054.41 &-91.97 &-9.86 &-58.05 &-804.17 \\\\ \\hline \\end{array}$\n$g(x)=$ [ANS].", "answer_v3": [ "-3.36*x^4+1.9*x^3+-3.99*x^2+0.880000000000001*x+-5.29" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0114", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "4", "keywords": [], "problem_v1": "Find the cubic polynomial $f(x)$ such that $f(-1)=1,$ $f'(-1)=2,$ $f''(-1)=-4,$ and $f'''(-1)=6.$\n$f(x)=$ [ANS].", "answer_v1": [ "x^3+x^2+x+2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the cubic polynomial $f(x)$ such that $f(2)=-12,$ $f'(2)=-15,$ $f''(2)=-18,$ and $f'''(2)=-12.$\n$f(x)=$ [ANS].", "answer_v2": [ "3*x^2-2*x^3-3*x-2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the cubic polynomial $f(x)$ such that $f(-1)=5,$ $f'(-1)=-7,$ $f''(-1)=8,$ and $f'''(-1)=-6.$\n$f(x)=$ [ANS].", "answer_v3": [ "x^2-x^3-2*x+1" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0115", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [], "problem_v1": "John and Tonya are brother and sister. John has five times as many sisters as brothers, and Tonya has twice as many sisters as brothers. How many boys and girls are there in this family?\nAnswer: [ANS] boys and [ANS] girls.", "answer_v1": [ "2", "5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Sasha and Mike are sister and brother. Sasha has as many brothers as sisters, and Mike has twice as many sisters as brothers. How many girls and boys are there in this family?\nAnswer: [ANS] girls and [ANS] boys.", "answer_v2": [ "4", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Sam and Tonya are brother and sister. Sam has three times as many sisters as brothers, and Tonya has as many brothers as sisters. How many boys and girls are there in this family?\nAnswer: [ANS] boys and [ANS] girls.", "answer_v3": [ "2", "3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0116", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "", "keywords": [], "problem_v1": "A dietitian is planning a meal that supplies certain quantities of vitamin C, calcium, and magnesium. Three foods will be used, their quantities measured in milligrams. The nutrients supplied by one unit of each food and the dietary requirements are given in the table below.\n$\\begin{array}{ccccc}\\hline Nutrient & Food 1 & Food 2 & Food 3 & Total Required (mg) \\\\ \\hline Vitamin C & 40 & 60 & 40 & 580 \\\\ \\hline Calcium & 20 & 45 & 40 & 430 \\\\ \\hline Magnesium & 20 & 45 & 30 & 390 \\\\ \\hline \\end{array}$\nWrite the augmented matrix for this problem.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array} What quantity (in units) of Food 1 is necesary to meet the dietary requirements? [ANS]\nWhat quantity (in units) of Food 2 is necesary to meet the dietary requirements? [ANS]\nWhat quantity (in units) of Food 3 is necesary to meet the dietary requirements? [ANS]", "answer_v1": [ "40", "60", "40", "580", "20", "45", "40", "430", "20", "45", "30", "390" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "A dietitian is planning a meal that supplies certain quantities of vitamin C, calcium, and magnesium. Three foods will be used, their quantities measured in milligrams. The nutrients supplied by one unit of each food and the dietary requirements are given in the table below.\n$\\begin{array}{ccccc}\\hline Nutrient & Food 1 & Food 2 & Food 3 & Total Required (mg) \\\\ \\hline Vitamin C & 20 & 10 & 10 & 115 \\\\ \\hline Calcium & 50 & 35 & 45 & 387.5 \\\\ \\hline Magnesium & 20 & 25 & 20 & 215 \\\\ \\hline \\end{array}$\nWrite the augmented matrix for this problem.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array} What quantity (in units) of Food 1 is necesary to meet the dietary requirements? [ANS]\nWhat quantity (in units) of Food 2 is necesary to meet the dietary requirements? [ANS]\nWhat quantity (in units) of Food 3 is necesary to meet the dietary requirements? [ANS]", "answer_v2": [ "20", "10", "10", "115", "50", "35", "45", "387.5", "20", "25", "20", "215" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "A dietitian is planning a meal that supplies certain quantities of vitamin C, calcium, and magnesium. Three foods will be used, their quantities measured in milligrams. The nutrients supplied by one unit of each food and the dietary requirements are given in the table below.\n$\\begin{array}{ccccc}\\hline Nutrient & Food 1 & Food 2 & Food 3 & Total Required (mg) \\\\ \\hline Vitamin C & 30 & 60 & 15 & 367.5 \\\\ \\hline Calcium & 20 & 60 & 20 & 350 \\\\ \\hline Magnesium & 20 & 60 & 30 & 375 \\\\ \\hline \\end{array}$\nWrite the augmented matrix for this problem.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array} What quantity (in units) of Food 1 is necesary to meet the dietary requirements? [ANS]\nWhat quantity (in units) of Food 2 is necesary to meet the dietary requirements? [ANS]\nWhat quantity (in units) of Food 3 is necesary to meet the dietary requirements? [ANS]", "answer_v3": [ "30", "60", "15", "367.5", "20", "60", "20", "350", "20", "60", "30", "375" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0117", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [], "problem_v1": "Consider a two-commodity market. When the unit prices of the products are $P_1$ and $P_2$, the quantities demanded, $D_1$ and $D_2$, and the quantities supplied, $S_1$ and $S_2$, are given by \\begin{array}{r@{}r@{}r@{}r@{}r@{}r@{}r} D_1 &=& 106 &-& 3 P_1 &+& P_2 \\\\ D_2 &=& 142 &+& P_1 &-& 3 P_2 \\\\ & & & & & & \\\\ S_1 &=&-33 &+& 3 P_1 & & \\\\ S_2 &=&-33 & & &+& 2 P_2 \\end{array}\n(a) What is the relationship between the two commodities? Do they compete, as do Volvos and BMWs, or do they complement one another, as do shirts and ties? [ANS]\n(b) Find the equilibrium prices (i.e. the prices for which supply equals demand), for both products. $P_1=$ [ANS] $P_2=$ [ANS]", "answer_v1": [ "compete", "30", "41" ], "answer_type_v1": [ "MCS", "NV", "NV" ], "options_v1": [ [ "compete", "complement" ], [], [] ], "problem_v2": "Consider a two-commodity market. When the unit prices of the products are $P_1$ and $P_2$, the quantities demanded, $D_1$ and $D_2$, and the quantities supplied, $S_1$ and $S_2$, are given by \\begin{array}{r@{}r@{}r@{}r@{}r@{}r@{}r} D_1 &=& 146 &-& 3 P_1 &-& P_2 \\\\ D_2 &=& 149 &-& P_1 &-& 2 P_2 \\\\ & & & & & & \\\\ S_1 &=&-32 &+& 2 P_1 & & \\\\ S_2 &=&-45 & & &+& 3 P_2 \\end{array}\n(a) What is the relationship between the two commodities? Do they compete, as do Volvos and BMWs, or do they complement one another, as do shirts and ties? [ANS]\n(b) Find the equilibrium prices (i.e. the prices for which supply equals demand), for both products. $P_1=$ [ANS] $P_2=$ [ANS]", "answer_v2": [ "complement", "29", "33" ], "answer_type_v2": [ "MCS", "NV", "NV" ], "options_v2": [ [ "compete", "complement" ], [], [] ], "problem_v3": "Consider a two-commodity market. When the unit prices of the products are $P_1$ and $P_2$, the quantities demanded, $D_1$ and $D_2$, and the quantities supplied, $S_1$ and $S_2$, are given by \\begin{array}{r@{}r@{}r@{}r@{}r@{}r@{}r} D_1 &=& 154 &-& 3 P_1 &-& P_2 \\\\ D_2 &=& 148 &-& P_1 &-& 2 P_2 \\\\ & & & & & & \\\\ S_1 &=&-72 &+& 3 P_1 & & \\\\ S_2 &=&-66 & & &+& 2 P_2 \\end{array}\n(a) What is the relationship between the two commodities? Do they compete, as do Volvos and BMWs, or do they complement one another, as do shirts and ties? [ANS]\n(b) Find the equilibrium prices (i.e. the prices for which supply equals demand), for both products. $P_1=$ [ANS] $P_2=$ [ANS]", "answer_v3": [ "complement", "30", "46" ], "answer_type_v3": [ "MCS", "NV", "NV" ], "options_v3": [ [ "compete", "complement" ], [], [] ] }, { "id": "Linear_algebra_0118", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "4", "keywords": [], "problem_v1": "Find the polynomial of degree 4 whose graph goes through the points $(-3,-218),$ $(-1,-2),$ $(0, 4),$ $(2, 22),$ and $(3,-26).$\n$f(x)=$ [ANS].", "answer_v1": [ "3*x^3-2*x^4+4*x^2+5*x+4" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the polynomial of degree 4 whose graph goes through the points $(-3,-299),$ $(-2,-58),$ $(0, 10),$ $(1, 17),$ and $(3,-113).$\n$f(x)=$ [ANS].", "answer_v2": [ "3*x^3-3*x^4+3*x^2+4*x+10" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the polynomial of degree 4 whose graph goes through the points $(-3,-309),$ $(-1,-5),$ $(0, 3),$ $(2, 1),$ and $(3,-117).$\n$f(x)=$ [ANS].", "answer_v3": [ "3*x^3-3*x^4+3*x^2+5*x+3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0119", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [], "problem_v1": "Consider the chemical reaction a CH_4+b O_2 \\rightarrow c CO_2+d H_2O, where $a,$ $b,$ $c,$ and $d$ are unknown positive integers. The reaction mush be balanced; that is, the number of atoms of each element must be the same before and after the reaction. For example, because the number of oxygen atoms must remain the same, 2b=2c+d. While there are many possible choices for $a,$ $b,$ $c,$ and $d$ that balance the reaction, it is customary to use the smallest possible integers. Balance this reaction. $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]\n$d=$ [ANS]", "answer_v1": [ "1", "2", "1", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the chemical reaction a NO_2+b H_2 O \\rightarrow c HNO_2+d HNO_3, where $a,$ $b,$ $c,$ and $d$ are unknown positive integers. The reaction mush be balanced; that is, the number of atoms of each element must be the same before and after the reaction. For example, because the number of oxygen atoms must remain the same, 2a+b=2c+3d. While there are many possible choices for $a,$ $b,$ $c,$ and $d$ that balance the reaction, it is customary to use the smallest possible integers. Balance this reaction. $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]\n$d=$ [ANS]", "answer_v2": [ "2", "1", "1", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the chemical reaction a NH_3+b O_2 \\rightarrow c NO+d H_2 O, where $a,$ $b,$ $c,$ and $d$ are unknown positive integers. The reaction mush be balanced; that is, the number of atoms of each element must be the same before and after the reaction. For example, because the number of oxygen atoms must remain the same, 2b=c+d. While there are many possible choices for $a,$ $b,$ $c,$ and $d$ that balance the reaction, it is customary to use the smallest possible integers. Balance this reaction. $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]\n$d=$ [ANS]", "answer_v3": [ "4", "5", "4", "6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0120", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "linear system of equations", "problem solving" ], "problem_v1": "One week, a computer store sold a total of 43 computers and hard drives. The revenue from these sales was \\$29,150.00. The computers sold for \\$950.00 per unit and the hard drives sold for \\$170.00 per unit. For the following problems, use $x$ for the number of computers sold and $y$ for the number of hard drives sold that week. a) Write a mathematical expression for the combined number of computers and hard drives sold that week. Answer: [ANS]\nc) How many computers did the store sell that week? Answer: [ANS]computers", "answer_v1": [ "x + y = 43", "950 * x + 170 * y = 29150", "28" ], "answer_type_v1": [ "EQ", "EQ", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "One week, a computer store sold a total of 39 computers and hard drives. The revenue from these sales was \\$14,280.00. The computers sold for \\$600.00 per unit and the hard drives sold for \\$120.00 per unit. For the following problems, use $x$ for the number of computers sold and $y$ for the number of hard drives sold that week. a) Write a mathematical expression for the combined number of computers and hard drives sold that week. Answer: [ANS]\nb) Write a mathematical expression for the amount of revenue from this week's sales. Answer: [ANS]\nc) How many hard drives did the store sell that week? Answer: [ANS] hard drives", "answer_v2": [ "x + y = 39", "600 * x + 120 * y = 14280", "19" ], "answer_type_v2": [ "EQ", "EQ", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "One week, a computer store sold a total of 39 computers and hard drives. The revenue from these sales was \\$18,500.00. The computers sold for \\$700.00 per unit and the hard drives sold for \\$150.00 per unit. For the following problems, use $x$ for the number of computers sold and $y$ for the number of hard drives sold that week. a) Write a mathematical expression for the combined number of computers and hard drives sold that week. Answer: [ANS]\nc) How many computers did the store sell that week? Answer: [ANS]computers", "answer_v3": [ "x + y = 39", "700 * x + 150 * y = 18500", "23" ], "answer_type_v3": [ "EQ", "EQ", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0121", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "linear system of equations", "problem solving" ], "problem_v1": "One positive number is 8 times another number. The difference between the two numbers is 112, find the numbers. Answer: [ANS].", "answer_v1": [ "(16, 128)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "One positive number is 11 times another number. The difference between the two numbers is 30, find the numbers. Answer: [ANS].", "answer_v2": [ "(3, 33)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "One positive number is 8 times another number. The difference between the two numbers is 49, find the numbers. Answer: [ANS].", "answer_v3": [ "(7, 56)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0122", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "linear system of equations", "problem solving" ], "problem_v1": "Three times a first number decreased by a second number is\n21. The first number increased by three times the second number\nis 47. Find the numbers. Answer: [ANS].", "answer_v1": [ "(11, 12)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Two times a first number decreased by a second number is\n2. The first number increased by twice the second number\nis 31. Find the numbers. Answer: [ANS].", "answer_v2": [ "(7, 12)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Two times a first number decreased by a second number is\n7. The first number increased by three times the second number\nis 35. Find the numbers. Answer: [ANS].", "answer_v3": [ "(8, 9)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0123", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "linear system of equations", "problem solving" ], "problem_v1": "A coin purse contains a mixture of 37 dimes and quarters. The coins have a total value of \\$6.55. For the following problems, Use $x$ for the number of dimes and $y$ for the number of quarters. a) Write a mathematical expression for the total number of coins. Answer: [ANS]\nc) How many quarters are in the coin purse? Answer: [ANS]quarters", "answer_v1": [ "x + y = 37", "0.10 * x + 0.25 * y = 6.55", "19" ], "answer_type_v1": [ "EQ", "EQ", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A coin purse contains a mixture of 31 dimes and quarters. The coins have a total value of \\$6.25. For the following problems, Use $x$ for the number of dimes and $y$ for the number of quarters. a) Write a mathematical expression for the total number of coins. Answer: [ANS]\nc) How many dimes are in the coin purse? Answer: [ANS]dimes", "answer_v2": [ "x + y = 31", "0.10 * x + 0.25 * y = 6.25", "10" ], "answer_type_v2": [ "EQ", "EQ", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A coin purse contains a mixture of 32 dimes and quarters. The coins have a total value of \\$6.05. For the following problems, Use $x$ for the number of dimes and $y$ for the number of quarters. a) Write a mathematical expression for the total number of coins. Answer: [ANS]\nc) How many dimes are in the coin purse? Answer: [ANS]dimes", "answer_v3": [ "x + y = 32", "0.10 * x + 0.25 * y = 6.05", "13" ], "answer_type_v3": [ "EQ", "EQ", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0124", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "linear system of equations", "problem solving" ], "problem_v1": "When a small plane flies with the wind, it can travel 3600 miles in 5 hours. When it flies against the wind, it takes 5.45454545454545 hours to travel the same distance. For the following problems, use $x$ for the rate of the airplane in still air and $y$ for the rate of the wind. a) Write a mathematical expression for the distance travelled with the wind. Answer: [ANS]\nc) What is the rate of the wind? (Note: The rate is measured in miles per hour: mph) Answer: [ANS]mph", "answer_v1": [ "3600 = (x+y)*5", "3600 = (x-y)*5.45454545454545", "30" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "When a small plane flies with the wind, it can travel 2040 miles in 3 hours. When it flies against the wind, it takes 3.4 hours to travel the same distance. For the following problems, use $x$ for the rate of the airplane in still air and $y$ for the rate of the wind. a) Write a mathematical expression for the distance travelled with the wind. Answer: [ANS]\nc) What is the rate of the plane in still air? (Note: The rate is measured in miles per hour: mph) Answer: [ANS]mph", "answer_v2": [ "2040 = (x+y)*3", "2040 = (x-y)*3.4", "640" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "When a small plane flies with the wind, it can travel 2780 miles in 4 hours. When it flies against the wind, it takes 4.448 hours to travel the same distance. For the following problems, use $x$ for the rate of the airplane in still air and $y$ for the rate of the wind. a) Write a mathematical expression for the distance travelled with the wind. Answer: [ANS]\nc) What is the rate of the wind? (Note: The rate is measured in miles per hour: mph) Answer: [ANS]mph", "answer_v3": [ "2780 = (x+y)*4", "2780 = (x-y)*4.448", "35" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0125", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "linear system of equations", "problem solving" ], "problem_v1": "Things did not go quite as planned. You invested \\$4,700.00, part of it in a stock that paid 6\\% annual interest. However, the rest of the money suffered a 4\\% loss. The total annual income from both investments was \\$62.00. For the following, use $x$ for the amount invested at 6\\% and $y$ for the amount that lost money. a) Write a mathematical expression for the total amount of money invested. Answer: [ANS]\nb) Write a mathematical expression for the total interest earned. Answer: [ANS]\nc) How much money was invested at 6\\%? Answer: $[ANS]", "answer_v1": [ "x + y = 4700", "6/100 * x - 4/100 * y = 62", "2500.00" ], "answer_type_v1": [ "EQ", "EQ", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Things did not go quite as planned. You invested \\$4,500.00, part of it in a stock that paid 4\\% annual interest. However, the rest of the money suffered a 3\\% loss. The total annual income from both investments was \\$68.00. For the following, use $x$ for the amount invested at 4\\% and $y$ for the amount that lost money. a) Write a mathematical expression for the total amount of money invested. Answer: [ANS]\nb) Write a mathematical expression for the total interest earned. Answer: [ANS]\nc) How much money was invested at a loss? Answer: $[ANS]", "answer_v2": [ "x + y = 4500", "4/100 * x - 3/100 * y = 68", "1600.00" ], "answer_type_v2": [ "EQ", "EQ", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Things did not go quite as planned. You invested \\$3,100.00, part of it in a stock that paid 7\\% annual interest. However, the rest of the money suffered a 5\\% loss. The total annual income from both investments was \\$13.00. For the following, use $x$ for the amount invested at 7\\% and $y$ for the amount that lost money. a) Write a mathematical expression for the total amount of money invested. Answer: [ANS]\nb) Write a mathematical expression for the total interest earned. Answer: [ANS]\nc) How much money was invested at a loss? Answer: $[ANS]", "answer_v3": [ "x + y = 3100", "7/100 * x - 5/100 * y = 13", "1700.00" ], "answer_type_v3": [ "EQ", "EQ", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0126", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "linear system of equations", "problem solving" ], "problem_v1": "With the current, you can canoe 30 miles in 6 hours. Against the same current, you can canoe only $ \\frac{3}{4} $ of this distance in 11.25 hours. For the following problems, use $x$ for the rate you canoe in still water and $y$ for the rate of the current. a) Write a mathematical expression for the distance you can canoe with the current. Answer: [ANS]\nc) What is the rate of the canoe in still water? (Note: The rate is measured in miles per hour: mph) Answer: [ANS]mph", "answer_v1": [ "30 = (x+y)*6", "0.75*30 = (x-y)*11.25", "3.5" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "With the current, you can canoe 2.625 miles in 0.75 hours. Against the same current, you can canoe only half of this distance in 2.625 hours. For the following problems, use $x$ for the rate you canoe in still water and $y$ for the rate of the current. a) Write a mathematical expression for the distance you can canoe with the current. Answer: [ANS]\nc) What is the rate of the current? (Note: The rate is measured in miles per hour: mph) Answer: [ANS]mph", "answer_v2": [ "2.625 = (x+y)*0.75", "0.5*2.625 = (x-y)*2.625", "1.5" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "With the current, you can canoe 12 miles in 3 hours. Against the same current, you can canoe only half of this distance in 6 hours. For the following problems, use $x$ for the rate you canoe in still water and $y$ for the rate of the current. a) Write a mathematical expression for the distance you can canoe with the current. Answer: [ANS]\nc) What is the rate of the canoe in still water? (Note: The rate is measured in miles per hour: mph) Answer: [ANS]mph", "answer_v3": [ "12 = (x+y)*3", "0.5*12 = (x-y)*6", "2.5" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0127", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "linear system of equations", "problem solving" ], "problem_v1": "An elementary school sells tickets to the school play to raise money for new \nplayground equipment. A total of 131 tickets are sold with \nticket prices of \\$10 per adult and \\$4 per child.\nThe school raised \\$1052 from ticket sales. For the following problems, Use $x$ for the number of adults and $y$ for the number of children. a) Write an equation that shows how $x$ and $y$ are related to the total number of tickets sold. The total number of tickets sold should be the right hand side of your equation. Answer: [ANS]\nc) How many adults attended the play? Answer: [ANS]adults", "answer_v1": [ "x + y = 131", "10 * x + 4 * y = 1052", "88" ], "answer_type_v1": [ "EQ", "EQ", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "An elementary school sells tickets to the school play to raise money for new \nplayground equipment. A total of 112 tickets are sold with \nticket prices of \\$7 per adult and \\$3 per child.\nThe school raised \\$552 from ticket sales. For the following problems, Use $x$ for the number of adults and $y$ for the number of children. a) Write an equation that shows how $x$ and $y$ are related to the total number of tickets sold. The total number of tickets sold should be the right hand side of your equation. Answer: [ANS]\nc) How many children attended the play? Answer: [ANS]children", "answer_v2": [ "x + y = 112", "7 * x + 3 * y = 552", "58" ], "answer_type_v2": [ "EQ", "EQ", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "An elementary school sells tickets to the school play to raise money for new \nplayground equipment. A total of 109 tickets are sold with \nticket prices of \\$7 per adult and \\$4 per child.\nThe school raised \\$631 from ticket sales. For the following problems, Use $x$ for the number of adults and $y$ for the number of children. a) Write an equation that shows how $x$ and $y$ are related to the total number of tickets sold. The total number of tickets sold should be the right hand side of your equation. Answer: [ANS]\nc) How many adults attended the play? Answer: [ANS]adults", "answer_v3": [ "x + y = 109", "7 * x + 4 * y = 631", "65" ], "answer_type_v3": [ "EQ", "EQ", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0128", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "3", "keywords": [ "algebra" ], "problem_v1": "You have boxes of five types. They are characterized by their length $L$, their width $W$, their height $H$, their age $A$, and their price $P$, as given in the following table: \\begin{array}{cccccc} \\hbox{type} & L & W & H & A & P \\\\ I & 1 & 1 & 1 & 1 & 1 \\\\ II & 1 & 1 & 2 & 3 & 2 \\\\ III & 1 & 2 & 3 & 2 & 2 \\\\ IV & 1 & 2 & 4 & 1 & 2 \\\\ V & 1 & 2 & 5 & 1 & 3 \\\\ \\end{array} So for example, boxes of type IV measure 1 by 2 by 4 feet, are a year old, and cost\\char 36 2.-each.\nSuppose the sum of the lengths of your boxes is 25, the sum of their widths 40, the sum of their heights 75, the sum of their ages 43, and the total price of those boxes 51. You have [ANS] boxes of type I, [ANS] boxes of type II, [ANS] boxes of type III, [ANS] boxes of type IV, [ANS] boxes of type V.", "answer_v1": [ "4", "6", "6", "4", "5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "You have boxes of five types. They are characterized by their length $L$, their width $W$, their height $H$, their age $A$, and their price $P$, as given in the following table: \\begin{array}{cccccc} \\hbox{type} & L & W & H & A & P \\\\ I & 1 & 1 & 1 & 1 & 1 \\\\ II & 1 & 1 & 2 & 3 & 2 \\\\ III & 1 & 2 & 3 & 2 & 2 \\\\ IV & 1 & 2 & 4 & 1 & 2 \\\\ V & 1 & 2 & 5 & 1 & 3 \\\\ \\end{array} So for example, boxes of type IV measure 1 by 2 by 4 feet, are a year old, and cost\\char 36 2.-each.\nSuppose the sum of the lengths of your boxes is 19, the sum of their widths 31, the sum of their heights 56, the sum of their ages 29, and the total price of those boxes 36. You have [ANS] boxes of type I, [ANS] boxes of type II, [ANS] boxes of type III, [ANS] boxes of type IV, [ANS] boxes of type V.", "answer_v2": [ "4", "3", "4", "6", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "You have boxes of five types. They are characterized by their length $L$, their width $W$, their height $H$, their age $A$, and their price $P$, as given in the following table: \\begin{array}{cccccc} \\hbox{type} & L & W & H & A & P \\\\ I & 1 & 1 & 1 & 1 & 1 \\\\ II & 1 & 1 & 2 & 3 & 2 \\\\ III & 1 & 2 & 3 & 2 & 2 \\\\ IV & 1 & 2 & 4 & 1 & 2 \\\\ V & 1 & 2 & 5 & 1 & 3 \\\\ \\end{array} So for example, boxes of type IV measure 1 by 2 by 4 feet, are a year old, and cost\\char 36 2.-each.\nSuppose the sum of the lengths of your boxes is 32, the sum of their widths 52, the sum of their heights 94, the sum of their ages 57, and the total price of those boxes 63. You have [ANS] boxes of type I, [ANS] boxes of type II, [ANS] boxes of type III, [ANS] boxes of type IV, [ANS] boxes of type V.", "answer_v3": [ "4", "8", "9", "8", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Linear_algebra_0129", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [], "problem_v1": "A coffee shop sells assorted types of coffees, and blends them into their \"signature blends\". Sri Lankan sells for \\$4.20 per pound, and Costa Rican sells for \\$7.20 per pound. A customer pays \\$20.85 for 3 pounds of a blend. How much of each type of coffee goes into the blend?\nAmount of Sri Lankan coffee=[ANS] pounds\nAmount of Costa Rican coffee=[ANS] pounds", "answer_v1": [ "0.25", "2.75" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A coffee shop sells assorted types of coffees, and blends them into their \"signature blends\". Kenyan sells for \\$3.20 per pound, and Honduran sells for \\$6.00 per pound. A customer pays \\$14.50 for 3 pounds of a blend. How much of each type of coffee goes into the blend?\nAmount of Kenyan coffee=[ANS] pounds\nAmount of Honduran coffee=[ANS] pounds", "answer_v2": [ "1.25", "1.75" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A coffee shop sells assorted types of coffees, and blends them into their \"signature blends\". Columbian sells for \\$3.60 per pound, and Peruvian sells for \\$6.60 per pound. A customer pays \\$25.65 for 4 pounds of a blend. How much of each type of coffee goes into the blend?\nAmount of Columbian coffee=[ANS] pounds\nAmount of Peruvian coffee=[ANS] pounds", "answer_v3": [ "0.25", "3.75" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0130", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [], "problem_v1": "Two solutions of salt water contain 0.07\\% and 0.19\\% salt respectively. A lab technician wants to make 1 liter of solution which contains 0.16\\% salt. How much of each solution should she use?\nAmount of 0.07\\% solution=[ANS] milliliters\nAmount of 0.19\\% solution=[ANS] milliliters", "answer_v1": [ "1000*(0.19-0.16)/(0.19-0.07)", "1000*(0.16-0.07)/(0.19-0.07)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Two solutions of salt water contain 0.1\\% and 0.3\\% salt respectively. A lab technician wants to make 1 liter of solution which contains 0.13\\% salt. How much of each solution should she use?\nAmount of 0.1\\% solution=[ANS] milliliters\nAmount of 0.3\\% solution=[ANS] milliliters", "answer_v2": [ "1000*(0.3-0.13)/(0.3-0.1)", "1000*(0.13-0.1)/(0.3-0.1)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Two solutions of salt water contain 0.07\\% and 0.17\\% salt respectively. A lab technician wants to make 1 liter of solution which contains 0.1\\% salt. How much of each solution should she use?\nAmount of 0.07\\% solution=[ANS] milliliters\nAmount of 0.17\\% solution=[ANS] milliliters", "answer_v3": [ "1000*(0.17-0.1)/(0.17-0.07)", "1000*(0.1-0.07)/(0.17-0.07)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0131", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "Algebra' 'Linear Equations" ], "problem_v1": "Find two numbers whose sum is 4 and whose difference is 2. Your answer is: The largest of the two numbers is: [ANS]\nThe smallest of the two number is: [ANS]", "answer_v1": [ "3", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find two numbers whose sum is 0 and whose difference is-10. Your answer is: The largest of the two numbers is: [ANS]\nThe smallest of the two number is: [ANS]", "answer_v2": [ "5", "-5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find two numbers whose sum is-1 and whose difference is-3. Your answer is: The largest of the two numbers is: [ANS]\nThe smallest of the two number is: [ANS]", "answer_v3": [ "1", "-2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0132", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "Algebra' 'Linear Equations" ], "problem_v1": "Susan places \\$6100 in three investments at rates of 9\\%, 10.8\\% and 16.2\\% per annum, respectively. The total income after one year is \\$ 788.40. If the amount placed in the third investment is \\$1200 more than the amount placed in the second, find the amount of each investment. Your answer is: Amount at 9\\% equals: \\$ [ANS]\nAmount at 10.8\\% equals: \\$ [ANS]\nAmount at 16.2\\% equals: \\$ [ANS]", "answer_v1": [ "1500", "1700", "2900" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Susan places \\$5900 in three investments at rates of 5\\%, 6\\% and 9\\% per annum, respectively. The total income after one year is \\$ 406.00. If the amount placed in the third investment is \\$900 more than the amount placed in the second, find the amount of each investment. Your answer is: Amount at 5\\% equals: \\$ [ANS]\nAmount at 6\\% equals: \\$ [ANS]\nAmount at 9\\% equals: \\$ [ANS]", "answer_v2": [ "2000", "1500", "2400" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Susan places \\$5500 in three investments at rates of 6\\%, 7.2\\% and 10.8\\% per annum, respectively. The total income after one year is \\$ 460.80. If the amount placed in the third investment is \\$600 more than the amount placed in the second, find the amount of each investment. Your answer is: Amount at 6\\% equals: \\$ [ANS]\nAmount at 7.2\\% equals: \\$ [ANS]\nAmount at 10.8\\% equals: \\$ [ANS]", "answer_v3": [ "1500", "1700", "2300" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0133", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "Algebra' 'Linear Equations" ], "problem_v1": "A woman has 31 coins in her pocket, all of which are dimes and quarters. If the total value of the coins is \\$ 5.20, how many dimes and how many quarters does she have? Your answer is: Number of dimes equals [ANS]\nNumber of quarters equals [ANS]", "answer_v1": [ "17", "14" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A woman has 25 coins in her pocket, all of which are dimes and quarters. If the total value of the coins is \\$ 5.35, how many dimes and how many quarters does she have? Your answer is: Number of dimes equals [ANS]\nNumber of quarters equals [ANS]", "answer_v2": [ "6", "19" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A woman has 24 coins in her pocket, all of which are dimes and quarters. If the total value of the coins is \\$ 4.50, how many dimes and how many quarters does she have? Your answer is: Number of dimes equals [ANS]\nNumber of quarters equals [ANS]", "answer_v3": [ "10", "14" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0134", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "Algebra' 'Linear Equations" ], "problem_v1": "A man invests his savings in two accounts, one paying 6\\% and the other paying 10\\% simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is \\$8822 dollars. How much did he invest at each rate? Your answer is: Amount invested at 6\\% equals \\$ [ANS]\nAmount invested at 10\\% equals \\$ [ANS]", "answer_v1": [ "80200", "40100" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A man invests his savings in two accounts, one paying 6\\% and the other paying 10\\% simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is \\$2926 dollars. How much did he invest at each rate? Your answer is: Amount invested at 6\\% equals \\$ [ANS]\nAmount invested at 10\\% equals \\$ [ANS]", "answer_v2": [ "26600", "13300" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A man invests his savings in two accounts, one paying 6\\% and the other paying 10\\% simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is \\$4950 dollars. How much did he invest at each rate? Your answer is: Amount invested at 6\\% equals \\$ [ANS]\nAmount invested at 10\\% equals \\$ [ANS]", "answer_v3": [ "45000", "22500" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0135", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "modeling", "percent", "interest" ], "problem_v1": "Alex splits an investment of \\$55000, a portion earning simple interest at a rate of 4.6 \\% per year and the rest earning at a rate of 8.2 \\% per year in such a way that the return on the total investment is 6.76 \\%. How much money was invested at each rate? (Round to the nearest 100 dollars.)\nAt rate 4.6 \\%: [ANS]\nAt rate 8.2 \\%: [ANS]", "answer_v1": [ "22000", "33000" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Laura splits an investment of \\$76000, a portion earning simple interest at a rate of 3.3 \\% per year and the rest earning at a rate of 7 \\% per year in such a way that the return on the total investment is 6.26 \\%. How much money was invested at each rate? (Round to the nearest 100 dollars.)\nAt rate 3.3 \\%: [ANS]\nAt rate 7 \\%: [ANS]", "answer_v2": [ "15200", "60800" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Cheryl splits an investment of \\$56000, a portion earning simple interest at a rate of 3.7 \\% per year and the rest earning at a rate of 7.7 \\% per year in such a way that the return on the total investment is 6.1 \\%. How much money was invested at each rate? (Round to the nearest 100 dollars.)\nAt rate 3.7 \\%: [ANS]\nAt rate 7.7 \\%: [ANS]", "answer_v3": [ "22400", "33600" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0136", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "functions", "algebra", "application of linear equations", "systems of linear equations" ], "problem_v1": "A concert venue sold 1950 tickets one evening. Tickets cost $\\$40.00$ for a covered pavilion seat and $\\$30.00$ for a lawn seat. Total receipts were $\\$65{,}750.00$. How many of each type of ticket were sold? Lawn tickets: [ANS]\nPavilion tickets: [ANS]", "answer_v1": [ "1225", "725" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "During a one-month promotional campaign, Tiger Films gave either a free DVD rental or a 12-serving box of microwave popcorn to new members. It cost the store \\$1 for each free rental and \\$2 for each box of popcorn. A total of 100 new members were signed up and the store's cost for the incentives was \\$150. How many of each incentive were given away? DVD Rentals: [ANS]\nBoxes of Popcorn: [ANS]", "answer_v2": [ "50", "50" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A concert venue sold 2050 tickets one evening. Tickets cost $\\$30.00$ for a covered pavilion seat and $\\$20.00$ for a lawn seat. Total receipts were $\\$50{,}000.00$. How many of each type of ticket were sold? Lawn tickets: [ANS]\nPavilion tickets: [ANS]", "answer_v3": [ "1150", "900" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0137", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "Two common names for streets are First Street and Main Street, with $13085$ streets bearing one of these names. There are $1925$ more streets named First Street than Main Street. How many streets bear each name? Number of First Streets: [ANS]\nNumber of Main Streets: [ANS]", "answer_v1": [ "7505", "5580" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Two common names for streets are First Street and Main Street, with $12100$ streets bearing one of these names. There are $230$ more streets named First Street than Main Street. How many streets bear each name? Number of First Streets: [ANS]\nNumber of Main Streets: [ANS]", "answer_v2": [ "6165", "5935" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Two common names for streets are First Street and Main Street, with $12230$ streets bearing one of these names. There are $1020$ more streets named First Street than Main Street. How many streets bear each name? Number of First Streets: [ANS]\nNumber of Main Streets: [ANS]", "answer_v3": [ "6625", "5605" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0138", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "A department store sold $63$ shirts one day. All short-sleeved shirts cost $\\$13.00$ each and all long-sleeved shirts cost $\\$23.00$ each. Total receipts for the day were $\\$1{,}099.00$. How many of each kind of shirt were sold? Short-Sleeve: [ANS]\nLong-Sleeve: [ANS]", "answer_v1": [ "35", "28" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A department store sold $33$ shirts one day. All short-sleeved shirts cost $\\$10.00$ each and all long-sleeved shirts cost $\\$17.00$ each. Total receipts for the day were $\\$414.00$. How many of each kind of shirt were sold? Short-Sleeve: [ANS]\nLong-Sleeve: [ANS]", "answer_v2": [ "21", "12" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A department store sold $44$ shirts one day. All short-sleeved shirts cost $\\$10.00$ each and all long-sleeved shirts cost $\\$19.00$ each. Total receipts for the day were $\\$602.00$. How many of each kind of shirt were sold? Short-Sleeve: [ANS]\nLong-Sleeve: [ANS]", "answer_v3": [ "26", "18" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0139", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "An apple contains 140 calories and 3 g of fiber. A banana contains 150 calories and 4 g of fiber. Set up a system of equations that could be solved to determine the number of apples ($A$) and bananas ($B$) that should be eaten to obtain 5930 calories and 142 g of fiber. Equation 1: [ANS]\nEquation 2: [ANS]", "answer_v1": [ "140*A+150*B = 5930", "3*A+4*B = 142" ], "answer_type_v1": [ "EQ", "EQ" ], "options_v1": [ [], [] ], "problem_v2": "An apple contains 90 calories and 5 g of fiber. A banana contains 110 calories and 3 g of fiber. Set up a system of equations that could be solved to determine the number of apples ($A$) and bananas ($B$) that should be eaten to obtain 3630 calories and 127 g of fiber. Equation 1: [ANS]\nEquation 2: [ANS]", "answer_v2": [ "90*A+110*B = 3630", "5*A+3*B = 127" ], "answer_type_v2": [ "EQ", "EQ" ], "options_v2": [ [], [] ], "problem_v3": "An apple contains 100 calories and 2 g of fiber. A banana contains 130 calories and 3 g of fiber. Set up a system of equations that could be solved to determine the number of apples ($A$) and bananas ($B$) that should be eaten to obtain 3970 calories and 87 g of fiber. Equation 1: [ANS]\nEquation 2: [ANS]", "answer_v3": [ "100*A+130*B = 3970", "2*A+3*B = 87" ], "answer_type_v3": [ "EQ", "EQ" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0140", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "functions", "algebra", "application of linear equations", "systems of linear equations" ], "problem_v1": "Justin's boat travels 144 km downstream in 4 hours and it travels 168 km upstream in 7 hours. Find the speed of the boat in still water and the speed of the stream's current. Boat Speed: [ANS] km/h Current Speed: [ANS] km/h", "answer_v1": [ "30", "6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Justin's boat travels 48 km downstream in 2 hours and it travels 32 km upstream in 4 hours. Find the speed of the boat in still water and the speed of the stream's current. Boat Speed: [ANS] km/h Current Speed: [ANS] km/h", "answer_v2": [ "16", "8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Justin's boat travels 81 km downstream in 3 hours and it travels 75 km upstream in 5 hours. Find the speed of the boat in still water and the speed of the stream's current. Boat Speed: [ANS] km/h Current Speed: [ANS] km/h", "answer_v3": [ "21", "6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0141", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "equations", "linear system of equations" ], "problem_v1": "Suppose that you invest \\$ $16700.00$ between two accounts. The first account is safer and yields 5.5\\% interest. The second account is riskier, but yields an interest rate of 8.8\\%. Letting $x$ denote the amount of money invested in the first, safer account, and letting $y$ denote the amount of money invested in the second, riskier account, setup a system of linear equations which you could solve to find out how much money you should invest in the accounts so that you earn \\$ $1202.30$ in interest per year. Equation 1: [ANS]\nEquation 2: [ANS]\nHelp: Make sure to enter big numbers WITHOUT commas.", "answer_v1": [ "x+y = 16700", "0.055*x+0.088*y = 1202.3" ], "answer_type_v1": [ "EQ", "EQ" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that you invest \\$ $12400.00$ between two accounts. The first account is safer and yields 4.1\\% interest. The second account is riskier, but yields an interest rate of 9.8\\%. Letting $x$ denote the amount of money invested in the first, safer account, and letting $y$ denote the amount of money invested in the second, riskier account, setup a system of linear equations which you could solve to find out how much money you should invest in the accounts so that you earn \\$ $890.30$ in interest per year. Equation 1: [ANS]\nEquation 2: [ANS]\nHelp: Make sure to enter big numbers WITHOUT commas.", "answer_v2": [ "x+y = 12400", "0.041*x+0.098*y = 890.3" ], "answer_type_v2": [ "EQ", "EQ" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that you invest \\$ $14100.00$ between two accounts. The first account is safer and yields 4.6\\% interest. The second account is riskier, but yields an interest rate of 8.8\\%. Letting $x$ denote the amount of money invested in the first, safer account, and letting $y$ denote the amount of money invested in the second, riskier account, setup a system of linear equations which you could solve to find out how much money you should invest in the accounts so that you earn \\$ $972.00$ in interest per year. Equation 1: [ANS]\nEquation 2: [ANS]\nHelp: Make sure to enter big numbers WITHOUT commas.", "answer_v3": [ "x+y = 14100", "0.046*x+0.088*y = 972" ], "answer_type_v3": [ "EQ", "EQ" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0142", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "2", "keywords": [ "algebra", "application of linear equations", "systems of linear equations", "equations" ], "problem_v1": "During a one-month promotional campaign, Tiger Films gave either a free DVD rental or a 12-serving box of microwave popcorn to new members. It cost the store \\$1 for each free rental and \\$2 for each box of popcorn. A total of 88 new members were signed up and the store's cost for the incentives was \\$131. How many of each incentive were given away? DVD Rentals: [ANS]\nBoxes of Popcorn: [ANS]", "answer_v1": [ "45", "43" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "During a one-month promotional campaign, Tiger Films gave either a free DVD rental or a 12-serving box of microwave popcorn to new members. It cost the store \\$1 for each free rental and \\$2 for each box of popcorn. A total of 89 new members were signed up and the store's cost for the incentives was \\$147. How many of each incentive were given away? DVD Rentals: [ANS]\nBoxes of Popcorn: [ANS]", "answer_v2": [ "31", "58" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "During a one-month promotional campaign, Tiger Films gave either a free DVD rental or a 12-serving box of microwave popcorn to new members. It cost the store \\$1 for each free rental and \\$2 for each box of popcorn. A total of 80 new members were signed up and the store's cost for the incentives was \\$124. How many of each incentive were given away? DVD Rentals: [ANS]\nBoxes of Popcorn: [ANS]", "answer_v3": [ "36", "44" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0143", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "algebra", "systems of equations", "substitution method" ], "problem_v1": "Eugene's financial advisor invests money for Eugene into two accounts. Account 1 earns a simple interest rate of $3.163\\%$. Account 2 earns $6.852\\%$ simple interest but is a riskier investment. Eugene isn't told how much was invested in each account, but he knows that he invested \\$6,773.00 total, and he earned \\$425.81 in interest (before commission) after 3 years. How much was invested in each account? Amount invested in Account 1=$[ANS]\nAmount invested in Account 2=$[ANS]", "answer_v1": [ "4030.00", "2743.00" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Eugene's financial advisor invests money for Eugene into two accounts. Account 1 earns a simple interest rate of $3.864\\%$. Account 2 earns $5.674\\%$ simple interest but is a riskier investment. Eugene isn't told how much was invested in each account, but he knows that he invested \\$7,270.00 total, and he earned \\$224.90 in interest (before commission) after 3 years. How much was invested in each account? Amount invested in Account 1=$[ANS]\nAmount invested in Account 2=$[ANS]", "answer_v2": [ "3365.00", "3905.00" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Eugene's financial advisor invests money for Eugene into two accounts. Account 1 earns a simple interest rate of $3.211\\%$. Account 2 earns $5.996\\%$ simple interest but is a riskier investment. Eugene isn't told how much was invested in each account, but he knows that he invested \\$6,306.00 total, and he earned \\$294.03 in interest (before commission) after 3 years. How much was invested in each account? Amount invested in Account 1=$[ANS]\nAmount invested in Account 2=$[ANS]", "answer_v3": [ "3735.00", "2571.00" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0144", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "algebra", "systems of equations", "substitution method" ], "problem_v1": "A merchandise vendor at a sports arena is given a large uncounted pile of hats and shirts to sell at a game. Hats cost \\$18.00 each, and shirts sell for \\$19.00 each. At the end of the night, the vendor realized he forgot to keep track of how many quantities of each item he sold. However, he does know that he sold 66 total items and sold \\$1,222.00 of merchandise. How many hats did he sell?\nNumber of hats=[ANS]", "answer_v1": [ "32" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A merchandise vendor at a sports arena is given a large uncounted pile of hats and shirts to sell at a game. Hats cost \\$12.00 each, and shirts sell for \\$22.00 each. At the end of the night, the vendor realized he forgot to keep track of how many quantities of each item he sold. However, he does know that he sold 54 total items and sold \\$898.00 of merchandise. How many hats did he sell?\nNumber of hats=[ANS]", "answer_v2": [ "29" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A merchandise vendor at a sports arena is given a large uncounted pile of hats and shirts to sell at a game. Hats cost \\$14.00 each, and shirts sell for \\$19.00 each. At the end of the night, the vendor realized he forgot to keep track of how many quantities of each item he sold. However, he does know that he sold 60 total items and sold \\$990.00 of merchandise. How many hats did he sell?\nNumber of hats=[ANS]", "answer_v3": [ "30" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0145", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "algebra", "systems of equations", "substitution method" ], "problem_v1": "When sold for $\\$776.00$, a certain desktop has an annual supply of $129$ million computers and an annual demand of $156.5$ million computers. When the price increases to $\\$863.00$, the annual supply increases to $148$ million computers, and the demand drops to $134$ million computers.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity (in billions). The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v1": [ "4.57895*q+185.316", "1381.13-3.86667*q", "833.65", "141.59" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "When sold for $\\$708.00$, a certain desktop has an annual supply of $134$ million computers and an annual demand of $151.5$ million computers. When the price increases to $\\$845.00$, the annual supply increases to $154.5$ million computers, and the demand drops to $133.5$ million computers.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity (in billions). The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v2": [ "6.68293*q-187.512", "1861.08-7.61111*q", "770.27", "143.318" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "When sold for $\\$731.00$, a certain desktop has an annual supply of $129$ million computers and an annual demand of $152.5$ million computers. When the price increases to $\\$855.00$, the annual supply increases to $147$ million computers, and the demand drops to $134$ million computers.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity (in billions). The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v3": [ "6.88889*q-157.667", "1753.16-6.7027*q", "810.84", "140.589" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0146", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "algebra", "systems of equations", "substitution method" ], "problem_v1": "When gas prices are $\\$2.96$ per gallon, the annual supply for gas in New York State is $136$ billion gallons and the annual demand is $143$ billion gallons. When the price increases to $\\$3.24$ per gallon, the annual supply increases to $145$ billion gallons and the demand decreases to $131$ billion gallons.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity (in billions). The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v1": [ "0.031*q-1.256", "6.249-0.023*q", "3.05", "138.981" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "When gas prices are $\\$2.72$ per gallon, the annual supply for gas in New York State is $140$ billion gallons and the annual demand is $140$ billion gallons. When the price increases to $\\$3.19$ per gallon, the annual supply increases to $153$ billion gallons and the demand decreases to $131$ billion gallons.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity (in billions). The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v2": [ "0.036*q-2.32", "10-0.052*q", "2.72", "140" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "When gas prices are $\\$2.80$ per gallon, the annual supply for gas in New York State is $136$ billion gallons and the annual demand is $141$ billion gallons. When the price increases to $\\$3.22$ per gallon, the annual supply increases to $144$ billion gallons and the demand decreases to $131$ billion gallons.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity (in billions). The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v3": [ "0.052*q-4.272", "8.722-0.042*q", "2.92", "138.234" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0147", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "algebra", "systems of equations", "substitution method" ], "problem_v1": "When sold for $\\$258{,}000.00$, Ferraris have an annual supply of $6633$ vehicles and an annual demand of $6648$ vehicles. When their price increases to $\\$277{,}000.00$, the annual supply increases to $6960$, and the demand decreases to $5899$ billion gallons.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity. The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v1": [ "58.104*q-127404", "426641-25.3672*q", "258264.87", "6637.56" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "When sold for $\\$250{,}000.00$, Ferraris have an annual supply of $6773$ vehicles and an annual demand of $6459$ vehicles. When their price increases to $\\$273{,}000.00$, the annual supply increases to $7090$, and the demand decreases to $5895$ billion gallons.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity. The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v2": [ "72.5552*q-241416", "513399-40.7801*q", "241802.50", "6660.02" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "When sold for $\\$253{,}000.00$, Ferraris have an annual supply of $6642$ vehicles and an annual demand of $6511$ vehicles. When their price increases to $\\$276{,}000.00$, the annual supply increases to $6941$, and the demand decreases to $5904$ billion gallons.\n(a) Assuming that the supply and demand equations are linear, \ffind the supply and demand equations. Supply Equation $p=$ [ANS]\nDemand Equation $p=$ [ANS]\n(Note: The equations should be in the form $p=mq+b$ where $p$ denotes the price (in dollars) and $q$ denotes the quantity. The slope and $y$-intercept should be accurate to two decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price $p=$ $[ANS]\nEquilibrium quantity $q=$ [ANS]\n.", "answer_v3": [ "76.9231*q-257923", "499710-37.8913*q", "249674.39", "6598.77" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0148", "subject": "Linear_algebra", "topic": "Systems of linear equations", "subtopic": "Applications", "level": "5", "keywords": [ "algebra", "systems of equations", "substitution method" ], "problem_v1": "A hockey arena merchandise shop sells two types of jerseys: practice jerseys and game-worn jerseys. Practice jerseys cost \\$40.99 while game-worn jerseys cost \\$216.99 each. If during one particular month, the merchandise shop sells 1003 total jerseys and earns a revenue of \\$114,504.97, how many of each type of jersey did they sell that month? Number of practice jerseys=[ANS]\nNumber of game-worn jerseys=[ANS]", "answer_v1": [ "586", "417" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A hockey arena merchandise shop sells two types of jerseys: practice jerseys and game-worn jerseys. Practice jerseys cost \\$26.99 while game-worn jerseys cost \\$244.99 each. If during one particular month, the merchandise shop sells 805 total jerseys and earns a revenue of \\$87,126.95, how many of each type of jersey did they sell that month? Number of practice jerseys=[ANS]\nNumber of game-worn jerseys=[ANS]", "answer_v2": [ "505", "300" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A hockey arena merchandise shop sells two types of jerseys: practice jerseys and game-worn jerseys. Practice jerseys cost \\$31.99 while game-worn jerseys cost \\$218.99 each. If during one particular month, the merchandise shop sells 892 total jerseys and earns a revenue of \\$96,790.08, how many of each type of jersey did they sell that month? Number of practice jerseys=[ANS]\nNumber of game-worn jerseys=[ANS]", "answer_v3": [ "527", "365" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0149", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "algebra", "matrix operation", "matrix" ], "problem_v1": "Given the matrix $A=\\left[\\begin{array}{cc} 3 &1\\cr 0 &1 \\end{array}\\right]$, find $A^3$.\n$A^3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "27", "13", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Given the matrix $A=\\left[\\begin{array}{cc}-5 &5\\cr 0 &-4 \\end{array}\\right]$, find $A^3$.\n$A^3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-125", "305", "0", "-64" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Given the matrix $A=\\left[\\begin{array}{cc}-2 &1\\cr 0 &-2 \\end{array}\\right]$, find $A^3$.\n$A^3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-8", "12", "0", "-8" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0150", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "algebra", "matrix operation", "matrix" ], "problem_v1": "If A=\\left[\\begin{array}{ccc} 3 &1 &1\\cr 2 &-2 &-2 \\end{array}\\right], B=\\left[\\begin{array}{ccc} 1 &1 &-1\\cr 0 &2 &-3 \\end{array}\\right] then $4 A+3 B=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "15", "7", "1", "8", "-2", "-17" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "If A=\\left[\\begin{array}{ccc}-5 &5 &-4\\cr-2 &5 &-2 \\end{array}\\right], B=\\left[\\begin{array}{ccc}-3 &-2 &1\\cr-5 &2 &-1 \\end{array}\\right] then $5 A+2 B=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-31", "21", "-18", "-20", "29", "-12" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "If A=\\left[\\begin{array}{ccc}-2 &1 &-2\\cr 1 &-3 &-2 \\end{array}\\right], B=\\left[\\begin{array}{ccc} 3 &5 &4\\cr-3 &-2 &-3 \\end{array}\\right] then $2 A+4 B=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "8", "22", "12", "-10", "-14", "-16" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0151", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "algebra", "matrix operation", "matrix" ], "problem_v1": "If A=\\left[\\begin{array}{cccc} 2 &4 &-4 &-3 \\cr 1 &1 &-2 &0 \\cr 3 &-4 &0 &-1 \\cr-2 &-1 &0 &-7 \\cr-6 &-2 &0 &5 \\cr-1 &-3 &-1 &7 \\cr 9 &-4 &-9 &-5 \\cr \\end{array}\\right] then $A_{4 1}$ is [ANS].", "answer_v1": [ "-2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If A=\\left[\\begin{array}{ccccc}-7 &-3 &8 &-3 &-6 \\cr-3 &1 &-8 &3 &-1 \\cr 6 &-6 &-6 &-4 &1 \\cr \\end{array}\\right] then $A_{1 2}$ is [ANS].", "answer_v2": [ "-3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If A=\\left[\\begin{array}{cccc}-4 &1 &-6 &-3 \\cr 6 &8 &7 &-6 \\cr-4 &-5 &-9 &1 \\cr 9 &6 &2 &-7 \\cr \\end{array}\\right] then $A_{2 3}$ is [ANS].", "answer_v3": [ "7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0152", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "3", "keywords": [ "algebra", "matrix operation", "matrix" ], "problem_v1": "Given the matrices B=\\left[\\begin{array}{ccc} 2 &1 &1\\cr 2 &-1 &-1 \\end{array}\\right], F=\\left[\\begin{array}{ccc} 0 &1 &-1\\cr 0 &1 &-2\\cr 0 &-1 &-1 \\end{array}\\right], can the operation $BF$ be performed? [ANS] If your answer is Yes, calculate $BF$. $BF=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0", "2", "-5", "0", "2", "1" ], "answer_type_v1": [ "TF", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [ "Yes", "No" ], [], [], [], [], [] ], "problem_v2": "Given the matrices B=\\left[\\begin{array}{ccc}-3 &3 &-2\\cr-1 &3 &-1 \\end{array}\\right], F=\\left[\\begin{array}{ccc}-2 &-1 &0\\cr-3 &1 &0\\cr 2 &-2 &-2 \\end{array}\\right], can the operation $BF$ be performed? [ANS] If your answer is Yes, calculate $BF$. $BF=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-7", "10", "4", "-9", "6", "2" ], "answer_type_v2": [ "TF", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [ "Yes", "No" ], [], [], [], [], [] ], "problem_v3": "Given the matrices B=\\left[\\begin{array}{ccc}-1 &1 &-2\\cr 0 &-2 &-1 \\end{array}\\right], F=\\left[\\begin{array}{ccc} 2 &3 &3\\cr-2 &-1 &-2\\cr-3 &1 &3 \\end{array}\\right], can the operation $BF$ be performed? [ANS] If your answer is Yes, calculate $BF$. $BF=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "2", "-6", "-11", "7", "1", "1" ], "answer_type_v3": [ "TF", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [ "Yes", "No" ], [], [], [], [], [] ] }, { "id": "Linear_algebra_0153", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "algebra", "matrix operation", "matrix" ], "problem_v1": "If A=\\left[\\begin{array}{ccc} 2 &4 &-4 \\cr-3 &1 &1 \\cr-2 &0 &3 \\cr-4 &0 &-1 \\cr \\end{array}\\right] B=\\left[\\begin{array}{cc} 0 &-7 \\cr-6 &-2 \\cr \\end{array}\\right] C=\\left[\\begin{array}{cccc}-1 &-3 &-1 &7 \\cr 9 &-4 &-9 &-5 \\cr 1 &-6 &-3 &5 \\cr \\end{array}\\right] then decide if each of the following operations is defined (answer yes or no) $A+B$ [ANS]\n$A+C$ [ANS]\n$B+C$ [ANS]\n$AB$ [ANS]\n$BA$ [ANS]\n$AC$ [ANS]\n$CA$ [ANS]\n$BC$ [ANS]\n$CB$ [ANS]", "answer_v1": [ "NO", "NO", "NO", "no", "no", "yes", "yes", "no", "no" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "If A=\\left[\\begin{array}{cccc}-7 &-3 &8 &-3 \\cr \\end{array}\\right] B=\\left[\\begin{array}{cc} 1 &-8 \\cr \\end{array}\\right] C=\\left[\\begin{array}{cc} 6 &-6 \\cr-6 &-4 \\cr 1 &-6 \\cr \\end{array}\\right] then decide if each of the following operations is defined (answer yes or no) $A+B$ [ANS]\n$A+C$ [ANS]\n$B+C$ [ANS]\n$AB$ [ANS]\n$BA$ [ANS]\n$AC$ [ANS]\n$CA$ [ANS]\n$BC$ [ANS]\n$CB$ [ANS]", "answer_v2": [ "NO", "NO", "NO", "no", "no", "no", "no", "no", "no" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "If A=\\left[\\begin{array}{ccc}-4 &1 &-6 \\cr-3 &6 &8 \\cr \\end{array}\\right] B=\\left[\\begin{array}{c}-4 \\cr-5 \\cr-9 \\cr 1 \\cr \\end{array}\\right] C=\\left[\\begin{array}{cccc} 2 &-7 &-4 &3 \\cr 4 &0 &-2 &8 \\cr 6 &7 &1 &2 \\cr 9 &5 &-6 &1 \\cr \\end{array}\\right] then decide if each of the following operations is defined (answer yes or no) $A+B$ [ANS]\n$A+C$ [ANS]\n$B+C$ [ANS]\n$AB$ [ANS]\n$BA$ [ANS]\n$AC$ [ANS]\n$CA$ [ANS]\n$BC$ [ANS]\n$CB$ [ANS]", "answer_v3": [ "NO", "NO", "NO", "no", "no", "no", "no", "no", "yes" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0156", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "linear algebra", "matrix" ], "problem_v1": "If $A=\\left[\\begin{array}{cc} 3 &1\\cr 2 &3 \\end{array}\\right]$ and $B=\\left[\\begin{array}{cc}-3 &-2\\cr 1 &1 \\end{array}\\right]$, then\n$AB=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} and\n$BA=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n[ANS] True or False: $AB=BA$ for any two square matrices $A$ and $B$ of the same size.", "answer_v1": [ "-8", "-5", "-3", "-1", "-13", "-9", "5", "4" ], "answer_type_v1": [ "NV", "NV", "TF", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [ "True", "False" ], [], [], [], [], [] ], "problem_v2": "If $A=\\left[\\begin{array}{cc}-5 &6\\cr-5 &-2 \\end{array}\\right]$ and $B=\\left[\\begin{array}{cc} 6 &-2\\cr-4 &-2 \\end{array}\\right]$, then\n$AB=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} and\n$BA=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n[ANS] True or False: $AB=BA$ for any two square matrices $A$ and $B$ of the same size.", "answer_v2": [ "-54", "-2", "-22", "14", "-20", "40", "30", "-20" ], "answer_type_v2": [ "NV", "NV", "TF", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [ "True", "False" ], [], [], [], [], [] ], "problem_v3": "If $A=\\left[\\begin{array}{cc}-2 &1\\cr-3 &1 \\end{array}\\right]$ and $B=\\left[\\begin{array}{cc}-4 &-2\\cr 4 &5 \\end{array}\\right]$, then\n$AB=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} and\n$BA=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n[ANS] True or False: $AB=BA$ for any two square matrices $A$ and $B$ of the same size.", "answer_v3": [ "12", "9", "16", "11", "14", "-6", "-23", "9" ], "answer_type_v3": [ "NV", "NV", "TF", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [ "True", "False" ], [], [], [], [], [] ] }, { "id": "Linear_algebra_0157", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "4", "keywords": [ "linear algebra", "matrix", "linear system" ], "problem_v1": "Find a non-zero $2 \\times 2$ matrix such that $\\left[\\begin{array}{cc} 5 &2\\cr-15 &-6 \\end{array}\\right]$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $=\\left[\\begin{array}{cc} 0 &0\\cr 0 &0 \\end{array}\\right]$.", "answer_v1": [ "0.2", "0.2", "-0.5", "-0.5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find a non-zero $2 \\times 2$ matrix such that $\\left[\\begin{array}{cc}-8 &8\\cr 40 &-40 \\end{array}\\right]$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $=\\left[\\begin{array}{cc} 0 &0\\cr 0 &0 \\end{array}\\right]$.", "answer_v2": [ "-0.125", "-0.125", "-0.125", "-0.125" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find a non-zero $2 \\times 2$ matrix such that $\\left[\\begin{array}{cc}-4 &2\\cr 16 &-8 \\end{array}\\right]$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $=\\left[\\begin{array}{cc} 0 &0\\cr 0 &0 \\end{array}\\right]$.", "answer_v3": [ "-0.25", "-0.25", "-0.5", "-0.5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0158", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "4", "keywords": [ "linear algebra", "matrix", "linear system" ], "problem_v1": "If A=\\left[\\begin{array}{cc} x &5\\cr y &2\\cr \\end{array}\\right]. determine the values of $x$ and $y$ for which $A^2=A$. $x=$ [ANS], $y=$ [ANS].", "answer_v1": [ "1-2", "2*(1-2)/5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If A=\\left[\\begin{array}{cc} x &-8\\cr y &8\\cr \\end{array}\\right]. determine the values of $x$ and $y$ for which $A^2=A$. $x=$ [ANS], $y=$ [ANS].", "answer_v2": [ "1-8", "8*(1-8)/-8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If A=\\left[\\begin{array}{cc} x &-4\\cr y &2\\cr \\end{array}\\right]. determine the values of $x$ and $y$ for which $A^2=A$. $x=$ [ANS], $y=$ [ANS].", "answer_v3": [ "1-2", "2*(1-2)/-4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0160", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "linear algebra", "matrix" ], "problem_v1": "Write a $2 \\times 3$ matrix $A$ with the following entries: $a_{22}=-5$, $a_{13}=2$, $a_{12}=1$, $a_{21}=6$, $a_{23}=-4$, $a_{11}=5$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "5", "1", "2", "6", "-5", "-4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Write a $2 \\times 3$ matrix $A$ with the following entries: $a_{11}=-9$, $a_{12}=9$, $a_{21}=-2$, $a_{22}=10$, $a_{13}=-7$, $a_{23}=-3$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-9", "9", "-7", "-2", "10", "-3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Write a $2 \\times 3$ matrix $A$ with the following entries: $a_{21}=1$, $a_{13}=-5$, $a_{23}=-2$, $a_{22}=-7$, $a_{12}=3$, $a_{11}=-4$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-4", "3", "-5", "1", "-7", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0161", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "3", "keywords": [ "linear algebra", "matrix", "transpose" ], "problem_v1": "Given the augmented matrix A=\\left\\lbrack \\begin{array}{rrr|r} 1 & 2 &-3 &-2 \\\\ 4 & 9 & 1 & 1 \\\\-3 &-9 &-2 & 2 \\end{array} \\right\\rbrack, perform each row operation in the order specified and enter the final result.\nFirst: $-4 R_1+R_2 \\rightarrow R_2$, Second: $3 R_1+R_3 \\rightarrow R_3$, Third: $3 R_2+R_3 \\rightarrow R_3$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "2", "-3", "-2", "0", "1", "13", "9", "0", "0", "28", "23" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Given the augmented matrix A=\\left\\lbrack \\begin{array}{rrr|r} 1 &-1 & 6 &-2 \\\\ 2 &-1 &-4 &-2 \\\\-4 & 2 & 1 &-6 \\end{array} \\right\\rbrack, perform each row operation in the order specified and enter the final result.\nFirst: $-2 R_1+R_2 \\rightarrow R_2$, Second: $4 R_1+R_3 \\rightarrow R_3$, Third: $2 R_2+R_3 \\rightarrow R_3$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "-1", "6", "-2", "0", "1", "-16", "2", "0", "0", "-7", "-10" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Given the augmented matrix A=\\left\\lbrack \\begin{array}{rrr|r} 1 &-2 &-2 & 4 \\\\ 2 &-3 & 5 & 5 \\\\-3 & 4 &-4 &-3 \\end{array} \\right\\rbrack, perform each row operation in the order specified and enter the final result.\nFirst: $-2 R_1+R_2 \\rightarrow R_2$, Second: $3 R_1+R_3 \\rightarrow R_3$, Third: $2 R_2+R_3 \\rightarrow R_3$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "-2", "-2", "4", "0", "1", "9", "-3", "0", "0", "8", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0165", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Let A=\\left(\\begin{array}{ccc} 0 & 8 & 6 \\cr 0 & 0 & 6 \\cr 0 & 0 & 0 \\end{array}\\right). Then A^2=\\left(\\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\cr a_{21} & a_{22} & a_{23} \\cr a_{31} & a_{32} & a_{33} \\end{array}\\right) where $a_{11}=$ [ANS], $a_{12}=$ [ANS], $a_{13}=$ [ANS], $a_{21}=$ [ANS], $a_{22}=$ [ANS], $a_{23}=$ [ANS], $a_{31}=$ [ANS], $a_{32}=$ [ANS], $a_{33}=$ [ANS],\nand A^3=\\left(\\begin{array}{ccc} c_{11} & c_{12} & c_{13} \\cr c_{21} & c_{22} & c_{23} \\cr c_{31} & c_{32} & c_{33} \\end{array}\\right) where $c_{11}=$ [ANS], $c_{12}=$ [ANS], $c_{13}=$ [ANS], $c_{21}=$ [ANS], $c_{22}=$ [ANS], $c_{23}=$ [ANS], $c_{31}=$ [ANS], $c_{32}=$ [ANS], $c_{33}=$ [ANS].", "answer_v1": [ "0", "0", "48", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let A=\\left(\\begin{array}{ccc} 0 & 2 & 9 \\cr 0 & 0 & 3 \\cr 0 & 0 & 0 \\end{array}\\right). Then A^2=\\left(\\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\cr a_{21} & a_{22} & a_{23} \\cr a_{31} & a_{32} & a_{33} \\end{array}\\right) where $a_{11}=$ [ANS], $a_{12}=$ [ANS], $a_{13}=$ [ANS], $a_{21}=$ [ANS], $a_{22}=$ [ANS], $a_{23}=$ [ANS], $a_{31}=$ [ANS], $a_{32}=$ [ANS], $a_{33}=$ [ANS],\nand A^3=\\left(\\begin{array}{ccc} c_{11} & c_{12} & c_{13} \\cr c_{21} & c_{22} & c_{23} \\cr c_{31} & c_{32} & c_{33} \\end{array}\\right) where $c_{11}=$ [ANS], $c_{12}=$ [ANS], $c_{13}=$ [ANS], $c_{21}=$ [ANS], $c_{22}=$ [ANS], $c_{23}=$ [ANS], $c_{31}=$ [ANS], $c_{32}=$ [ANS], $c_{33}=$ [ANS].", "answer_v2": [ "0", "0", "6", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let A=\\left(\\begin{array}{ccc} 0 & 4 & 6 \\cr 0 & 0 & 4 \\cr 0 & 0 & 0 \\end{array}\\right). Then A^2=\\left(\\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\cr a_{21} & a_{22} & a_{23} \\cr a_{31} & a_{32} & a_{33} \\end{array}\\right) where $a_{11}=$ [ANS], $a_{12}=$ [ANS], $a_{13}=$ [ANS], $a_{21}=$ [ANS], $a_{22}=$ [ANS], $a_{23}=$ [ANS], $a_{31}=$ [ANS], $a_{32}=$ [ANS], $a_{33}=$ [ANS],\nand A^3=\\left(\\begin{array}{ccc} c_{11} & c_{12} & c_{13} \\cr c_{21} & c_{22} & c_{23} \\cr c_{31} & c_{32} & c_{33} \\end{array}\\right) where $c_{11}=$ [ANS], $c_{12}=$ [ANS], $c_{13}=$ [ANS], $c_{21}=$ [ANS], $c_{22}=$ [ANS], $c_{23}=$ [ANS], $c_{31}=$ [ANS], $c_{32}=$ [ANS], $c_{33}=$ [ANS].", "answer_v3": [ "0", "0", "16", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0166", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "Algebra' 'Matrix' 'Matrices' 'True' 'False" ], "problem_v1": "If A is $5 \\times 6,$ B is $7 \\times 5$ and C is $6 \\times 7$ then: a) BA has dimension [ANS] $\\times$ [ANS]\nb) CB has dimension [ANS] $\\times$ [ANS]\nc) AC has dimension [ANS] $\\times$ [ANS]", "answer_v1": [ "7", "6", "6", "5", "5", "7" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "If A is $2 \\times 9,$ B is $3 \\times 2$ and C is $9 \\times 3$ then: a) BA has dimension [ANS] $\\times$ [ANS]\nb) CB has dimension [ANS] $\\times$ [ANS]\nc) AC has dimension [ANS] $\\times$ [ANS]", "answer_v2": [ "3", "9", "9", "2", "2", "3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "If A is $3 \\times 6,$ B is $4 \\times 3$ and C is $6 \\times 4$ then: a) BA has dimension [ANS] $\\times$ [ANS]\nb) CB has dimension [ANS] $\\times$ [ANS]\nc) AC has dimension [ANS] $\\times$ [ANS]", "answer_v3": [ "4", "6", "6", "3", "3", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0167", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "Algebra' 'Matrix' 'Matrices" ], "problem_v1": "Determine the value(s) of $x$ such that $\\left[\\begin{array}{ccc} x &2 &1\\cr \\end{array}\\right] \\left[\\begin{array}{ccc} 1 &-2 &-1\\cr-2 &2 &-1\\cr-1 &-16 &-1 \\end{array}\\right] \\left[\\begin{array}{c} x\\cr-1\\cr 2\\cr \\end{array}\\right]=[0]$ $x$=[ANS]\nNote: If there is more than one value separate them by commas.", "answer_v1": [ "(2, 3)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Determine the value(s) of $x$ such that $\\left[\\begin{array}{ccc} x &2 &1\\cr \\end{array}\\right] \\left[\\begin{array}{ccc} 6 &5 &-6\\cr 5 &-1 &3\\cr-6 &-53 &-1 \\end{array}\\right] \\left[\\begin{array}{c} x\\cr-1\\cr-6\\cr \\end{array}\\right]=[0]$ $x$=[ANS]\nNote: If there is more than one value separate them by commas.", "answer_v2": [ "(-5/6, -5)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Determine the value(s) of $x$ such that $\\left[\\begin{array}{ccc} x &2 &1\\cr \\end{array}\\right] \\left[\\begin{array}{ccc} 1 &3 &-1\\cr 3 &0 &-2\\cr-1 &9 &-1 \\end{array}\\right] \\left[\\begin{array}{c} x\\cr-1\\cr-3\\cr \\end{array}\\right]=[0]$ $x$=[ANS]\nNote: If there is more than one value separate them by commas.", "answer_v3": [ "(-3, -2)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0168", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "Algebra' 'Matrix' 'Matrices" ], "problem_v1": "Determine $x$ and $y$ such that \\left[\\begin{array}{ccc} 2& 1&-2 \\cr 1&-1& 1 \\end{array}\\right]+\\left[\\begin{array}{ccc} x-y & 2 &-2 \\\\ 1 & x &-2 \\end{array} \\right]=\\left[\\begin{array}{ccc} 3 & 3 &-4 \\\\ 2 & 2x+y &-1 \\end{array}\\right] $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "0", "-1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Determine $x$ and $y$ such that \\left[\\begin{array}{ccc}-4&-3& 4 \\cr-3& 1& 1 \\end{array}\\right]+\\left[\\begin{array}{ccc} x-y &-1 &-2 \\\\-2 & x &-1 \\end{array} \\right]=\\left[\\begin{array}{ccc} 0 &-4 & 2 \\\\-5 & 2x+y & 0 \\end{array}\\right] $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "2.5", "-1.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Determine $x$ and $y$ such that \\left[\\begin{array}{ccc}-2&-2&-3 \\cr 3& 3&-2 \\end{array}\\right]+\\left[\\begin{array}{ccc} x-y & 0 &-1 \\\\ 4 & x &-2 \\end{array} \\right]=\\left[\\begin{array}{ccc}-1 &-2 &-4 \\\\ 7 & 2x+y &-4 \\end{array}\\right] $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "2", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0169", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "3", "keywords": [ "Algebra' 'Matrix' 'Matrices' 'True' 'False" ], "problem_v1": "Sally went to the store and purchased 4 skirts, 4 dresses and 6 shirts. Anna went to the store and purchased 4 skirts, 3 dresses and 4 shirts. Each skirt costs \\$21, each dress cost \\$22, and each shirt costs \\$10. a) Write a $2 \\times 3$ matrix summarizing the purchases made by Sally and Anna. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ b) Write a $3 \\times 1$ matrix summarizing the cost (in dollars) of the items. (Keep the order of information).\n$\\begin{array}{ccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] \\\\ \\hline [ANS] \\\\ \\hline \\end{array}$ c) Use matrix multiplication to find a $2 \\times 1$ matrix summarizing the total amount (in dollars) spent by Sally and Anna.\n$\\begin{array}{cccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] & [ANS] \\\\ \\hline [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "4", "4", "6", "4", "3", "4", "21", "22", "10", "232", "190" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Sally went to the store and purchased 2 skirts, 5 dresses and 3 shirts. Anna went to the store and purchased 3 skirts, 5 dresses and 4 shirts. Each skirt costs \\$13, each dress cost \\$16, and each shirt costs \\$14. a) Write a $2 \\times 3$ matrix summarizing the purchases made by Sally and Anna. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ b) Write a $3 \\times 1$ matrix summarizing the cost (in dollars) of the items. (Keep the order of information).\n$\\begin{array}{ccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] \\\\ \\hline [ANS] \\\\ \\hline \\end{array}$ c) Use matrix multiplication to find a $2 \\times 1$ matrix summarizing the total amount (in dollars) spent by Sally and Anna.\n$\\begin{array}{cccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] & [ANS] \\\\ \\hline [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "2", "5", "3", "3", "5", "4", "13", "16", "14", "148", "175" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Sally went to the store and purchased 2 skirts, 4 dresses and 4 shirts. Anna went to the store and purchased 4 skirts, 2 dresses and 4 shirts. Each skirt costs \\$26, each dress cost \\$29, and each shirt costs \\$18. a) Write a $2 \\times 3$ matrix summarizing the purchases made by Sally and Anna. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ b) Write a $3 \\times 1$ matrix summarizing the cost (in dollars) of the items. (Keep the order of information).\n$\\begin{array}{ccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] \\\\ \\hline [ANS] \\\\ \\hline \\end{array}$ c) Use matrix multiplication to find a $2 \\times 1$ matrix summarizing the total amount (in dollars) spent by Sally and Anna.\n$\\begin{array}{cccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] & [ANS] \\\\ \\hline [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "2", "4", "4", "4", "2", "4", "26", "29", "18", "240", "234" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0170", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "3", "keywords": [ "Algebra' 'Matrix' 'Matrices' 'True' 'False" ], "problem_v1": "During the month of January, \"ABC Appliances\" sold 53 microwaves, 22 refrigerators and 39 stoves, while \"XYZ Appliances\" sold 47 microwaves, 22 refrigerators and 31 stoves. During the month of February, \"ABC Appliances\" sold 47 microwaves, 20 refrigerators and 25 stoves, while \"XYZ Appliances\" sold 40 microwaves, 30 refrigerators and 23 stoves. a) Write a $2 \\times 3$ matrix summarizing the sales for the month of January. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ b) Write a $2 \\times 3$ matrix summarizing the sales for the month of February. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ c) Use matrix addition to find a $2 \\times 3$ matrix summarizing the total sales for the months of January and February.\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "53", "22", "39", "47", "22", "31", "47", "20", "25", "40", "30", "23", "100", "42", "64", "87", "52", "54" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "During the month of January, \"ABC Appliances\" sold 32 microwaves, 29 refrigerators and 24 stoves, while \"XYZ Appliances\" sold 35 microwaves, 16 refrigerators and 37 stoves. During the month of February, \"ABC Appliances\" sold 35 microwaves, 39 refrigerators and 24 stoves, while \"XYZ Appliances\" sold 27 microwaves, 30 refrigerators and 28 stoves. a) Write a $2 \\times 3$ matrix summarizing the sales for the month of January. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ b) Write a $2 \\times 3$ matrix summarizing the sales for the month of February. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ c) Use matrix addition to find a $2 \\times 3$ matrix summarizing the total sales for the months of January and February.\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "32", "29", "24", "35", "16", "37", "35", "39", "24", "27", "30", "28", "67", "68", "48", "62", "46", "65" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "During the month of January, \"ABC Appliances\" sold 39 microwaves, 22 refrigerators and 28 stoves, while \"XYZ Appliances\" sold 55 microwaves, 29 refrigerators and 47 stoves. During the month of February, \"ABC Appliances\" sold 42 microwaves, 17 refrigerators and 25 stoves, while \"XYZ Appliances\" sold 31 microwaves, 20 refrigerators and 22 stoves. a) Write a $2 \\times 3$ matrix summarizing the sales for the month of January. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ b) Write a $2 \\times 3$ matrix summarizing the sales for the month of February. (Keep the order of information).\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$ c) Use matrix addition to find a $2 \\times 3$ matrix summarizing the total sales for the months of January and February.\n$\\begin{array}{cccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "39", "22", "28", "55", "29", "47", "42", "17", "25", "31", "20", "22", "81", "39", "53", "86", "49", "69" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0171", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "5", "keywords": [ "linear algebra", "matrix equation" ], "problem_v1": "An investment advisor currently has two types of investments available for clients: a conservative investment A that pays $8$ \\% per year and investment B of higher risk that pays $14$ \\%. Clients may divide their investments between the two to achieve any total return desired between $8$ \\% and $14$ \\%. However, the higher the desired return, the higher the risk. How should each client listed in the table invest to achieve the desired return?\n$\\begin{array}{ccccc}\\hline & Client 1 & Client 2 & Client 3 & k \\\\ \\hline Total Investment & \\$ 27000 & \\$ 45000 & \\$ 36000 & k_1 \\\\ \\hline Annual Return Desired & \\$ 3060 & \\$ 4920 & \\$ 4260 & k_2 \\\\ \\hline \\end{array}$\nHow much money should Client 1 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]\nHow much money should Client 2 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]\nHow much money should Client 3 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]", "answer_v1": [ "12000", "15000", "23000", "22000", "13000", "23000" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "An investment advisor currently has two types of investments available for clients: a conservative investment A that pays $6$ \\% per year and investment B of higher risk that pays $13$ \\%. Clients may divide their investments between the two to achieve any total return desired between $6$ \\% and $13$ \\%. However, the higher the desired return, the higher the risk. How should each client listed in the table invest to achieve the desired return?\n$\\begin{array}{ccccc}\\hline & Client 1 & Client 2 & Client 3 & k \\\\ \\hline Total Investment & \\$ 20000 & \\$ 49000 & \\$ 31000 & k_1 \\\\ \\hline Annual Return Desired & \\$ 2040 & \\$ 3920 & \\$ 3120 & k_2 \\\\ \\hline \\end{array}$\nHow much money should Client 1 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]\nHow much money should Client 2 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]\nHow much money should Client 3 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]", "answer_v2": [ "8000", "12000", "35000", "14000", "13000", "18000" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "An investment advisor currently has two types of investments available for clients: a conservative investment A that pays $9$ \\% per year and investment B of higher risk that pays $15$ \\%. Clients may divide their investments between the two to achieve any total return desired between $9$ \\% and $15$ \\%. However, the higher the desired return, the higher the risk. How should each client listed in the table invest to achieve the desired return?\n$\\begin{array}{ccccc}\\hline & Client 1 & Client 2 & Client 3 & k \\\\ \\hline Total Investment & \\$ 23000 & \\$ 46000 & \\$ 32000 & k_1 \\\\ \\hline Annual Return Desired & \\$ 2790 & \\$ 5640 & \\$ 4020 & k_2 \\\\ \\hline \\end{array}$\nHow much money should Client 1 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]\nHow much money should Client 2 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]\nHow much money should Client 3 invest in each account to achieved the desired return? Amount in investment A: [ANS]\nAmount in investment B: [ANS]", "answer_v3": [ "11000", "12000", "21000", "25000", "13000", "19000" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0172", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "linear algebra", "matrix" ], "problem_v1": "Compute the following: $\\left[\\begin{array}{ccc} 5 &2 &2\\cr 4 &-4 &-3\\cr 1 &1 &-2 \\end{array}\\right]+\\left[\\begin{array}{ccc} 3 &-4 &-1\\cr-2 &-1 &-7\\cr-6 &-2 &5 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "8", "-2", "1", "2", "-5", "-10", "-5", "-1", "3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the following: $\\left[\\begin{array}{ccc}-8 &8 &-7\\cr-3 &8 &-3\\cr-6 &-3 &1 \\end{array}\\right]+\\left[\\begin{array}{ccc}-8 &3 &-1\\cr 6 &-6 &-6\\cr-4 &1 &-6 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-16", "11", "-8", "3", "2", "-9", "-10", "-2", "-5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the following: $\\left[\\begin{array}{ccc}-4 &2 &-4\\cr 1 &-6 &-3\\cr 6 &8 &7 \\end{array}\\right]+\\left[\\begin{array}{ccc}-6 &-4 &-5\\cr-9 &1 &9\\cr 6 &2 &-7 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-10", "-2", "-9", "-8", "-5", "6", "12", "10", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0173", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "linear algebra", "matrix operations" ], "problem_v1": "Let $A$ and $B$ be the following matrices.\nA=\\left[\\begin{array}{cccc} 5 &2 &2 &4 \\end{array}\\right], \\qquad B=\\left[\\begin{array}{c}-4\\cr-3\\cr 1\\cr 1 \\end{array}\\right] Perform the following operations: $A \\cdot B=$ $\\left[\\right.$ [ANS] $\\left.\\right]$ $B \\cdot A=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "-20", "-8", "-8", "-16", "-15", "-6", "-6", "-12", "5", "2", "2", "4", "5", "2", "2", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $A$ and $B$ be the following matrices.\nA=\\left[\\begin{array}{cccc}-8 &8 &-7 &-3 \\end{array}\\right], \\qquad B=\\left[\\begin{array}{c} 8\\cr-3\\cr-6\\cr-3 \\end{array}\\right] Perform the following operations: $A \\cdot B=$ $\\left[\\right.$ [ANS] $\\left.\\right]$ $B \\cdot A=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-64", "64", "-56", "-24", "24", "-24", "21", "9", "48", "-48", "42", "18", "24", "-24", "21", "9" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $A$ and $B$ be the following matrices.\nA=\\left[\\begin{array}{cccc}-4 &2 &-4 &1 \\end{array}\\right], \\qquad B=\\left[\\begin{array}{c}-6\\cr-3\\cr 6\\cr 8 \\end{array}\\right] Perform the following operations: $A \\cdot B=$ $\\left[\\right.$ [ANS] $\\left.\\right]$ $B \\cdot A=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "24", "-12", "24", "-6", "12", "-6", "12", "-3", "-24", "12", "-24", "6", "-32", "16", "-32", "8" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0174", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "2", "keywords": [ "algebra", "matrix operation", "matrix" ], "problem_v1": "Given the matrices B=\\left[\\begin{array}{rrr} 3 & 1 & 1\\\\ 2 &-2 &-2\\\\ \\end{array}\\right],\\qquad C=\\left[\\begin{array}{rrr} 1 & 1 &-1\\\\ 0 & 2 &-3\\\\ \\end{array}\\right], find $3B+2C$. Write $3B+2C$ as \\left[\\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23}\\\\ \\end{array}\\right], input your answer below: $a_{11}=$ [ANS]\n$a_{12}=$ [ANS]\n$a_{13}=$ [ANS]\n$a_{21}=$ [ANS]\n$a_{22}=$ [ANS]\n$a_{23}=$ [ANS]", "answer_v1": [ "11", "5", "1", "6", "-2", "-12" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Given the matrices B=\\left[\\begin{array}{rrr}-5 & 5 &-4\\\\-2 & 5 &-2\\\\ \\end{array}\\right],\\qquad C=\\left[\\begin{array}{rrr}-3 &-2 & 1\\\\-5 & 2 &-1\\\\ \\end{array}\\right], find $3B+2C$. Write $3B+2C$ as \\left[\\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23}\\\\ \\end{array}\\right], input your answer below: $a_{11}=$ [ANS]\n$a_{12}=$ [ANS]\n$a_{13}=$ [ANS]\n$a_{21}=$ [ANS]\n$a_{22}=$ [ANS]\n$a_{23}=$ [ANS]", "answer_v2": [ "-21", "11", "-10", "-16", "19", "-8" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Given the matrices B=\\left[\\begin{array}{rrr}-2 & 1 &-2\\\\ 1 &-3 &-2\\\\ \\end{array}\\right],\\qquad C=\\left[\\begin{array}{rrr} 3 & 5 & 4\\\\-3 &-2 &-3\\\\ \\end{array}\\right], find $3B+2C$. Write $3B+2C$ as \\left[\\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23}\\\\ \\end{array}\\right], input your answer below: $a_{11}=$ [ANS]\n$a_{12}=$ [ANS]\n$a_{13}=$ [ANS]\n$a_{21}=$ [ANS]\n$a_{22}=$ [ANS]\n$a_{23}=$ [ANS]", "answer_v3": [ "0", "13", "2", "-3", "-13", "-12" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0175", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "3", "keywords": [ "matrix' 'sum' 'vector" ], "problem_v1": "If A\\vec{u}=\\left[\\begin{array}{c} 3\\cr 1\\cr-2 \\end{array}\\right] \\ \\mbox{and} \\ A\\vec{v}=\\left[\\begin{array}{c} 1\\cr 2\\cr-2 \\end{array}\\right] then\n$A(-2 \\vec{u}-3 \\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-9", "-8", "10" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If A\\vec{u}=\\left[\\begin{array}{c}-5\\cr-4\\cr 5 \\end{array}\\right] \\ \\mbox{and} \\ A\\vec{v}=\\left[\\begin{array}{c} 5\\cr-2\\cr-2 \\end{array}\\right] then\n$A(-3 \\vec{u}-3 \\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "0", "18", "-9" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If A\\vec{u}=\\left[\\begin{array}{c}-2\\cr-2\\cr-3 \\end{array}\\right] \\ \\mbox{and} \\ A\\vec{v}=\\left[\\begin{array}{c} 1\\cr 1\\cr-2 \\end{array}\\right] then\n$A(-3 \\vec{u}-2 \\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "4", "4", "13" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0176", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "3", "keywords": [ "matrix' 'product" ], "problem_v1": "Solve for $X$.\n\\left[\\begin{array}{cc} 5 &2\\cr 2 &4 \\end{array}\\right]-3 X=\\left[\\begin{array}{cc}-4 &-3\\cr 1 &1 \\end{array}\\right]. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "3", "5/3", "1/3", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Solve for $X$.\n\\left[\\begin{array}{cc}-8 &8\\cr-7 &-3 \\end{array}\\right]-4 X=\\left[\\begin{array}{cc} 8 &-3\\cr-6 &-3 \\end{array}\\right]. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-4", "11/4", "-1/4", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Solve for $X$.\n\\left[\\begin{array}{cc}-4 &2\\cr-4 &1 \\end{array}\\right]-5 X=\\left[\\begin{array}{cc}-6 &-3\\cr 6 &8 \\end{array}\\right]. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "2/5", "1", "-2", "-7/5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0177", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Matrix algebra", "level": "3", "keywords": [ "transformations", "matrices", "matrix multiplication" ], "problem_v1": "Suppose a computer program needs to apply an affine transformation to a complex three-dimensional object made up of 7000 points. The transformation is composed of 10 matrices (call them $M_1$ through $M_{10}$), so for each point $(x, y, z)$ in the object, the following operation is performed. $\\left[\\begin{array}{c} ~ \\\\ ~ M_1 ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_2 ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_3 ~ \\\\ ~ \\end{array}\\right] ~ \\cdots \\left[\\begin{array}{c} ~ \\\\ ~ M_{9} ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_{10} ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} x \\\\ y \\\\ z \\\\ 1 \\end{array}\\right]$ Each multiplication of a matrix times a column vector involves 16 multiplications (of one number by another) and 12 additions, for a total of 28 arithmetic operations. Each multiplication of a matrix times another matrix involves 64 multiplications and 48 additions, for a total of 112 arithmetic operations. (These numbers are not made up or chosen randomly; they are facts about $4\\times 4$ matrix multiplication.)\nThe most inefficient way of applying the transformation to the 7000 points would be to begin on the left, multiplying $M_1$ by $M_2$, then that result by $M_3$, and so on along the list from left to right, and doing the same 10 multiplications again for each of the 7000 points. How many arithmetic operations would this require for transforming one point? Answer: [ANS]\nHow many would it require in total for transforming all 7000 points? Answer: [ANS]\nA more efficient method would be to start on the right, first multiplying $M_{10}$ by the column vector, then multiplying $M_{9}$ by that result, and so on, proceeding along the list from right to left. How many arithmetic operations would this require for transforming one point? Answer: [ANS]\nHow many would it require in total for transforming all 7000 points? Answer: [ANS]\nUsing this method would therefore save what percentage of the time of the previous method? Answer: [ANS] \\%\nThe most efficient method would be to multiply the 10 matrices together and call that result $A$ (without yet multiplying $A$ by any column vector). After computing $A$ (just once of course), multiply $A$ by each of the 7000 points, each represented as a column vector. How many arithmetic operations would it take to compute $A$? Answer: [ANS]\nHow many arithmetic operations would it require to multiply $A$ by one point? Answer: [ANS]\nHow many would this method require in total? Answer: [ANS]\nUsing this method would therefore save what percentage of the time of the previous method? Answer: [ANS] \\%", "answer_v1": [ "1036", "7.252E+6", "280", "1.96E+6", "72.973", "1008", "28", "197008", "89.9486" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose a computer program needs to apply an affine transformation to a complex three-dimensional object made up of 10000 points. The transformation is composed of 7 matrices (call them $M_1$ through $M_{7}$), so for each point $(x, y, z)$ in the object, the following operation is performed. $\\left[\\begin{array}{c} ~ \\\\ ~ M_1 ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_2 ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_3 ~ \\\\ ~ \\end{array}\\right] ~ \\cdots \\left[\\begin{array}{c} ~ \\\\ ~ M_{6} ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_{7} ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} x \\\\ y \\\\ z \\\\ 1 \\end{array}\\right]$ Each multiplication of a matrix times a column vector involves 16 multiplications (of one number by another) and 12 additions, for a total of 28 arithmetic operations. Each multiplication of a matrix times another matrix involves 64 multiplications and 48 additions, for a total of 112 arithmetic operations. (These numbers are not made up or chosen randomly; they are facts about $4\\times 4$ matrix multiplication.)\nThe most inefficient way of applying the transformation to the 10000 points would be to begin on the left, multiplying $M_1$ by $M_2$, then that result by $M_3$, and so on along the list from left to right, and doing the same 7 multiplications again for each of the 10000 points. How many arithmetic operations would this require for transforming one point? Answer: [ANS]\nHow many would it require in total for transforming all 10000 points? Answer: [ANS]\nA more efficient method would be to start on the right, first multiplying $M_{7}$ by the column vector, then multiplying $M_{6}$ by that result, and so on, proceeding along the list from right to left. How many arithmetic operations would this require for transforming one point? Answer: [ANS]\nHow many would it require in total for transforming all 10000 points? Answer: [ANS]\nUsing this method would therefore save what percentage of the time of the previous method? Answer: [ANS] \\%\nThe most efficient method would be to multiply the 7 matrices together and call that result $A$ (without yet multiplying $A$ by any column vector). After computing $A$ (just once of course), multiply $A$ by each of the 10000 points, each represented as a column vector. How many arithmetic operations would it take to compute $A$? Answer: [ANS]\nHow many arithmetic operations would it require to multiply $A$ by one point? Answer: [ANS]\nHow many would this method require in total? Answer: [ANS]\nUsing this method would therefore save what percentage of the time of the previous method? Answer: [ANS] \\%", "answer_v2": [ "700", "7E+6", "196", "1.96E+6", "72", "672", "28", "280672", "85.68" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose a computer program needs to apply an affine transformation to a complex three-dimensional object made up of 7000 points. The transformation is composed of 8 matrices (call them $M_1$ through $M_{8}$), so for each point $(x, y, z)$ in the object, the following operation is performed. $\\left[\\begin{array}{c} ~ \\\\ ~ M_1 ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_2 ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_3 ~ \\\\ ~ \\end{array}\\right] ~ \\cdots \\left[\\begin{array}{c} ~ \\\\ ~ M_{7} ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} ~ \\\\ ~ M_{8} ~ \\\\ ~ \\end{array}\\right] ~ \\left[\\begin{array}{c} x \\\\ y \\\\ z \\\\ 1 \\end{array}\\right]$ Each multiplication of a matrix times a column vector involves 16 multiplications (of one number by another) and 12 additions, for a total of 28 arithmetic operations. Each multiplication of a matrix times another matrix involves 64 multiplications and 48 additions, for a total of 112 arithmetic operations. (These numbers are not made up or chosen randomly; they are facts about $4\\times 4$ matrix multiplication.)\nThe most inefficient way of applying the transformation to the 7000 points would be to begin on the left, multiplying $M_1$ by $M_2$, then that result by $M_3$, and so on along the list from left to right, and doing the same 8 multiplications again for each of the 7000 points. How many arithmetic operations would this require for transforming one point? Answer: [ANS]\nHow many would it require in total for transforming all 7000 points? Answer: [ANS]\nA more efficient method would be to start on the right, first multiplying $M_{8}$ by the column vector, then multiplying $M_{7}$ by that result, and so on, proceeding along the list from right to left. How many arithmetic operations would this require for transforming one point? Answer: [ANS]\nHow many would it require in total for transforming all 7000 points? Answer: [ANS]\nUsing this method would therefore save what percentage of the time of the previous method? Answer: [ANS] \\%\nThe most efficient method would be to multiply the 8 matrices together and call that result $A$ (without yet multiplying $A$ by any column vector). After computing $A$ (just once of course), multiply $A$ by each of the 7000 points, each represented as a column vector. How many arithmetic operations would it take to compute $A$? Answer: [ANS]\nHow many arithmetic operations would it require to multiply $A$ by one point? Answer: [ANS]\nHow many would this method require in total? Answer: [ANS]\nUsing this method would therefore save what percentage of the time of the previous method? Answer: [ANS] \\%", "answer_v3": [ "812", "5.684E+6", "224", "1.568E+6", "72.4138", "784", "28", "196784", "87.45" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0181", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Row operations", "level": "2", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Give the result of applying the row operation $R_{1} \\leftrightarrow R_{3}$ to the given matrix.\n$\\left[\\begin{array}{ccc} 3 &1 &1\\cr 2 &-2 &-2\\cr 1 &1 &-1 \\end{array}\\right] \\mathop{\\longrightarrow}^{R_{1} \\leftrightarrow R_{3}}$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "1", "-1", "2", "-2", "-2", "3", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Give the result of applying the row operation $R_{2} \\leftrightarrow R_{3}$ to the given matrix.\n$\\left[\\begin{array}{ccc}-5 &5 &-4\\cr-2 &5 &-2\\cr-3 &-2 &1 \\end{array}\\right] \\mathop{\\longrightarrow}^{R_{2} \\leftrightarrow R_{3}}$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-5", "5", "-4", "-3", "-2", "1", "-2", "5", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Give the result of applying the row operation $R_{2} \\leftrightarrow R_{3}$ to the given matrix.\n$\\left[\\begin{array}{ccc}-2 &1 &-2\\cr 1 &-3 &-2\\cr 3 &5 &4 \\end{array}\\right] \\mathop{\\longrightarrow}^{R_{2} \\leftrightarrow R_{3}}$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-2", "1", "-2", "3", "5", "4", "1", "-3", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0182", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Row operations", "level": "2", "keywords": [ "linear algebra", "augmented", "row operation" ], "problem_v1": "Consider the matrix\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[-4-41122-3-300 \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]33-5-500-2-2-2-2-1-100-8-8-7-7-2-20066-1-1-4-4-1-1 &-4 & 1 & 2 &-3 & 0 & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]33-5-500-2-2-2-2-1-100-8-8-7-7-2-20066-1-1-4-4-1-1 & 3 &-5 & 0 &-2 &-2 &-1 & 0 &-8 &-7 &-2 & 0 & 6 &-1 &-4 &-1 \\\\ \\hline 3 &-5 & 0 &-2 &-2 \\\\ \\hline-1 & 0 &-8 &-7 &-2 \\\\ \\hline 0 & 6 &-1 &-4 &-1 \\\\ \\hline \\end{array}$\n(a) On the matrix above, perform the row operation $6 R_{3}+R_{1} \\to R_{1}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) On the original matrix, perform the row operation $3 R_{3} \\to R_{3}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(c) On the original matrix, perform the row operation $R_{1} \\leftrightarrow R_{3}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "-10", "1", "-46", "-45", "-12", "3", "-5", "0", "-2", "-2", "-1", "0", "-8", "-7", "-2", "0", "6", "-1", "-4", "-1", "-4", "1", "2", "-3", "0", "3", "-5", "0", "-2", "-2", "-3", "0", "-24", "-21", "-6", "0", "6", "-1", "-4", "-1", "-1", "0", "-8", "-7", "-2", "3", "-5", "0", "-2", "-2", "-4", "1", "2", "-3", "0", "0", "6", "-1", "-4", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the matrix\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[-4-4-7-7-4-411-9-9 \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]33-1-177-6-6-7-7-5-522-7-7-4-411-8-8-8-8-8-8-4-400 &-4 &-7 &-4 & 1 &-9 & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]33-1-177-6-6-7-7-5-522-7-7-4-411-8-8-8-8-8-8-4-400 & 3 &-1 & 7 &-6 &-7 &-5 & 2 &-7 &-4 & 1 &-8 &-8 &-8 &-4 & 0 \\\\ \\hline 3 &-1 & 7 &-6 &-7 \\\\ \\hline-5 & 2 &-7 &-4 & 1 \\\\ \\hline-8 &-8 &-8 &-4 & 0 \\\\ \\hline \\end{array}$\n(a) On the matrix above, perform the row operation $-17 R_{1}+R_{3} \\to R_{3}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) On the original matrix, perform the row operation $9 R_{2} \\to R_{2}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(c) On the original matrix, perform the row operation $R_{3} \\leftrightarrow R_{4}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "-4", "-7", "-4", "1", "-9", "3", "-1", "7", "-6", "-7", "63", "121", "61", "-21", "154", "-8", "-8", "-8", "-4", "0", "-4", "-7", "-4", "1", "-9", "27", "-9", "63", "-54", "-63", "-5", "2", "-7", "-4", "1", "-8", "-8", "-8", "-4", "0", "-4", "-7", "-4", "1", "-9", "3", "-1", "7", "-6", "-7", "-8", "-8", "-8", "-4", "0", "-5", "2", "-7", "-4", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the matrix\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[-3-3669988-6-6 \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]-4-4-5-5-10-102210106622-8-8-4-4334400-2-29966 &-3 & 6 & 9 & 8 &-6 & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]-4-4-5-5-10-102210106622-8-8-4-4334400-2-29966 &-4 &-5 &-10 & 2 & 10 & 6 & 2 &-8 &-4 & 3 & 4 & 0 &-2 & 9 & 6 \\\\ \\hline-4 &-5 &-10 & 2 & 10 \\\\ \\hline 6 & 2 &-8 &-4 & 3 \\\\ \\hline 4 & 0 &-2 & 9 & 6 \\\\ \\hline \\end{array}$\n(a) On the matrix above, perform the row operation $-4 R_{1}+R_{4} \\to R_{4}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) On the original matrix, perform the row operation $4 R_{3} \\to R_{3}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(c) On the original matrix, perform the row operation $R_{1} \\leftrightarrow R_{3}$. The new matrix is:\n$\\begin{array}{cccccccccccccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS] [ANS] [ANS] [ANS] [ANS] \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "-3", "6", "9", "8", "-6", "-4", "-5", "-10", "2", "10", "6", "2", "-8", "-4", "3", "16", "-24", "-38", "-23", "30", "-3", "6", "9", "8", "-6", "-4", "-5", "-10", "2", "10", "24", "8", "-32", "-16", "12", "4", "0", "-2", "9", "6", "6", "2", "-8", "-4", "3", "-4", "-5", "-10", "2", "10", "-3", "6", "9", "8", "-6", "4", "0", "-2", "9", "6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0183", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Echelon form", "level": "3", "keywords": [ "linear algebra", "matrix", "echelon" ], "problem_v1": "Reduce the matrix A=\\left[\\begin{array}{ccc} 2 &1 &7\\cr-1 &-1 &-4 \\end{array}\\right] to reduced row-echelon form.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "0", "3", "0", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Reduce the matrix A=\\left[\\begin{array}{ccc}-4 &-1 &15\\cr 3 &-1 &-20 \\end{array}\\right] to reduced row-echelon form.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "-5", "0", "1", "5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Reduce the matrix A=\\left[\\begin{array}{ccc}-2 &1 &5\\cr-3 &-1 &5 \\end{array}\\right] to reduced row-echelon form.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "0", "-2", "0", "1", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0184", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Echelon form", "level": "2", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Given \\left[\\begin{array}{ccccc} 1 &-3 &1 &-2 &1\\cr 1 &-3 &0 &-1 &-1\\cr 1 &-3 &1 &-2 &1 \\end{array}\\right] \\sim \\left[\\begin{array}{ccccc} 1 &-3 &0 &-1 &-1\\cr 0 &0 &1 &-1 &2\\cr 0 &0 &0 &0 &0 \\end{array}\\right], use the reduced row echelon form above to solve the system \\left\\lbrace \\begin{array}{rcr} w-3x+y-2z &=& 1 \\\\ w-3x-z &=&-1 \\\\ w-3x+y-2z &=& 1 \\end{array} \\right. If necessary, parametrize your answer using the free variables of the system.\n$\\left\\lbrack \\begin{array}{c} w \\\\ x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-1-(-1)*z-(-3)*x", "x", "2-(-1)*z", "z" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Given \\left[\\begin{array}{ccccc}-2 &-4 &2 &8 &-12\\cr 1 &2 &0 &-1 &1\\cr-1 &-2 &1 &4 &-6 \\end{array}\\right] \\sim \\left[\\begin{array}{ccccc} 1 &2 &0 &-1 &1\\cr 0 &0 &1 &3 &-5\\cr 0 &0 &0 &0 &0 \\end{array}\\right], use the reduced row echelon form above to solve the system \\left\\lbrace \\begin{array}{rcr}-2w-4x+2y+8z &=&-12 \\\\ w+2x-z &=& 1 \\\\-w-2x+y+4z &=&-6 \\end{array} \\right. If necessary, parametrize your answer using the free variables of the system.\n$\\left\\lbrack \\begin{array}{c} w \\\\ x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "1-(-1)*z-2*x", "x", "-5-3*z", "z" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Given \\left[\\begin{array}{ccccc}-1 &2 &-1 &8 &-1\\cr 1 &-2 &0 &-3 &4\\cr-1 &2 &-1 &8 &-1 \\end{array}\\right] \\sim \\left[\\begin{array}{ccccc} 1 &-2 &0 &-3 &4\\cr 0 &0 &1 &-5 &-3\\cr 0 &0 &0 &0 &0 \\end{array}\\right], use the reduced row echelon form above to solve the system \\left\\lbrace \\begin{array}{rcr}-w+2x-y+8z &=&-1 \\\\ w-2x-3z &=& 4 \\\\-w+2x-y+8z &=&-1 \\end{array} \\right. If necessary, parametrize your answer using the free variables of the system.\n$\\left\\lbrack \\begin{array}{c} w \\\\ x \\\\ y \\\\ z \\end{array} \\right\\rbrack=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "4-(-3)*z-(-2)*x", "x", "-3-(-5)*z", "z" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0185", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Echelon form", "level": "2", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "Consider the system \\left\\lbrace \\begin{array}{rcr} 2x_{1}+x_{2}+x_{3} &=& 2 \\\\-x_{1}-x_{2}+x_{3} &=&-1 \\\\ x_{1}-2x_{2}-x_{3} &=&-1 \\end{array} \\right.\n(a) Find the reduced row echelon form of the augmented matrix for this system. Your answers must be fractions (decimals are not allowed). You should be able to do this exercise without a calculator.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}.\n(b) Solve the original system of equations. Your answers must be fractions (decimals are not allowed).\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "1", "0", "0", "5/9", "0", "1", "0", "2/3", "0", "0", "1", "2/9", "5/9", "2/3", "2/9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the system \\left\\lbrace \\begin{array}{rcr}-2x_{1}-2x_{2}+x_{3} &=&-2 \\\\-x_{1}-3x_{2}-3x_{3} &=&-3 \\\\-x_{1}-3x_{2}+3x_{3} &=&-1 \\end{array} \\right.\n(a) Find the reduced row echelon form of the augmented matrix for this system. Your answers must be fractions (decimals are not allowed). You should be able to do this exercise without a calculator.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}.\n(b) Solve the original system of equations. Your answers must be fractions (decimals are not allowed).\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "1", "0", "0", "3/4", "0", "1", "0", "5/12", "0", "0", "1", "1/3", "3/4", "5/12", "1/3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the system \\left\\lbrace \\begin{array}{rcr}-x_{1}+x_{2}-2x_{3} &=&-2 \\\\-x_{1}+2x_{2}+3x_{3} &=& 3 \\\\-2x_{1}-x_{2}-2x_{3} &=&-3 \\end{array} \\right.\n(a) Find the reduced row echelon form of the augmented matrix for this system. Your answers must be fractions (decimals are not allowed). You should be able to do this exercise without a calculator.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}.\n(b) Solve the original system of equations. Your answers must be fractions (decimals are not allowed).\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "1", "0", "0", "7/17", "0", "1", "0", "5/17", "0", "0", "1", "16/17", "7/17", "5/17", "16/17" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0186", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Echelon form", "level": "3", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices" ], "problem_v1": "(a) Perform the indicated row operations on the matrix $A$ successively in the order they are given until a matrix in row echelon form is produced. A=\\left[\\begin{array}{ccc} 5 &5 &5\\cr 3 &4 &2 \\end{array}\\right] Apply $(1/5) R_1 \\to R_1$ to $A$.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} Apply $R_2-3 R_1 \\to R_2$ to the previous result.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(b) Solve the system \\left\\lbrace \\begin{array}{rcl} 5x_1+5x_2 &=& 5 \\\\ 3x_1+4x_2 &=& 2 \\\\ \\end{array} \\right. $\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "1", "1", "1", "3", "4", "2", "1", "1", "1", "0", "1", "-1", "2", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "(a) Perform the indicated row operations on the matrix $A$ successively in the order they are given until a matrix in row echelon form is produced. A=\\left[\\begin{array}{ccc} 7 &-35 &35\\cr 3 &-14 &11 \\end{array}\\right] Apply $(1/7) R_1 \\to R_1$ to $A$.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} Apply $R_2-3 R_1 \\to R_2$ to the previous result.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(b) Solve the system \\left\\lbrace \\begin{array}{rcl} 7x_1-35x_2 &=& 35 \\\\ 3x_1-14x_2 &=& 11 \\\\ \\end{array} \\right. $\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "1", "-5", "5", "3", "-14", "11", "1", "-5", "5", "0", "1", "-4", "-15", "-4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "(a) Perform the indicated row operations on the matrix $A$ successively in the order they are given until a matrix in row echelon form is produced. A=\\left[\\begin{array}{ccc} 2 &-4 &2\\cr 5 &-9 &3 \\end{array}\\right] Apply $(1/2) R_1 \\to R_1$ to $A$.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} Apply $R_2-5 R_1 \\to R_2$ to the previous result.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(b) Solve the system \\left\\lbrace \\begin{array}{rcl} 2x_1-4x_2 &=& 2 \\\\ 5x_1-9x_2 &=& 3 \\\\ \\end{array} \\right. $\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "1", "-2", "1", "5", "-9", "3", "1", "-2", "1", "0", "1", "-2", "-3", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0187", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Echelon form", "level": "2", "keywords": [ "matrix' 'augmented' 'Echelon" ], "problem_v1": "Determine whether the following matrices are in echelon form, reduced echelon form or not in echelon form.\n[ANS] $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &0 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{ccccc} 8 &7 &-4 &7 &-5\\cr 0 &-7 &1 &1 &1\\cr 0 &0 &1 &0 &-6\\cr 0 &0 &0 &1 &0 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{ccc}-10 &0 &1\\cr 0 &-2 &0 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{cccc} 1 &0 &0 &-6\\cr 0 &0 &0 &0\\cr 0 &1 &0 &6 \\end{array}\\right]$", "answer_v1": [ "Reduced Echelon Form", "Echelon Form", "Echelon Form", "Not in Echelon Form" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ] ], "problem_v2": "Determine whether the following matrices are in echelon form, reduced echelon form or not in echelon form.\n[ANS] $\\left[\\begin{array}{ccc}-10 &0 &1\\cr 0 &5 &0 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{ccc} 1 &1 &-3\\cr 1 &0 &-7 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{cccc} 1 &0 &0 &-2\\cr 0 &0 &0 &0\\cr 0 &1 &0 &-6 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &0 \\end{array}\\right]$", "answer_v2": [ "Echelon Form", "Not in Echelon Form", "Not in Echelon Form", "Reduced Echelon Form" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ] ], "problem_v3": "Determine whether the following matrices are in echelon form, reduced echelon form or not in echelon form.\n[ANS] $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &0 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{cccc} 1 &0 &0 &4\\cr 0 &1 &0 &8\\cr 0 &0 &0 &0 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{ccccc} 4 &4 &-3 &9 &-9\\cr 0 &-4 &1 &1 &1\\cr 0 &0 &1 &0 &2\\cr 0 &0 &0 &1 &0 \\end{array}\\right]$\n[ANS] $\\left[\\begin{array}{ccc} 1 &1 &10\\cr 1 &0 &-7 \\end{array}\\right]$", "answer_v3": [ "Reduced Echelon Form", "Reduced Echelon Form", "Echelon Form", "Not in Echelon Form" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ], [ "Echelon Form", "Reduced Echelon Form", "Not in Echelon Form" ] ] }, { "id": "Linear_algebra_0188", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Rank", "level": "3", "keywords": [ "linear algebra", "subspaces", "basis", "rank" ], "problem_v1": "Suppose that $A$ is a 8 $\\times$ 7 matrix which has a null space of dimension 3. The rank of $A$ is $\\text{rank}(A)=$ [ANS]", "answer_v1": [ "4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $A$ is a 5 $\\times$ 9 matrix which has a null space of dimension 2. The rank of $A$ is $\\text{rank}(A)=$ [ANS]", "answer_v2": [ "7" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $A$ is a 6 $\\times$ 8 matrix which has a null space of dimension 2. The rank of $A$ is $\\text{rank}(A)=$ [ANS]", "answer_v3": [ "6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0189", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Rank", "level": "2", "keywords": [ "linear algebra", "matrix" ], "problem_v1": "If A=\\left[\\begin{array}{ccc} 2 &1 &1\\cr 0 &-1 &-1\\cr 0 &-1 &0 \\end{array}\\right], then $\\text{rank}(A)=$ [ANS] and\n$A^2=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "4", "0", "1", "0", "2", "1", "0", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "If A=\\left[\\begin{array}{ccc}-3 &3 &-2\\cr 0 &3 &-1\\cr 6 &-9 &5 \\end{array}\\right], then $\\text{rank}(A)=$ [ANS] and\n$A^2=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-3", "18", "-7", "-6", "18", "-8", "12", "-54", "22" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "If A=\\left[\\begin{array}{ccc}-1 &1 &-2\\cr 0 &-2 &-1\\cr-2 &-4 &-7 \\end{array}\\right], then $\\text{rank}(A)=$ [ANS] and\n$A^2=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "5", "5", "15", "2", "8", "9", "16", "34", "57" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0190", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Rank", "level": "2", "keywords": [ "linear algebra", "matrix", "rank" ], "problem_v1": "Find the ranks of the following matrices.\n$\\text{rank} \\left[\\begin{array}{ccc} 7 &1 &-7\\cr 0 &-4 &0\\cr-5 &0 &5 \\end{array}\\right]=$ [ANS]\n$\\text{rank} \\left[\\begin{array}{ccc}-1 &-1 &-5\\cr 1 &7 &0\\cr 4 &0 &0 \\end{array}\\right]=$ [ANS]\n$\\text{rank} \\left[\\begin{array}{cc} 0 &-4\\cr 0 &-1\\cr 0 &-3 \\end{array}\\right]=$ [ANS]", "answer_v1": [ "2", "3", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the ranks of the following matrices.\n$\\text{rank} \\left[\\begin{array}{cc} 2 &-9\\cr-2 &9 \\end{array}\\right]=$ [ANS]\n$\\text{rank} \\left[\\begin{array}{ccc} 9 &1 &-9\\cr 0 &8 &0\\cr-2 &0 &2 \\end{array}\\right]=$ [ANS]\n$\\text{rank} \\left[\\begin{array}{cc} 0 &8\\cr 0 &1\\cr 0 &-3 \\end{array}\\right]=$ [ANS]", "answer_v2": [ "1", "2", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the ranks of the following matrices.\n$\\text{rank} \\left[\\begin{array}{cccc} 0 &9 &0 &0\\cr-2 &0 &0 &0\\cr 0 &0 &1 &0\\cr 0 &0 &0 &2 \\end{array}\\right]=$ [ANS]\n$\\text{rank} \\left[\\begin{array}{ccc} 0 &2 &-3\\cr 0 &7 &0\\cr 6 &0 &0 \\end{array}\\right]=$ [ANS]\n$\\text{rank} \\left[\\begin{array}{ccc} 7 &1 &-7\\cr 0 &-6 &0\\cr-3 &0 &3 \\end{array}\\right]=$ [ANS]", "answer_v3": [ "4", "3", "2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0191", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Rank", "level": "3", "keywords": [ "linear algebra", "matrix", "rank" ], "problem_v1": "Find the value of $k$ for which the matrix A=\\left[\\begin{array}{ccc} 5 &2 &-3\\cr 2 &4 &2\\cr-4 &-3 &k\\cr \\end{array}\\right] has rank $2$.\n$k=$ [ANS]", "answer_v1": [ "1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the value of $k$ for which the matrix A=\\left[\\begin{array}{ccc}-8 &8 &8\\cr-7 &-3 &17\\cr 8 &-3 &k\\cr \\end{array}\\right] has rank $2$.\n$k=$ [ANS]", "answer_v2": [ "-13" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the value of $k$ for which the matrix A=\\left[\\begin{array}{ccc}-4 &2 &-4\\cr-4 &1 &-6\\cr-6 &-3 &k\\cr \\end{array}\\right] has rank $2$.\n$k=$ [ANS]", "answer_v3": [ "-18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0192", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Transpose and trace", "level": "3", "keywords": [], "problem_v1": "Enter a $3 \\times 3$ skew-symmetric matrix $A$ that has entries $a_{2 1}=5$, $a_{3 1}=0$, and $a_{2 3}=1$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0", "-5", "0", "5", "0", "1", "0", "-1", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Enter a $3 \\times 3$ skew-symmetric matrix $A$ that has entries $a_{1 2}=3$, $a_{1 3}=4$, and $a_{3 2}=2$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "0", "3", "4", "-3", "0", "-2", "-4", "2", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Enter a $3 \\times 3$ skew-symmetric matrix $A$ that has entries $a_{2 1}=3$, $a_{3 1}=0$, and $a_{3 2}=5$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "0", "-3", "0", "3", "0", "-5", "0", "5", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0193", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Transpose and trace", "level": "", "keywords": [ "linear algebra", "matrix", "trace" ], "problem_v1": "Find the trace of the matrix A=\\left[\\begin{array}{cc} 3 &1\\cr 1 &2 \\end{array}\\right]. $tr(A)=$ [ANS]", "answer_v1": [ "5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the trace of the matrix A=\\left[\\begin{array}{cc}-5 &5\\cr-4 &-2 \\end{array}\\right]. $tr(A)=$ [ANS]", "answer_v2": [ "-7" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the trace of the matrix A=\\left[\\begin{array}{cc}-2 &1\\cr-2 &1 \\end{array}\\right]. $tr(A)=$ [ANS]", "answer_v3": [ "-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0194", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Transpose and trace", "level": "2", "keywords": [ "linear algebra", "matrix", "multiplication" ], "problem_v1": "Let A=\\left[\\begin{array}{ccc} 3 &1 &1\\cr 2 &-2 &-2\\cr 1 &1 &-1 \\end{array}\\right]. Compute the following.\n$A+A^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n$A-A^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nFor any square matrix $A$, is the matrix $A+A^T$ lower triangular, upper triangular, symmetric, skew-symmetric, or none of these? [ANS]\nFor any square matrix $A$, is the matrix $A-A^T$ lower triangular, upper triangular, symmetric, skew-symmetric, or none of these? [ANS]", "answer_v1": [ "6", "3", "2", "3", "-4", "-1", "2", "-1", "-2", "0", "-1", "0", "1", "0", "-3", "0", "3", "0" ], "answer_type_v1": [ "NV", "NV", "MCS", "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [ "Lower triangular", "Upper triangular", "Symmetric", "Skew-symmetric", "None of these" ], [ "Lower triangular", "Upper triangular", "Symmetric", "Skew-symmetric", "None of these" ], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{ccc}-5 &5 &-4\\cr-2 &5 &-2\\cr-3 &-2 &1 \\end{array}\\right]. Compute the following.\n$A+A^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n$A-A^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nFor any square matrix $A$, is the matrix $A+A^T$ lower triangular, upper triangular, symmetric, skew-symmetric, or none of these? [ANS]\nFor any square matrix $A$, is the matrix $A-A^T$ lower triangular, upper triangular, symmetric, skew-symmetric, or none of these? [ANS]", "answer_v2": [ "-10", "3", "-7", "3", "10", "-4", "-7", "-4", "2", "0", "7", "-1", "-7", "0", "0", "1", "0", "0" ], "answer_type_v2": [ "NV", "NV", "MCS", "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [ "Lower triangular", "Upper triangular", "Symmetric", "Skew-symmetric", "None of these" ], [ "Lower triangular", "Upper triangular", "Symmetric", "Skew-symmetric", "None of these" ], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{ccc}-2 &1 &-2\\cr 1 &-3 &-2\\cr 3 &5 &4 \\end{array}\\right]. Compute the following.\n$A+A^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n$A-A^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nFor any square matrix $A$, is the matrix $A+A^T$ lower triangular, upper triangular, symmetric, skew-symmetric, or none of these? [ANS]\nFor any square matrix $A$, is the matrix $A-A^T$ lower triangular, upper triangular, symmetric, skew-symmetric, or none of these? [ANS]", "answer_v3": [ "-4", "2", "1", "2", "-6", "3", "1", "3", "8", "0", "0", "-5", "0", "0", "-7", "5", "7", "0" ], "answer_type_v3": [ "NV", "NV", "MCS", "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [ "Lower triangular", "Upper triangular", "Symmetric", "Skew-symmetric", "None of these" ], [ "Lower triangular", "Upper triangular", "Symmetric", "Skew-symmetric", "None of these" ], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0195", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Transpose and trace", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Let $A$ be a 5 by 8 matrix. Then $-A$ is a [ANS] by [ANS] matrix, and $A^T$ is a [ANS] by [ANS] matrix.", "answer_v1": [ "5", "8", "8", "5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $A$ be a 2 by 9 matrix. Then $-A$ is a [ANS] by [ANS] matrix, and $A^T$ is a [ANS] by [ANS] matrix.", "answer_v2": [ "2", "9", "9", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $A$ be a 3 by 8 matrix. Then $-A$ is a [ANS] by [ANS] matrix, and $A^T$ is a [ANS] by [ANS] matrix.", "answer_v3": [ "3", "8", "8", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0196", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Transpose and trace", "level": "2", "keywords": [ "matrix' 'product" ], "problem_v1": "If u=\\left[\\begin{array}{c} 5\\cr 2\\cr 2 \\end{array}\\right] then\n$u^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} and $u^T u=$ \\begin {array}{c} [ANS] \\end{array}.", "answer_v1": [ "25", "10", "10", "10", "4", "4", "10", "4", "4", "33" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "If u=\\left[\\begin{array}{c}-8\\cr 8\\cr-7 \\end{array}\\right] then\n$u^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} and $u^T u=$ \\begin {array}{c} [ANS] \\end{array}.", "answer_v2": [ "64", "-64", "56", "-64", "64", "-56", "56", "-56", "49", "177" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "If u=\\left[\\begin{array}{c}-4\\cr 2\\cr-4 \\end{array}\\right] then\n$u^T=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} and $u^T u=$ \\begin {array}{c} [ANS] \\end{array}.", "answer_v3": [ "16", "-8", "16", "-8", "4", "-8", "16", "-8", "16", "36" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0197", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "3", "keywords": [], "problem_v1": "For the encoding matrix $A=\\left[\\begin{array}{ccc} 2 &3 &1\\cr 3 &7 &3\\cr-8 &-14 &-5 \\end{array}\\right]$ decode the message $\\left[\\left[\\begin{array}{c} 26\\cr 56\\cr-117 \\end{array}\\right],\\left[\\begin{array}{c} 57\\cr 144\\cr-271 \\end{array}\\right]\\right]$, which gives a six character word: [ANS]", "answer_v1": [ "decals" ], "answer_type_v1": [ "OE" ], "options_v1": [ [] ], "problem_v2": "For the encoding matrix $A=\\left[\\begin{array}{ccc} 41 &-5 &5\\cr-105 &11 &-14\\cr 215 &-22 &29 \\end{array}\\right]$ decode the message $\\left[\\left[\\begin{array}{c} 1151\\cr-2995\\cr 6146 \\end{array}\\right],\\left[\\begin{array}{c} 1031\\cr-2668\\cr 5471 \\end{array}\\right]\\right]$, which gives a six character word: [ANS]", "answer_v2": [ "zizzle" ], "answer_type_v2": [ "OE" ], "options_v2": [ [] ], "problem_v3": "For the encoding matrix $A=\\left[\\begin{array}{ccc}-8 &-2 &1\\cr 1 &-1 &-1\\cr-5 &2 &3 \\end{array}\\right]$ decode the message $\\left[\\left[\\begin{array}{c}-144\\cr 15\\cr-82 \\end{array}\\right],\\left[\\begin{array}{c}-14\\cr-27\\cr 68 \\end{array}\\right]\\right]$, which gives a six character word: [ANS]", "answer_v3": [ "rabbit" ], "answer_type_v3": [ "OE" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0198", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "3", "keywords": [], "problem_v1": "Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro, Parker’s, and Bleyer’s. The amount of nitrogen, phosphoric acid, and potash per 100 pounds for each brand is given by the nutrient vectors \u0016\u0017\n$\\begin{array}{ccc}\\hline v=\\left[\\begin{array}{c} 15\\cr 51\\cr 52 \\end{array}\\right] & p=\\left[\\begin{array}{c} 5\\cr 16\\cr 16 \\end{array}\\right] & b=\\left[\\begin{array}{c} 4\\cr 16\\cr 17 \\end{array}\\right] \\\\ \\hline Vigoro & Parker’s & Bleyer’s \\\\ \\hline \\end{array}$\nDetermine the linear transformation $T: \\mathbf{R}^3 \\rightarrow \\mathbf{R}^3$ that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output. $T(\\textbf{x})$=\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $\\textbf {x}$ Then find a formula for $T^{−1}$, and use it to determine the amount of Vigoro, Parker’s, and Bleyer’s required to produce $139$ pounds of nitrogen, $482$ pounds of phosphoric acid and $494$ pounds of potash. $T^{-1}(\\textbf{y})$=\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $\\textbf {y}$ $T^{-1}\\left(\\left[\\begin{array}{c} 139\\cr 482\\cr 494 \\end{array}\\right]\\right)$=\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "15", "5", "4", "51", "16", "16", "52", "16", "17", "16", "-21", "16", "-35", "47", "-36", "-16", "20", "-15", "6", "5", "6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro, Parker’s, and Bleyer’s. The amount of nitrogen, phosphoric acid, and potash per 100 pounds for each brand is given by the nutrient vectors \u0016\u0017\n$\\begin{array}{ccc}\\hline v=\\left[\\begin{array}{c} 18\\cr 43\\cr 46 \\end{array}\\right] & p=\\left[\\begin{array}{c} 2\\cr 5\\cr 5 \\end{array}\\right] & b=\\left[\\begin{array}{c} 5\\cr 12\\cr 13 \\end{array}\\right] \\\\ \\hline Vigoro & Parker’s & Bleyer’s \\\\ \\hline \\end{array}$\nDetermine the linear transformation $T: \\mathbf{R}^3 \\rightarrow \\mathbf{R}^3$ that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output. $T(\\textbf{x})$=\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $\\textbf {x}$ Then find a formula for $T^{−1}$, and use it to determine the amount of Vigoro, Parker’s, and Bleyer’s required to produce $114$ pounds of nitrogen, $274$ pounds of phosphoric acid and $291$ pounds of potash. $T^{-1}(\\textbf{y})$=\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $\\textbf {y}$ $T^{-1}\\left(\\left[\\begin{array}{c} 114\\cr 274\\cr 291 \\end{array}\\right]\\right)$=\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "18", "2", "5", "43", "5", "12", "46", "5", "13", "5", "-1", "-1", "-7", "4", "-1", "-15", "2", "4", "5", "7", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro, Parker’s, and Bleyer’s. The amount of nitrogen, phosphoric acid, and potash per 100 pounds for each brand is given by the nutrient vectors \u0016\u0017\n$\\begin{array}{ccc}\\hline v=\\left[\\begin{array}{c} 14\\cr 34\\cr 69 \\end{array}\\right] & p=\\left[\\begin{array}{c} 3\\cr 7\\cr 14 \\end{array}\\right] & b=\\left[\\begin{array}{c} 4\\cr 11\\cr 23 \\end{array}\\right] \\\\ \\hline Vigoro & Parker’s & Bleyer’s \\\\ \\hline \\end{array}$\nDetermine the linear transformation $T: \\mathbf{R}^3 \\rightarrow \\mathbf{R}^3$ that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output. $T(\\textbf{x})$=\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $\\textbf {x}$ Then find a formula for $T^{−1}$, and use it to determine the amount of Vigoro, Parker’s, and Bleyer’s required to produce $151$ pounds of nitrogen, $368$ pounds of phosphoric acid and $747$ pounds of potash. $T^{-1}(\\textbf{y})$=\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} $\\textbf {y}$ $T^{-1}\\left(\\left[\\begin{array}{c} 151\\cr 368\\cr 747 \\end{array}\\right]\\right)$=\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "14", "3", "4", "34", "7", "11", "69", "14", "23", "7", "-13", "5", "-23", "46", "-18", "-7", "11", "-4", "8", "9", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0199", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "3", "keywords": [ "matrix", "invertible" ], "problem_v1": "The matrix $\\left[\\begin{array}{cc} 8 &2\\cr 3 &k\\cr \\end{array}\\right]$ is invertible if and only if $k\\ne$ [ANS].", "answer_v1": [ "2*3/8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The matrix $\\left[\\begin{array}{cc} 2 &8\\cr-7 &k\\cr \\end{array}\\right]$ is invertible if and only if $k\\ne$ [ANS].", "answer_v2": [ "8*-7/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The matrix $\\left[\\begin{array}{cc} 4 &2\\cr-4 &k\\cr \\end{array}\\right]$ is invertible if and only if $k\\ne$ [ANS].", "answer_v3": [ "2*-4/4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0200", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "2", "keywords": [], "problem_v1": "Are the following matrices invertible?\n[ANS] 1. $\\left[\\begin{array}{cc} 0 &-7\\cr 0 &35 \\end{array}\\right]$\n[ANS] 2. $\\left[\\begin{array}{cc} 0 &-1\\cr-1 &-1 \\end{array}\\right]$\n[ANS] 3. $\\left[\\begin{array}{cc}-1 &0\\cr 2 &-3 \\end{array}\\right]$\n[ANS] 4. $\\left[\\begin{array}{cc} 0 &-7\\cr 0 &0 \\end{array}\\right]$", "answer_v1": [ "NOT INVERTIBLE", "INVERTIBLE", "INVERTIBLE", "Not invertible" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ] ], "problem_v2": "Are the following matrices invertible?\n[ANS] 1. $\\left[\\begin{array}{cc}-5 &5\\cr-4 &-2 \\end{array}\\right]$\n[ANS] 2. $\\left[\\begin{array}{cc} 5 &-2\\cr-3 &-2 \\end{array}\\right]$\n[ANS] 3. $\\left[\\begin{array}{cc} 1 &-8\\cr-2 &16 \\end{array}\\right]$\n[ANS] 4. $\\left[\\begin{array}{cc}-2 &-8\\cr 0 &0 \\end{array}\\right]$", "answer_v2": [ "INVERTIBLE", "INVERTIBLE", "NOT INVERTIBLE", "Not invertible" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ] ], "problem_v3": "Are the following matrices invertible?\n[ANS] 1. $\\left[\\begin{array}{cc} 4 &-3\\cr-2 &-3 \\end{array}\\right]$\n[ANS] 2. $\\left[\\begin{array}{cc} 2 &-7\\cr-8 &28 \\end{array}\\right]$\n[ANS] 3. $\\left[\\begin{array}{cc}-5 &1\\cr 5 &3 \\end{array}\\right]$\n[ANS] 4. $\\left[\\begin{array}{cc}-8 &-7\\cr 0 &0 \\end{array}\\right]$", "answer_v3": [ "INVERTIBLE", "NOT INVERTIBLE", "INVERTIBLE", "Not invertible" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ], [ "Invertible", "Not invertible" ] ] }, { "id": "Linear_algebra_0201", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "3", "keywords": [ "matrix", "inverse" ], "problem_v1": "If A=\\left[\\begin{array}{cccc} 3 &1 &0 &0\\cr 2 &1 &0 &0\\cr 0 &0 &1 &-2\\cr 0 &0 &-2 &5 \\end{array}\\right], then\n$A^{-1}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v1": [ "1", "-1", "0", "0", "-2", "3", "0", "0", "0", "0", "5", "2", "0", "0", "2", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "If A=\\left[\\begin{array}{cccc}-5 &-1 &0 &0\\cr 24 &5 &0 &0\\cr 0 &0 &-1 &5\\cr 0 &0 &-2 &11 \\end{array}\\right], then\n$A^{-1}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v2": [ "-5", "-1", "0", "0", "24", "5", "0", "0", "0", "0", "-11", "5", "0", "0", "-2", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "If A=\\left[\\begin{array}{cccc}-2 &1 &0 &0\\cr-1 &1 &0 &0\\cr 0 &0 &1 &-3\\cr 0 &0 &-2 &7 \\end{array}\\right], then\n$A^{-1}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v3": [ "-1", "1", "0", "0", "-1", "2", "0", "0", "0", "0", "7", "3", "0", "0", "2", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0202", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "3", "keywords": [ "algebra", "matrix operation", "matrix", "inverse" ], "problem_v1": "Given the matrix \\left[\\begin{array}{ccc} 2 &-1 &1\\cr-1 &1 &-1\\cr 1 &1 &0 \\end{array}\\right]\n(a) does the inverse of the matrix exist? [ANS] (b) if your answer is yes, enter the inverse of the matrix below.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "1", "0", "-1", "-1", "1", "-2", "-3", "1" ], "answer_type_v1": [ "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [ "Yes", "No" ], [], [], [], [], [], [], [], [] ], "problem_v2": "Given the matrix \\left[\\begin{array}{ccc}-3 &-14 &1\\cr-1 &1 &-1\\cr 1 &3 &0 \\end{array}\\right]\n(a) does the inverse of the matrix exist? [ANS] (b) if your answer is yes, enter the inverse of the matrix below.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "3", "3", "13", "-1", "-1", "-4", "-4", "-5", "-17" ], "answer_type_v2": [ "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [ "Yes", "No" ], [], [], [], [], [], [], [], [] ], "problem_v3": "Given the matrix \\left[\\begin{array}{ccc}-1 &-4 &1\\cr-1 &1 &-1\\cr 1 &1 &0 \\end{array}\\right]\n(a) does the inverse of the matrix exist? [ANS] (b) if your answer is yes, enter the inverse of the matrix below.\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "1", "3", "-1", "-1", "-2", "-2", "-3", "-5" ], "answer_type_v3": [ "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [ "Yes", "No" ], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0203", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "3", "keywords": [ "algebra", "inverse", "matrix" ], "problem_v1": "Solve the system of equations \\begin{array}{r} 3x-6 y+z=-10 \\\\-x+y-z=2 \\\\ x-2 y=-3 \\\\ \\end{array} by converting to a matrix equation and using the inverse of the coefficient matrix.\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v1": [ "1", "2", "-1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the system of equations \\begin{array}{r} 2x-4 y+z=3 \\\\-x+y-z=-2 \\\\ x-2 y=0 \\\\ \\end{array} by converting to a matrix equation and using the inverse of the coefficient matrix.\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v2": [ "-2", "-1", "3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the system of equations \\begin{array}{r} 2x-4 y+z=-6 \\\\-x+y-z=4 \\\\ x-2 y=-2 \\\\ \\end{array} by converting to a matrix equation and using the inverse of the coefficient matrix.\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v3": [ "-2", "0", "-2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0204", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "3", "keywords": [ "algebra", "inverse", "solving system of equations" ], "problem_v1": "Solve the system of equations \\begin{array}{l} 5x+3y=13 \\\\ 3x+2y=8 \\\\ \\end{array} by converting to a matrix equation and using the inverse of the coefficient matrix. $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "2", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Solve the system of equations \\begin{array}{l} 5x+3y=-8 \\\\ 3x+2y=-4 \\\\ \\end{array} by converting to a matrix equation and using the inverse of the coefficient matrix. $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "-4", "4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Solve the system of equations \\begin{array}{l} 5x+3y=-7 \\\\ 3x+2y=-4 \\\\ \\end{array} by converting to a matrix equation and using the inverse of the coefficient matrix. $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "-2", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0205", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "3", "keywords": [ "algebra" ], "problem_v1": "Let A=\\left(\\begin{array}{cc} 5 &-4 \\cr 4 & 4 \\end{array}\\right) Find $A^{-1}$ and use it to solve AX=B where B=\\left(\\begin{array}{c} 3 \\cr 3 \\end{array}\\right) X=\\left(\\begin{array}{c} x_{1} \\cr x_{2} \\end{array}\\right) where $x_{1}=$ [ANS] and $x_{2}=$ [ANS].", "answer_v1": [ "0.666666666666667", "0.0833333333333333" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let A=\\left(\\begin{array}{cc} 2 &-5 \\cr 2 & 3 \\end{array}\\right) Find $A^{-1}$ and use it to solve AX=B where B=\\left(\\begin{array}{c} 5 \\cr 3 \\end{array}\\right) X=\\left(\\begin{array}{c} x_{1} \\cr x_{2} \\end{array}\\right) where $x_{1}=$ [ANS] and $x_{2}=$ [ANS].", "answer_v2": [ "1.875", "-0.25" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let A=\\left(\\begin{array}{cc} 3 &-4 \\cr 3 & 4 \\end{array}\\right) Find $A^{-1}$ and use it to solve AX=B where B=\\left(\\begin{array}{c} 2 \\cr 3 \\end{array}\\right) X=\\left(\\begin{array}{c} x_{1} \\cr x_{2} \\end{array}\\right) where $x_{1}=$ [ANS] and $x_{2}=$ [ANS].", "answer_v3": [ "0.833333333333333", "0.125" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0206", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Inverses", "level": "2", "keywords": [ "matrix' 'equation" ], "problem_v1": "In each part, find the matrix X solving the given equation. a. $\\left[\\begin{array}{cc} 5 &0\\cr 0 &1 \\end{array}\\right] X=\\left[\\begin{array}{cc} 2 &3\\cr 5 &-4 \\end{array}\\right].$ $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. $\\left[\\begin{array}{cc} 0 &1\\cr 1 &0 \\end{array}\\right] X=\\left[\\begin{array}{cc}-4 &1\\cr 2 &-3 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. $\\left[\\begin{array}{cc} 1 &3\\cr 0 &1 \\end{array}\\right] X=\\left[\\begin{array}{cc}-5 &-2\\cr-2 &-1 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. $\\left[\\begin{array}{cc} 1 &-3\\cr 2 &-5 \\end{array}\\right] X=\\left[\\begin{array}{cc}-2 &6\\cr-1 &-4 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &-1 \\end{array}\\right] X=\\left[\\begin{array}{ccc} 8 &10 &-5\\cr-10 &-5 &1\\cr-7 &-3 &5 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} f. $\\left[\\begin{array}{ccc} 0 &0 &1\\cr 0 &1 &0\\cr 1 &0 &0 \\end{array}\\right] X=\\left[\\begin{array}{ccc} 6 &-1 &-7\\cr-3 &6 &-7\\cr-5 &-6 &-2 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} g. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &-5 &1 \\end{array}\\right] X=\\left[\\begin{array}{ccc}-6 &1 &1\\cr-8 &-8 &-1\\cr 9 &7 &-4 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0.4", "0.6", "5", "-4", "2", "-3", "-4", "1", "1", "1", "-2", "-1", "7", "-42", "3", "-16", "8", "10", "-5", "-10", "-5", "1", "7", "3", "-5", "-5", "-6", "-2", "-3", "6", "-7", "6", "-1", "-7", "-6", "1", "1", "-8", "-8", "-1", "-31", "-33", "-9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "In each part, find the matrix X solving the given equation. a. $\\left[\\begin{array}{cc}-9 &0\\cr 0 &1 \\end{array}\\right] X=\\left[\\begin{array}{cc} 9 &-7\\cr-3 &9 \\end{array}\\right].$ $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. $\\left[\\begin{array}{cc} 0 &1\\cr 1 &0 \\end{array}\\right] X=\\left[\\begin{array}{cc}-4 &-7\\cr-4 &1 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. $\\left[\\begin{array}{cc} 1 &-9\\cr 0 &1 \\end{array}\\right] X=\\left[\\begin{array}{cc} 3 &-1\\cr 7 &-6 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. $\\left[\\begin{array}{cc} 1 &-3\\cr 3 &-8 \\end{array}\\right] X=\\left[\\begin{array}{cc} 2 &-7\\cr-4 &1 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &-8 \\end{array}\\right] X=\\left[\\begin{array}{ccc}-8 &-8 &-4\\cr-9 &8 &-4\\cr-2 &4 &9 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} f. $\\left[\\begin{array}{ccc} 0 &0 &1\\cr 0 &1 &0\\cr 1 &0 &0 \\end{array}\\right] X=\\left[\\begin{array}{ccc}-1 &1 &8\\cr-9 &-2 &-2\\cr 5 &9 &-2 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} g. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &-9 &1 \\end{array}\\right] X=\\left[\\begin{array}{ccc}-3 &5 &-9\\cr-6 &-5 &-9\\cr 10 &5 &-1 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-1", "0.777778", "-3", "9", "-4", "1", "-4", "-7", "66", "-55", "7", "-6", "-28", "59", "-10", "22", "-8", "-8", "-4", "-9", "8", "-4", "0.25", "-0.5", "-1.125", "5", "9", "-2", "-9", "-2", "-2", "-1", "1", "8", "-3", "5", "-9", "-6", "-5", "-9", "-44", "-40", "-82" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "In each part, find the matrix X solving the given equation. a. $\\left[\\begin{array}{cc}-4 &0\\cr 0 &1 \\end{array}\\right] X=\\left[\\begin{array}{cc} 2 &-5\\cr 1 &-6 \\end{array}\\right].$ $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. $\\left[\\begin{array}{cc} 0 &1\\cr 1 &0 \\end{array}\\right] X=\\left[\\begin{array}{cc}-3 &6\\cr 9 &8 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. $\\left[\\begin{array}{cc} 1 &-6\\cr 0 &1 \\end{array}\\right] X=\\left[\\begin{array}{cc}-4 &-5\\cr-10 &2 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. $\\left[\\begin{array}{cc} 1 &4\\cr 5 &21 \\end{array}\\right] X=\\left[\\begin{array}{cc} 2 &-8\\cr-4 &3 \\end{array}\\right]$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &4 \\end{array}\\right] X=\\left[\\begin{array}{ccc}-2 &9 &6\\cr 7 &1 &2\\cr 10 &6 &-7 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} f. $\\left[\\begin{array}{ccc} 0 &0 &1\\cr 0 &1 &0\\cr 1 &0 &0 \\end{array}\\right] X=\\left[\\begin{array}{ccc} 1 &9 &5\\cr-6 &3 &-1\\cr-7 &-1 &-7 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} g. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &-3 &1 \\end{array}\\right] X=\\left[\\begin{array}{ccc} 1 &-9 &-9\\cr 6 &-7 &-5\\cr-10 &-2 &-2 \\end{array}\\right]$. $X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-0.5", "1.25", "1", "-6", "9", "8", "-3", "6", "-64", "7", "-10", "2", "58", "-180", "-14", "43", "-2", "9", "6", "7", "1", "2", "2.5", "1.5", "-1.75", "-7", "-1", "-7", "-6", "3", "-1", "1", "9", "5", "1", "-9", "-9", "6", "-7", "-5", "8", "-23", "-17" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0207", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Elementary matrices", "level": "2", "keywords": [], "problem_v1": "Assume that A is a matrix with three rows. Find the matrix B such that BA gives the matrix resulting from A after the given row operations are performed. $4 R_{3}+R_{2}\\Rightarrow R_{2}$ $-4 R_{1}\\Rightarrow R_{1}$ $B=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "-4", "0", "0", "0", "1", "4", "0", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Assume that A is a matrix with three rows. Find the matrix B such that BA gives the matrix resulting from A after the given row operations are performed. $-3 R_{1}+R_{3}\\Rightarrow R_{3}$ $8 R_{2}\\Rightarrow R_{2}$ $B=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "0", "0", "8", "0", "-3", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Assume that A is a matrix with three rows. Find the matrix B such that BA gives the matrix resulting from A after the given row operations are performed. $R_{1}+R_{3}\\Rightarrow R_{3}$ $-6 R_{2}\\Rightarrow R_{2}$ $B=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "0", "0", "0", "-6", "0", "1", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0208", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Elementary matrices", "level": "3", "keywords": [], "problem_v1": "Assume that A is a matrix with four rows. Find the elementary matrix E such that E A gives the matrix resulting from A after the given row operation is performed. Then find $E^{−1}$ and give the elementary row operation corresponding to $E^{−1}$. $R_{4}\\Leftrightarrow R_{2}$\n$E=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}\n$E^{-1}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}\nThe elementary row operation corresponding to $E^{-1}$ is: [ANS] A\\. $R_{4}\\Leftrightarrow R_{3}$ B\\. $R_{2}\\Leftrightarrow R_{3}$ C\\. $R_{4}\\Leftrightarrow R_{2}$", "answer_v1": [ "1", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "1", "0", "0" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [ "A", "B", "C" ], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Assume that A is a matrix with four rows. Find the elementary matrix E such that E A gives the matrix resulting from A after the given row operation is performed. Then find $E^{−1}$ and give the elementary row operation corresponding to $E^{−1}$. $R_{1}\\Leftrightarrow R_{4}$\n$E=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}\n$E^{-1}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}\nThe elementary row operation corresponding to $E^{-1}$ is: [ANS] A\\. $R_{4}\\Leftrightarrow R_{2}$ B\\. $R_{1}\\Leftrightarrow R_{2}$ C\\. $R_{1}\\Leftrightarrow R_{4}$", "answer_v2": [ "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [ "A", "B", "C" ], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Assume that A is a matrix with four rows. Find the elementary matrix E such that E A gives the matrix resulting from A after the given row operation is performed. Then find $E^{−1}$ and give the elementary row operation corresponding to $E^{−1}$. $R_{2}\\Leftrightarrow R_{3}$\n$E=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}\n$E^{-1}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}\nThe elementary row operation corresponding to $E^{-1}$ is: [ANS] A\\. $R_{3}\\Leftrightarrow R_{1}$ B\\. $R_{2}\\Leftrightarrow R_{1}$ C\\. $R_{2}\\Leftrightarrow R_{3}$", "answer_v3": [ "1", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "1", "1", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "1" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [ "A", "B", "C" ], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0209", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Elementary matrices", "level": "2", "keywords": [ "linear algebra", "matrix" ], "problem_v1": "Let A=\\left[\\begin{array}{ccc} 5 &1 &2\\cr 2 &-4 &2 \\end{array}\\right]. Perform the following row operations sequentially (applying each new operation to the previous result).\nFirst, swap the first and second rows:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nSecond, multiply every element of the first row by $ \\frac{1}{2} $:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nThird: add to the elements of the second row, $-5$ times the corresponding elements of the first row:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "2", "-4", "2", "5", "1", "2", "1", "-2", "1", "5", "1", "2", "1", "-2", "1", "0", "11", "-3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{ccc} 2 &-1 &-2\\cr 3 &-6 &-9 \\end{array}\\right]. Perform the following row operations sequentially (applying each new operation to the previous result).\nFirst, swap the first and second rows:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nSecond, multiply every element of the first row by $ \\frac{1}{3} $:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nThird: add to the elements of the second row, $-2$ times the corresponding elements of the first row:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "3", "-6", "-9", "2", "-1", "-2", "1", "-2", "-3", "2", "-1", "-2", "1", "-2", "-3", "0", "3", "4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{ccc} 3 &-1 &1\\cr 2 &-4 &8 \\end{array}\\right]. Perform the following row operations sequentially (applying each new operation to the previous result).\nFirst, swap the first and second rows:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nSecond, multiply every element of the first row by $ \\frac{1}{2} $:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nThird: add to the elements of the second row, $-3$ times the corresponding elements of the first row:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "2", "-4", "8", "3", "-1", "1", "1", "-2", "4", "3", "-1", "1", "1", "-2", "4", "0", "5", "-11" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0210", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Elementary matrices", "level": "3", "keywords": [ "Algebra' 'Linear Equations' 'Matrix' 'Matrices", "Elementary matrices" ], "problem_v1": "Give a $3 \\times 3$ elementary matrix $E$ which will carry out the row operation $R_{1} \\leftrightarrow R_{3}$.\n$E=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nTest that $E$ actually works for carrying out this row operation by computing the product $E A$ for the matrix A=\\left[\\begin{array}{cccc} 3 &1 &1 &2\\cr-2 &-2 &1 &1\\cr-1 &2 &-3 &-1 \\end{array}\\right]. $EA=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0", "0", "1", "0", "1", "0", "1", "0", "0", "-1", "2", "-3", "-1", "-2", "-2", "1", "1", "3", "1", "1", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Give a $3 \\times 3$ elementary matrix $E$ which will carry out the row operation $R_{2} \\leftrightarrow R_{3}$.\n$E=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nTest that $E$ actually works for carrying out this row operation by computing the product $E A$ for the matrix A=\\left[\\begin{array}{cccc}-5 &5 &-4 &-2\\cr 5 &-2 &-3 &-2\\cr 1 &-5 &2 &-1 \\end{array}\\right]. $EA=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "0", "0", "0", "1", "0", "1", "0", "-5", "5", "-4", "-2", "1", "-5", "2", "-1", "5", "-2", "-3", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Give a $3 \\times 3$ elementary matrix $E$ which will carry out the row operation $R_{1} \\leftrightarrow R_{2}$.\n$E=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nTest that $E$ actually works for carrying out this row operation by computing the product $E A$ for the matrix A=\\left[\\begin{array}{cccc}-2 &1 &-2 &1\\cr-3 &-2 &3 &5\\cr 4 &-3 &-2 &-3 \\end{array}\\right]. $EA=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "0", "1", "0", "1", "0", "0", "0", "0", "1", "-3", "-2", "3", "5", "-2", "1", "-2", "1", "4", "-3", "-2", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0211", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Elementary matrices", "level": "3", "keywords": [ "linear algebra", "elementery matrix", "multiplication" ], "problem_v1": "Consider the following Gauss-Jordan reduction:\n\\underbrace{\\left[\\begin{array}{ccc} 1 &0 &2\\cr 0 &0 &1\\cr 10 &5 &20 \\end{array}\\right]}_A \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &2\\cr 10 &5 &20\\cr 0 &0 &1 \\end{array}\\right]}_{E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &2\\cr 2 &1 &4\\cr 0 &0 &1 \\end{array}\\right]}_{E_2E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &2\\cr 0 &1 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_3E_2E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_4E_3E_2E_1 A}=I Find $E_1=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_2=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_3=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. $E_4=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.\nWrite $A$ as a product $A=E_1^{-1}E_2^{-1}E_3^{-1}E_4^{-1}$ of elementary matrices:\n$\\left[\\begin{array}{ccc} 1 &0 &2\\cr 0 &0 &1\\cr 10 &5 &20 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0.2", "0", "0", "0", "1", "1", "0", "0", "-2", "1", "0", "0", "0", "1", "1", "0", "-2", "0", "1", "0", "0", "0", "1", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "5", "0", "0", "0", "1", "1", "0", "0", "2", "1", "0", "0", "0", "1", "1", "0", "2", "0", "1", "0", "0", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the following Gauss-Jordan reduction:\n\\underbrace{\\left[\\begin{array}{ccc} 0 &1 &0\\cr-8 &-7 &-64\\cr 0 &0 &1 \\end{array}\\right]}_A \\to \\underbrace{\\left[\\begin{array}{ccc} 0 &1 &0\\cr-8 &0 &-64\\cr 0 &0 &1 \\end{array}\\right]}_{E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 0 &1 &0\\cr 1 &0 &8\\cr 0 &0 &1 \\end{array}\\right]}_{E_2E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &8\\cr 0 &1 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_3E_2E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_4E_3E_2E_1 A}=I Find $E_1=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_2=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_3=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. $E_4=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.\nWrite $A$ as a product $A=E_1^{-1}E_2^{-1}E_3^{-1}E_4^{-1}$ of elementary matrices:\n$\\left[\\begin{array}{ccc} 0 &1 &0\\cr-8 &-7 &-64\\cr 0 &0 &1 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "0", "7", "1", "0", "0", "0", "1", "1", "0", "0", "0", "-0.125", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "1", "0", "-8", "0", "1", "0", "0", "0", "1", "1", "0", "0", "-7", "1", "0", "0", "0", "1", "1", "0", "0", "0", "-8", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "1", "0", "8", "0", "1", "0", "0", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the following Gauss-Jordan reduction:\n\\underbrace{\\left[\\begin{array}{ccc} 0 &1 &2\\cr-4 &16 &0\\cr 0 &0 &1 \\end{array}\\right]}_A \\to \\underbrace{\\left[\\begin{array}{ccc} 0 &1 &2\\cr 1 &-4 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 0 &1 &0\\cr 1 &-4 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_2E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 0 &1 &0\\cr 1 &0 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_3E_2E_1 A} \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_4E_3E_2E_1 A}=I Find $E_1=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_2=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_3=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. $E_4=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.\nWrite $A$ as a product $A=E_1^{-1}E_2^{-1}E_3^{-1}E_4^{-1}$ of elementary matrices:\n$\\left[\\begin{array}{ccc} 0 &1 &2\\cr-4 &16 &0\\cr 0 &0 &1 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "0", "0", "0", "-0.25", "0", "0", "0", "1", "1", "0", "-2", "0", "1", "0", "0", "0", "1", "1", "0", "0", "4", "1", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "1", "0", "0", "0", "-4", "0", "0", "0", "1", "1", "0", "2", "0", "1", "0", "0", "0", "1", "1", "0", "0", "-4", "1", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0214", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Elementary matrices", "level": "2", "keywords": [ "matrix' 'inverse" ], "problem_v1": "a. Suppose that $E_1 \\left[\\begin{array}{cc}-1 &2\\cr-3 &-1 \\end{array}\\right]=\\left[\\begin{array}{cc}-5 &10\\cr-3 &-1 \\end{array}\\right]$. Find $E_1$ and $E_1^{-1}$. $E_1=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_1^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. b. Suppose that $E_2 \\left[\\begin{array}{cc}-1 &2\\cr-3 &-1 \\end{array}\\right]=\\left[\\begin{array}{cc}-3 &-1\\cr-1 &2 \\end{array}\\right]$. Find $E_2$ and $E_2^{-1}$. $E_2=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_2^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. c. Suppose that $E_3 \\left[\\begin{array}{cc}-1 &2\\cr-3 &-1 \\end{array}\\right]=\\left[\\begin{array}{cc}-16 &-3\\cr-3 &-1 \\end{array}\\right]$. Find $E_3$ and $E_3^{-1}$. $E_3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_3^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. d. Suppose that $E_4 \\left[\\begin{array}{ccc}-1 &-1 &-4\\cr-4 &-1 &3\\cr-2 &4 &5 \\end{array}\\right]=\\left[\\begin{array}{ccc}-1 &-1 &-4\\cr-20 &-5 &15\\cr-2 &4 &5 \\end{array}\\right]$. Find $E_4$ and $E_4^{-1}$. $E_4=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_4^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. e. Suppose that $E_5 \\left[\\begin{array}{ccc}-1 &-1 &-4\\cr-4 &-1 &3\\cr-2 &4 &5 \\end{array}\\right]=\\left[\\begin{array}{ccc}-1 &-1 &-4\\cr-2 &4 &5\\cr-4 &-1 &3 \\end{array}\\right]$. Find $E_5$ and $E_5^{-1}$. $E_5=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_5^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. f. Suppose that $E_6 \\left[\\begin{array}{ccc}-1 &-1 &-4\\cr-4 &-1 &3\\cr-2 &4 &5 \\end{array}\\right]=\\left[\\begin{array}{ccc}-1 &-1 &-4\\cr-4 &-1 &3\\cr-8 &-2 &-19 \\end{array}\\right]$. Find $E_6$ and $E_6^{-1}$. $E_6=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_6^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v1": [ "5", "0", "0", "1", "0.2", "0", "0", "1", "0", "1", "1", "0", "0", "1", "1", "0", "1", "5", "0", "1", "1", "-5", "0", "1", "1", "0", "0", "0", "5", "0", "0", "0", "1", "1", "0", "0", "0", "0.2", "0", "0", "0", "1", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "1", "0", "6", "0", "1", "1", "0", "0", "0", "1", "0", "-6", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "a. Suppose that $E_1 \\left[\\begin{array}{cc} 5 &-2\\cr-3 &-2 \\end{array}\\right]=\\left[\\begin{array}{cc} 10 &-4\\cr-3 &-2 \\end{array}\\right]$. Find $E_1$ and $E_1^{-1}$. $E_1=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_1^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. b. Suppose that $E_2 \\left[\\begin{array}{cc} 5 &-2\\cr-3 &-2 \\end{array}\\right]=\\left[\\begin{array}{cc}-3 &-2\\cr 5 &-2 \\end{array}\\right]$. Find $E_2$ and $E_2^{-1}$. $E_2=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_2^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. c. Suppose that $E_3 \\left[\\begin{array}{cc} 5 &-2\\cr-3 &-2 \\end{array}\\right]=\\left[\\begin{array}{cc}-16 &-16\\cr-3 &-2 \\end{array}\\right]$. Find $E_3$ and $E_3^{-1}$. $E_3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_3^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. d. Suppose that $E_4 \\left[\\begin{array}{ccc} 1 &-5 &2\\cr-1 &3 &-3\\cr-3 &-3 &1 \\end{array}\\right]=\\left[\\begin{array}{ccc} 1 &-5 &2\\cr-2 &6 &-6\\cr-3 &-3 &1 \\end{array}\\right]$. Find $E_4$ and $E_4^{-1}$. $E_4=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_4^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. e. Suppose that $E_5 \\left[\\begin{array}{ccc} 1 &-5 &2\\cr-1 &3 &-3\\cr-3 &-3 &1 \\end{array}\\right]=\\left[\\begin{array}{ccc} 1 &-5 &2\\cr-3 &-3 &1\\cr-1 &3 &-3 \\end{array}\\right]$. Find $E_5$ and $E_5^{-1}$. $E_5=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_5^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. f. Suppose that $E_6 \\left[\\begin{array}{ccc} 1 &-5 &2\\cr-1 &3 &-3\\cr-3 &-3 &1 \\end{array}\\right]=\\left[\\begin{array}{ccc} 1 &-5 &2\\cr-1 &3 &-3\\cr 1 &-23 &9 \\end{array}\\right]$. Find $E_6$ and $E_6^{-1}$. $E_6=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_6^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v2": [ "2", "0", "0", "1", "0.5", "0", "0", "1", "0", "1", "1", "0", "0", "1", "1", "0", "1", "7", "0", "1", "1", "-7", "0", "1", "1", "0", "0", "0", "2", "0", "0", "0", "1", "1", "0", "0", "0", "0.5", "0", "0", "0", "1", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "1", "0", "4", "0", "1", "1", "0", "0", "0", "1", "0", "-4", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "a. Suppose that $E_1 \\left[\\begin{array}{cc}-3 &-2\\cr 3 &5 \\end{array}\\right]=\\left[\\begin{array}{cc}-9 &-6\\cr 3 &5 \\end{array}\\right]$. Find $E_1$ and $E_1^{-1}$. $E_1=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_1^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. b. Suppose that $E_2 \\left[\\begin{array}{cc}-3 &-2\\cr 3 &5 \\end{array}\\right]=\\left[\\begin{array}{cc} 3 &5\\cr-3 &-2 \\end{array}\\right]$. Find $E_2$ and $E_2^{-1}$. $E_2=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_2^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. c. Suppose that $E_3 \\left[\\begin{array}{cc}-3 &-2\\cr 3 &5 \\end{array}\\right]=\\left[\\begin{array}{cc} 15 &28\\cr 3 &5 \\end{array}\\right]$. Find $E_3$ and $E_3^{-1}$. $E_3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_3^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}. d. Suppose that $E_4 \\left[\\begin{array}{ccc} 4 &-3 &-2\\cr-3 &-5 &1\\cr 5 &3 &1 \\end{array}\\right]=\\left[\\begin{array}{ccc} 4 &-3 &-2\\cr-9 &-15 &3\\cr 5 &3 &1 \\end{array}\\right]$. Find $E_4$ and $E_4^{-1}$. $E_4=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_4^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. e. Suppose that $E_5 \\left[\\begin{array}{ccc} 4 &-3 &-2\\cr-3 &-5 &1\\cr 5 &3 &1 \\end{array}\\right]=\\left[\\begin{array}{ccc} 4 &-3 &-2\\cr 5 &3 &1\\cr-3 &-5 &1 \\end{array}\\right]$. Find $E_5$ and $E_5^{-1}$. $E_5=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_5^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}. f. Suppose that $E_6 \\left[\\begin{array}{ccc} 4 &-3 &-2\\cr-3 &-5 &1\\cr 5 &3 &1 \\end{array}\\right]=\\left[\\begin{array}{ccc} 4 &-3 &-2\\cr-3 &-5 &1\\cr 25 &-12 &-9 \\end{array}\\right]$. Find $E_6$ and $E_6^{-1}$. $E_6=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_6^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v3": [ "3", "0", "0", "1", "0.333333", "0", "0", "1", "0", "1", "1", "0", "0", "1", "1", "0", "1", "6", "0", "1", "1", "-6", "0", "1", "1", "0", "0", "0", "3", "0", "0", "0", "1", "1", "0", "0", "0", "0.333333", "0", "0", "0", "1", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "1", "0", "5", "0", "1", "1", "0", "0", "0", "1", "0", "-5", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0215", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Elementary matrices", "level": "2", "keywords": [ "matrix' 'inverse" ], "problem_v1": "Suppose that: A=$\\left[\\begin{array}{cc} 1 &1\\cr-1 &2 \\end{array}\\right]$ and B=$\\left[\\begin{array}{ccc}-3 &-1 &-1\\cr-1 &-4 &-4\\cr-1 &3 &-2 \\end{array}\\right]$ Given the following descriptions, determine the following elementary matrices and their inverses. a. The elementary matrix $E_1$ multiplies the first row of A by $1/5$. $E_1=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_1^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. The elementary matrix $E_2$ multiplies the second row of A by $-4$. $E_2=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_2^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. The elementary matrix $E_3$ switches the first and second rows of A. $E_3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_3^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. The elementary matrix $E_4$ adds $6$ times the first row of A to the second row of A. $E_4=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_4^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. The elementary matrix $E_5$ multiplies the second row of B by $1/5$. $E_5=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_5^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} f. The elementary matrix $E_6$ multiplies the third row of B by $-5$. $E_6=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_6^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} g. The elementary matrix $E_7$ switches the first and third rows of B. $E_7=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_7^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} h. The elementary matrix $E_8$ adds $4$ times the third row of B to the second row of B. $E_8=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_8^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0.2", "0", "0", "1", "5", "0", "0", "1", "1", "0", "0", "-4", "1", "0", "0", "-0.25", "0", "1", "1", "0", "0", "1", "1", "0", "1", "0", "6", "1", "1", "0", "-6", "1", "1", "0", "0", "0", "0.2", "0", "0", "0", "1", "1", "0", "0", "0", "5", "0", "0", "0", "1", "1", "0", "0", "0", "1", "0", "0", "0", "-5", "1", "0", "0", "0", "1", "0", "0", "0", "-0.2", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "1", "0", "0", "0", "1", "4", "0", "0", "1", "1", "0", "0", "0", "1", "-4", "0", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose that: A=$\\left[\\begin{array}{cc}-3 &-2\\cr 1 &-5 \\end{array}\\right]$ and B=$\\left[\\begin{array}{ccc} 2 &-1 &3\\cr-3 &-3 &-3\\cr 1 &-3 &-2 \\end{array}\\right]$ Given the following descriptions, determine the following elementary matrices and their inverses. a. The elementary matrix $E_1$ multiplies the first row of A by $1/2$. $E_1=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_1^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. The elementary matrix $E_2$ multiplies the second row of A by $-2$. $E_2=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_2^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. The elementary matrix $E_3$ switches the first and second rows of A. $E_3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_3^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. The elementary matrix $E_4$ adds $3$ times the first row of A to the second row of A. $E_4=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_4^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. The elementary matrix $E_5$ multiplies the second row of B by $1/3$. $E_5=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_5^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} f. The elementary matrix $E_6$ multiplies the third row of B by $-2$. $E_6=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_6^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} g. The elementary matrix $E_7$ switches the first and third rows of B. $E_7=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_7^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} h. The elementary matrix $E_8$ adds $4$ times the third row of B to the second row of B. $E_8=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_8^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "0.5", "0", "0", "1", "2", "0", "0", "1", "1", "0", "0", "-2", "1", "0", "0", "-0.5", "0", "1", "1", "0", "0", "1", "1", "0", "1", "0", "3", "1", "1", "0", "-3", "1", "1", "0", "0", "0", "0.333333", "0", "0", "0", "1", "1", "0", "0", "0", "3", "0", "0", "0", "1", "1", "0", "0", "0", "1", "0", "0", "0", "-2", "1", "0", "0", "0", "1", "0", "0", "0", "-0.5", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "1", "0", "0", "0", "1", "4", "0", "0", "1", "1", "0", "0", "0", "1", "-4", "0", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose that: A=$\\left[\\begin{array}{cc} 3 &5\\cr 4 &-3 \\end{array}\\right]$ and B=$\\left[\\begin{array}{ccc}-2 &-3 &-5\\cr 1 &5 &3\\cr 1 &-4 &-2 \\end{array}\\right]$ Given the following descriptions, determine the following elementary matrices and their inverses. a. The elementary matrix $E_1$ multiplies the first row of A by $1/3$. $E_1=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_1^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. The elementary matrix $E_2$ multiplies the second row of A by $-3$. $E_2=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_2^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. The elementary matrix $E_3$ switches the first and second rows of A. $E_3=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_3^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. The elementary matrix $E_4$ adds $4$ times the first row of A to the second row of A. $E_4=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $E_4^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. The elementary matrix $E_5$ multiplies the second row of B by $1/4$. $E_5=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_5^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} f. The elementary matrix $E_6$ multiplies the third row of B by $-5$. $E_6=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_6^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} g. The elementary matrix $E_7$ switches the first and third rows of B. $E_7=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_7^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} h. The elementary matrix $E_8$ adds $4$ times the third row of B to the second row of B. $E_8=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}, $E_8^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "0.333333", "0", "0", "1", "3", "0", "0", "1", "1", "0", "0", "-3", "1", "0", "0", "-0.333333", "0", "1", "1", "0", "0", "1", "1", "0", "1", "0", "4", "1", "1", "0", "-4", "1", "1", "0", "0", "0", "0.25", "0", "0", "0", "1", "1", "0", "0", "0", "4", "0", "0", "0", "1", "1", "0", "0", "0", "1", "0", "0", "0", "-5", "1", "0", "0", "0", "1", "0", "0", "0", "-0.2", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "1", "0", "0", "0", "1", "4", "0", "0", "1", "1", "0", "0", "0", "1", "-4", "0", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0216", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Markov chains", "level": "3", "keywords": [], "problem_v1": "Let $A=\\left[\\begin{array}{ccc} 1 & 0 & 1/3 \\\\ 0 & 1 & 1/3 \\\\ 0 & 0 & 1/3 \\end{array}\\right]$ Numerically verify that each initial state vector $x_0$ has the given steady-state vector x. $x_0=\\left[\\begin{array}{c} 0.38\\cr 0.3\\cr 0.32 \\end{array}\\right]\\Rightarrow \\left[\\begin{array}{c} 0.54\\cr 0.46\\cr 0 \\end{array}\\right]$ $x_1=\\left[\\begin{array}{c} 0.32\\cr 0.37\\cr 0.31 \\end{array}\\right]\\Rightarrow \\left[\\begin{array}{c} 0.475\\cr 0.525\\cr 0 \\end{array}\\right]$\n$Ax_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $Ax_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$A^4x_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $A^4x_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$A^8x_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $A^8x_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.486667", "0.406667", "0.106667", "0.423333", "0.473333", "0.103333", "0.538025", "0.458025", "0.00395062", "0.473086", "0.523086", "0.00382716", "0.539976", "0.459976", "4.87731E-05", "0.474976", "0.524976", "4.72489E-05" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $A=\\left[\\begin{array}{ccc} 1 & 0 & 1/3 \\\\ 0 & 1 & 1/3 \\\\ 0 & 0 & 1/3 \\end{array}\\right]$ Numerically verify that each initial state vector $x_0$ has the given steady-state vector x. $x_0=\\left[\\begin{array}{c} 0.05\\cr 0.47\\cr 0.48 \\end{array}\\right]\\Rightarrow \\left[\\begin{array}{c} 0.29\\cr 0.71\\cr 0 \\end{array}\\right]$ $x_1=\\left[\\begin{array}{c} 0.08\\cr 0.17\\cr 0.75 \\end{array}\\right]\\Rightarrow \\left[\\begin{array}{c} 0.455\\cr 0.545\\cr 0 \\end{array}\\right]$\n$Ax_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $Ax_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$A^4x_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $A^4x_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$A^8x_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $A^8x_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "0.21", "0.63", "0.16", "0.33", "0.42", "0.25", "0.287037", "0.707037", "0.00592593", "0.45037", "0.54037", "0.00925926", "0.289963", "0.709963", "7.31596E-05", "0.454943", "0.544943", "0.000114312" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $A=\\left[\\begin{array}{ccc} 1 & 0 & 1/3 \\\\ 0 & 1 & 1/3 \\\\ 0 & 0 & 1/3 \\end{array}\\right]$ Numerically verify that each initial state vector $x_0$ has the given steady-state vector x. $x_0=\\left[\\begin{array}{c} 0.16\\cr 0.31\\cr 0.53 \\end{array}\\right]\\Rightarrow \\left[\\begin{array}{c} 0.425\\cr 0.575\\cr 0 \\end{array}\\right]$ $x_1=\\left[\\begin{array}{c} 0.14\\cr 0.28\\cr 0.58 \\end{array}\\right]\\Rightarrow \\left[\\begin{array}{c} 0.43\\cr 0.57\\cr 0 \\end{array}\\right]$\n$Ax_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $Ax_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$A^4x_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $A^4x_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$A^8x_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $A^8x_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "0.336667", "0.486667", "0.176667", "0.333333", "0.473333", "0.193333", "0.421728", "0.571728", "0.00654321", "0.42642", "0.56642", "0.00716049", "0.42496", "0.57496", "8.07804E-05", "0.429956", "0.569956", "8.84012E-05" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0217", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Markov chains", "level": "3", "keywords": [], "problem_v1": "Consumers in Shelbyville have a choice of one of two fast food restaurants, Krusty's and McDonald's. Both have trouble keeping customers. Of those who last went to Krusty's, 63\\% will go to McDonald's next time, and of those who last went to McDonald's, 81\\% will go to Krusty's next time.\n(a) Find the transition matrix describing this situation. (Assume the components of the state vector are in this order [Krusty's customers, McDonald's customers]).\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} (b) A customer goes out for fast food every Sunday, and just went to Krusty's. i. What is the probability that two Sundays from now she will go to McDonald's? [ANS]\nii. What is the probability that three Sundays from now she will go to McDonald's? [ANS]\n(c) Suppose a consumer has just moved to Shelbyville, and there is a 39\\% chance that he will go to Krusty's for his first fast food outing. What is the probability that his third fast food experience will be at Krusty's? [ANS]\n(d) Find the steady-state vector.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.37", "0.81", "0.63", "0.19", "0.5625", "0.4375" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Consumers in Shelbyville have a choice of one of two fast food restaurants, Krusty's and McDonald's. Both have trouble keeping customers. Of those who last went to Krusty's, 55\\% will go to McDonald's next time, and of those who last went to McDonald's, 85\\% will go to Krusty's next time.\n(a) Find the transition matrix describing this situation. (Assume the components of the state vector are in this order [Krusty's customers, McDonald's customers]).\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} (b) A customer goes out for fast food every Sunday, and just went to Krusty's. i. What is the probability that two Sundays from now she will go to McDonald's? [ANS]\nii. What is the probability that three Sundays from now she will go to McDonald's? [ANS]\n(c) Suppose a consumer has just moved to Shelbyville, and there is a 32\\% chance that he will go to Krusty's for his first fast food outing. What is the probability that his third fast food experience will be at Krusty's? [ANS]\n(d) Find the steady-state vector.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "0.45", "0.85", "0.55", "0.15", "0.607143", "0.392857" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Consumers in Shelbyville have a choice of one of two fast food restaurants, Krusty's and McDonald's. Both have trouble keeping customers. Of those who last went to Krusty's, 58\\% will go to McDonald's next time, and of those who last went to McDonald's, 81\\% will go to Krusty's next time.\n(a) Find the transition matrix describing this situation. (Assume the components of the state vector are in this order [Krusty's customers, McDonald's customers]).\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} (b) A customer goes out for fast food every Sunday, and just went to Krusty's. i. What is the probability that two Sundays from now she will go to McDonald's? [ANS]\nii. What is the probability that three Sundays from now she will go to McDonald's? [ANS]\n(c) Suppose a consumer has just moved to Shelbyville, and there is a 34\\% chance that he will go to Krusty's for his first fast food outing. What is the probability that his third fast food experience will be at Krusty's? [ANS]\n(d) Find the steady-state vector.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "0.42", "0.81", "0.58", "0.19", "0.582734", "0.417266" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0218", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Markov chains", "level": "3", "keywords": [], "problem_v1": "In an office complex of 1130 employees, on any given day some are at work and the rest are absent. It is known that if an employee is at work today, there is an 83\\% chance that she will be at work tomorrow, and if the employee is absent today, there is a 59\\% chance that she will be absent tomorrow. Suppose that today there are 926 employees at work.\n(a) Find the transition matrix for this scenario. (Assume the components of the state vector are listed in this order: [number at work, number absent]).\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} (b) Predict the number that will be at work five days from now. (Your answer should be an integer.) [ANS]\n(c) Find the steady-state vector. (The components need not be integers.)\n\\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.83", "0.41", "0.17", "0.59", "798.793", "331.207" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "In an office complex of 1030 employees, on any given day some are at work and the rest are absent. It is known that if an employee is at work today, there is an 75\\% chance that she will be at work tomorrow, and if the employee is absent today, there is a 64\\% chance that she will be absent tomorrow. Suppose that today there are 803 employees at work.\n(a) Find the transition matrix for this scenario. (Assume the components of the state vector are listed in this order: [number at work, number absent]).\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} (b) Predict the number that will be at work five days from now. (Your answer should be an integer.) [ANS]\n(c) Find the steady-state vector. (The components need not be integers.)\n\\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "0.75", "0.36", "0.25", "0.64", "607.869", "422.131" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "In an office complex of 1050 employees, on any given day some are at work and the rest are absent. It is known that if an employee is at work today, there is an 78\\% chance that she will be at work tomorrow, and if the employee is absent today, there is a 59\\% chance that she will be absent tomorrow. Suppose that today there are 840 employees at work.\n(a) Find the transition matrix for this scenario. (Assume the components of the state vector are listed in this order: [number at work, number absent]).\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} (b) Predict the number that will be at work five days from now. (Your answer should be an integer.) [ANS]\n(c) Find the steady-state vector. (The components need not be integers.)\n\\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "0.78", "0.41", "0.22", "0.59", "683.333", "366.667" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0219", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Markov chains", "level": "3", "keywords": [], "problem_v1": "Find all steady-state vectors for the given stochastic matrix. $A=\\left[\\begin{array}{rr}-0.07 & 0.06 \\\\ 0.07 &-0.06 \\end{array}\\right]$ $x=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.461538", "0.538462" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find all steady-state vectors for the given stochastic matrix. $A=\\left[\\begin{array}{rr}-0.02 & 0.03 \\\\ 0.02 &-0.03 \\end{array}\\right]$ $x=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "0.6", "0.4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find all steady-state vectors for the given stochastic matrix. $A=\\left[\\begin{array}{rr}-0.08 & 0.09 \\\\ 0.08 &-0.09 \\end{array}\\right]$ $x=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "0.529412", "0.470588" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0220", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Markov chains", "level": "2", "keywords": [], "problem_v1": "If possible, fill in the missing values to make the following matrix a doubly stochastic matrix. If it is not possible put-1 in each blank.\n$\\begin{array}{cccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS]0.76 0.76 \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.76 0.76 [ANS] & [ANS] & 0.76 & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.76 0.76 [ANS] & 0.76 & [ANS] \\\\ \\hline 0.76 & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "1-0.76", "1-0.76" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If possible, fill in the missing values to make the following matrix a doubly stochastic matrix. If it is not possible put-1 in each blank.\n$\\begin{array}{cccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS]0.09 0.09 \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.09 0.09 [ANS] & [ANS] & 0.09 & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.09 0.09 [ANS] & 0.09 & [ANS] \\\\ \\hline 0.09 & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "1-0.09", "1-0.09" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If possible, fill in the missing values to make the following matrix a doubly stochastic matrix. If it is not possible put-1 in each blank.\n$\\begin{array}{cccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[[ANS]0.32 0.32 \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.32 0.32 [ANS] & [ANS] & 0.32 & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.32 0.32 [ANS] & 0.32 & [ANS] \\\\ \\hline 0.32 & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "1-0.32", "1-0.32" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0221", "subject": "Linear_algebra", "topic": "Matrices", "subtopic": "Markov chains", "level": "2", "keywords": [], "problem_v1": "If possible, fill in the missing values to make the following matrix a doubly stochastic matrix. If it is not possible, put in values to make the matrix stochastic.\n$\\begin{array}{ccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[{\\rm a} {\\rm a}0.62 0.62{\\rm b} {\\rm b} \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.76 0.76{\\rm c} {\\rm c}0.22 0.220.14 0.140.28 0.28{\\rm d} {\\rm d} & {\\rm a} & 0.62 & {\\rm b} & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.76 0.76{\\rm c} {\\rm c}0.22 0.220.14 0.140.28 0.28{\\rm d} {\\rm d} & 0.76 & {\\rm c} & 0.22 & 0.14 & 0.28 & {\\rm d} \\\\ \\hline 0.76 & {\\rm c} & 0.22 \\\\ \\hline 0.14 & 0.28 & {\\rm d} \\\\ \\hline \\end{array}$\nIn order that the matrix be stochastic, we must have a=[ANS]\nIf the matrix is to be regular to stochastic, we must have b=[ANS]. Which means that if the matrix is to be stochastic c=[ANS] and d=[ANS]\nThe sum of the entries in the second row is [ANS]\nThe sum of the entries in the third row is [ANS]\n(T/F) This matrix is doubly stochastic [ANS]", "answer_v1": [ "1-0.76-0.14", "1-0.1-0.62", "1-0.62-0.28", "1-0.28-0.22", "0.76+0.1+0.22", "0.14+0.28+0.5", "False" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [], [ "True", "False" ] ], "problem_v2": "If possible, fill in the missing values to make the following matrix a doubly stochastic matrix. If it is not possible, put in values to make the matrix stochastic.\n$\\begin{array}{ccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[{\\rm a} {\\rm a}0.15 0.15{\\rm b} {\\rm b} \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.09 0.09{\\rm c} {\\rm c}0.2 0.20.85 0.850.29 0.29{\\rm d} {\\rm d} & {\\rm a} & 0.15 & {\\rm b} & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.09 0.09{\\rm c} {\\rm c}0.2 0.20.85 0.850.29 0.29{\\rm d} {\\rm d} & 0.09 & {\\rm c} & 0.2 & 0.85 & 0.29 & {\\rm d} \\\\ \\hline 0.09 & {\\rm c} & 0.2 \\\\ \\hline 0.85 & 0.29 & {\\rm d} \\\\ \\hline \\end{array}$\nIn order that the matrix be stochastic, we must have a=[ANS]\nIf the matrix is to be regular to stochastic, we must have b=[ANS]. Which means that if the matrix is to be stochastic c=[ANS] and d=[ANS]\nThe sum of the entries in the second row is [ANS]\nThe sum of the entries in the third row is [ANS]\n(T/F) This matrix is doubly stochastic [ANS]", "answer_v2": [ "1-0.09-0.85", "1-0.06-0.15", "1-0.15-0.29", "1-0.79-0.2", "0.09+0.56+0.2", "0.85+0.29+0.01", "False" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [], [ "True", "False" ] ], "problem_v3": "If possible, fill in the missing values to make the following matrix a doubly stochastic matrix. If it is not possible, put in values to make the matrix stochastic.\n$\\begin{array}{ccccccccccc}\\hline \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right[{\\rm a} {\\rm a}0.28 0.28{\\rm b} {\\rm b} \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.32 0.32{\\rm c} {\\rm c}0.12 0.120.42 0.420.4 0.4{\\rm d} {\\rm d} & {\\rm a} & 0.28 & {\\rm b} & \\left.\\vphantom{\\begin{array}{c}\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\!\\strut\\\\\\end{array}}\\right]0.32 0.32{\\rm c} {\\rm c}0.12 0.120.42 0.420.4 0.4{\\rm d} {\\rm d} & 0.32 & {\\rm c} & 0.12 & 0.42 & 0.4 & {\\rm d} \\\\ \\hline 0.32 & {\\rm c} & 0.12 \\\\ \\hline 0.42 & 0.4 & {\\rm d} \\\\ \\hline \\end{array}$\nIn order that the matrix be stochastic, we must have a=[ANS]\nIf the matrix is to be regular to stochastic, we must have b=[ANS]. Which means that if the matrix is to be stochastic c=[ANS] and d=[ANS]\nThe sum of the entries in the second row is [ANS]\nThe sum of the entries in the third row is [ANS]\n(T/F) This matrix is doubly stochastic [ANS]", "answer_v3": [ "1-0.32-0.42", "1-0.26-0.28", "1-0.28-0.4", "1-0.46-0.12", "0.32+0.32+0.12", "0.42+0.4+0.42", "False" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [], [ "True", "False" ] ] }, { "id": "Linear_algebra_0222", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Diagonalization", "level": "4", "keywords": [], "problem_v1": "Let A=\\left[\\begin{array}{cc} 2 &0.25\\cr 25 &2 \\end{array}\\right]. Find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.\n$S=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$D=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "1", "-10", "10", "-0.5", "0", "0", "4.5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{cc} 8 &-0.25\\cr 221 &-7 \\end{array}\\right]. Find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.\n$S=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$D=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-1", "-1", "-34", "-26", "-0.5", "0", "0", "1.5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{cc} 2 &-0.25\\cr 27 &-4 \\end{array}\\right]. Find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.\n$S=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$D=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-1", "-1", "-18", "-6", "-2.5", "0", "0", "0.5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0223", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Diagonalization", "level": "4", "keywords": [], "problem_v1": "Let M=\\left[\\begin{array}{cc} 5 &1\\cr-2 &8 \\end{array}\\right]. Find formulas for the entries of $M^n$, where $n$ is a positive integer.\n$M^n=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "2*6^n-7^n", "1*7^n-1*6^n", "2*6^n-2*7^n", "2*7^n-6^n" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let M=\\left[\\begin{array}{cc} 2 &-2\\cr 1 &5 \\end{array}\\right]. Find formulas for the entries of $M^n$, where $n$ is a positive integer.\n$M^n=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "2*3^n-4^n", "(-2)*4^n-(-2)*3^n", "(-1)*3^n-(-1)*4^n", "2*4^n-3^n" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let M=\\left[\\begin{array}{cc} 2 &-2\\cr 4 &8 \\end{array}\\right]. Find formulas for the entries of $M^n$, where $n$ is a positive integer.\n$M^n=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "2*4^n-6^n", "(-1)*6^n-(-1)*4^n", "(-2)*4^n-(-2)*6^n", "2*6^n-4^n" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0224", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Diagonalization", "level": "3", "keywords": [], "problem_v1": "Find a $2\\times 2$ matrix $A$ such that \\left[\\begin{array}{c} 3\\cr 1 \\end{array}\\right], \\ \\ \\ \\mbox{and} \\ \\ \\ \\left[\\begin{array}{c} 1\\cr 2 \\end{array}\\right] are eigenvectors of $A$ with eigenvalues $4$ and $-7$, respectively.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "6.2", "-6.6", "4.4", "-9.2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find a $2\\times 2$ matrix $A$ such that \\left[\\begin{array}{c}-5\\cr-4 \\end{array}\\right], \\ \\ \\ \\mbox{and} \\ \\ \\ \\left[\\begin{array}{c} 5\\cr-2 \\end{array}\\right] are eigenvectors of $A$ with eigenvalues $10$ and $-7$, respectively.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-1.33333", "14.1667", "4.53333", "4.33333" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find a $2\\times 2$ matrix $A$ such that \\left[\\begin{array}{c} 0\\cr-2 \\end{array}\\right], \\ \\ \\ \\mbox{and} \\ \\ \\ \\left[\\begin{array}{c} 1\\cr 3 \\end{array}\\right] are eigenvectors of $A$ with eigenvalues $3$ and $-7$, respectively.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-7", "0", "-30", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0225", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Diagonalization", "level": "3", "keywords": [], "problem_v1": "The matrix A=\\left[\\begin{array}{cc} 4 &1\\cr-1 &6 \\end{array}\\right] has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of its associated eigenspace.\nThe eigenvalue [ANS] has an associated eigenspace with dimension [ANS]. Is the matrix $A$ defective? [ANS]", "answer_v1": [ "5", "1", "defective" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "defective", "not defective" ] ], "problem_v2": "The matrix A=\\left[\\begin{array}{cc}-9 &-1\\cr 1 &-7 \\end{array}\\right] has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of its associated eigenspace.\nThe eigenvalue [ANS] has an associated eigenspace with dimension [ANS]. Is the matrix $A$ defective? [ANS]", "answer_v2": [ "-8", "1", "defective" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "defective", "not defective" ] ], "problem_v3": "The matrix A=\\left[\\begin{array}{cc}-6 &-2\\cr 2 &-2 \\end{array}\\right] has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of its associated eigenspace.\nThe eigenvalue [ANS] has an associated eigenspace with dimension [ANS]. Is the matrix $A$ defective? [ANS]", "answer_v3": [ "-4", "1", "defective" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "defective", "not defective" ] ] }, { "id": "Linear_algebra_0226", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Diagonalization", "level": "4", "keywords": [], "problem_v1": "The matrix C=\\left[\\begin{array}{ccc} 7 &4 &4\\cr-4 &-1 &-4\\cr-4 &-4 &-1 \\end{array}\\right] has two distinct eigenvalues with $\\lambda_1 < \\lambda_2$.\nThe smaller eigenvalue $\\lambda_1=$ [ANS] has multiplicity [ANS] and the dimension of the corresponding eigenspace is [ANS].\nThe larger eigenvalue $\\lambda_2=$ [ANS] has multiplicity [ANS] and the dimension of the corresponding eigenspace is [ANS].\nIs the matrix $C$ diagonalizable? [ANS]", "answer_v1": [ "-1", "1", "1", "3", "2", "2", "diagonalizable" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [], [ "diagonalizable", "not diagonalizable" ] ], "problem_v2": "The matrix C=\\left[\\begin{array}{ccc}-2 &0 &3\\cr 10 &1 &-10\\cr-4 &0 &5 \\end{array}\\right] has two distinct eigenvalues with $\\lambda_1 < \\lambda_2$.\nThe smaller eigenvalue $\\lambda_1=$ [ANS] has multiplicity [ANS] and the dimension of the corresponding eigenspace is [ANS].\nThe larger eigenvalue $\\lambda_2=$ [ANS] has multiplicity [ANS] and the dimension of the corresponding eigenspace is [ANS].\nIs the matrix $C$ diagonalizable? [ANS]", "answer_v2": [ "1", "2", "2", "2", "1", "1", "diagonalizable" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [], [ "diagonalizable", "not diagonalizable" ] ], "problem_v3": "The matrix C=\\left[\\begin{array}{ccc} 1 &0 &0\\cr 2 &-1 &0\\cr 0 &0 &-1 \\end{array}\\right] has two distinct eigenvalues with $\\lambda_1 < \\lambda_2$.\nThe smaller eigenvalue $\\lambda_1=$ [ANS] has multiplicity [ANS] and the dimension of the corresponding eigenspace is [ANS].\nThe larger eigenvalue $\\lambda_2=$ [ANS] has multiplicity [ANS] and the dimension of the corresponding eigenspace is [ANS].\nIs the matrix $C$ diagonalizable? [ANS]", "answer_v3": [ "-1", "2", "2", "1", "1", "1", "diagonalizable" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [], [ "diagonalizable", "not diagonalizable" ] ] }, { "id": "Linear_algebra_0227", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Diagonalization", "level": "3", "keywords": [ "matrix exponential" ], "problem_v1": "Compute the matrix exponential $e^A$ for the matrix $A=\\left[\\begin{array}{cc} 2 &0\\cr 0 &1 \\end{array}\\right]$.\n$e^A={}$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "7.38906", "0", "0", "2.71828" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Compute the matrix exponential $e^A$ for the matrix $A=\\left[\\begin{array}{cc}-3 &0\\cr 0 &3 \\end{array}\\right]$.\n$e^A={}$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "0.0497871", "0", "0", "20.0855" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Compute the matrix exponential $e^A$ for the matrix $A=\\left[\\begin{array}{cc}-1 &0\\cr 0 &1 \\end{array}\\right]$.\n$e^A={}$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "0.367879", "0", "0", "2.71828" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0228", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Diagonalization", "level": "3", "keywords": [], "problem_v1": "Let A=\\left[\\begin{array}{cc} 8 &6\\cr 0 &8 \\end{array}\\right]. If possible, find an invertible matrix $P$ so that $D=P^{-1} A P$ is a diagonal matrix. If it is not possible, enter the identity matrix for $P$ and the matrix $A$ for $D$. You must enter a number in every answer blank for the answer evaluator to work properly.\n$P=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n$D=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\nIs $A$ diagonalizable over $\\mathbb{R}$? [ANS] Be sure you can explain why or why not.", "answer_v1": [ "1", "0", "0", "1", "8", "6", "0", "8" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [ "diagonalizable", "not diagonalizable" ], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{cc} 2 &9\\cr 0 &2 \\end{array}\\right]. If possible, find an invertible matrix $P$ so that $D=P^{-1} A P$ is a diagonal matrix. If it is not possible, enter the identity matrix for $P$ and the matrix $A$ for $D$. You must enter a number in every answer blank for the answer evaluator to work properly.\n$P=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n$D=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\nIs $A$ diagonalizable over $\\mathbb{R}$? [ANS] Be sure you can explain why or why not.", "answer_v2": [ "1", "0", "0", "1", "2", "9", "0", "2" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [ "diagonalizable", "not diagonalizable" ], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{cc} 4 &6\\cr 0 &4 \\end{array}\\right]. If possible, find an invertible matrix $P$ so that $D=P^{-1} A P$ is a diagonal matrix. If it is not possible, enter the identity matrix for $P$ and the matrix $A$ for $D$. You must enter a number in every answer blank for the answer evaluator to work properly.\n$P=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n$D=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\nIs $A$ diagonalizable over $\\mathbb{R}$? [ANS] Be sure you can explain why or why not.", "answer_v3": [ "1", "0", "0", "1", "4", "6", "0", "4" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [ "diagonalizable", "not diagonalizable" ], [], [], [], [], [] ] }, { "id": "Linear_algebra_0229", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Diagonalization", "level": "4", "keywords": [], "problem_v1": "Suppose A=\\left[\\begin{array}{cc} 6 &-4\\cr 8 &-6 \\end{array}\\right]. Find an invertible matrix $P$ and a diagonal matrix $D$ so that $A=P D P^{-1}$. Use your answer to find an expression for $A^{7}$ in terms of $P$, a power of $D$, and $P^{-1}$ in that order.\n$A^{7}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v1": [ "1", "1", "1", "2", "128", "0", "0", "-128", "2", "-1", "-1", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose A=\\left[\\begin{array}{cc} 33 &15\\cr-70 &-32 \\end{array}\\right]. Find an invertible matrix $P$ and a diagonal matrix $D$ so that $A=P D P^{-1}$. Use your answer to find an expression for $A^{5}$ in terms of $P$, a power of $D$, and $P^{-1}$ in that order.\n$A^{5}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v2": [ "-1", "3", "2", "-7", "243", "0", "0", "-32", "-7", "-3", "-2", "-1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose A=\\left[\\begin{array}{cc}-2 &-4\\cr 0 &2 \\end{array}\\right]. Find an invertible matrix $P$ and a diagonal matrix $D$ so that $A=P D P^{-1}$. Use your answer to find an expression for $A^{8}$ in terms of $P$, a power of $D$, and $P^{-1}$ in that order.\n$A^{8}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v3": [ "-1", "1", "1", "0", "256", "0", "0", "256", "0", "1", "1", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0233", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "LU factorization", "level": "2", "keywords": [], "problem_v1": "A few terms from the given LU factorization of A are missing. Find them. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 4 &1 &0\\cr a &-4 &1\\cr \\end{array}\\right] \\left[\\begin{array}{cc}-3 &2\\cr 0 &b\\cr 0 &0\\cr \\end{array}\\right]=\\left[\\begin{array}{cc}-3 &c\\cr-12 &10\\cr-15 &2\\cr \\end{array}\\right]$ $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v1": [ "5", "2", "2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A few terms from the given LU factorization of A are missing. Find them. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr-3 &1 &0\\cr a &8 &1\\cr \\end{array}\\right] \\left[\\begin{array}{cc}-3 &-7\\cr 0 &b\\cr 0 &0\\cr \\end{array}\\right]=\\left[\\begin{array}{cc}-3 &c\\cr 9 &29\\cr 24 &120\\cr \\end{array}\\right]$ $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v2": [ "-8", "8", "-7" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A few terms from the given LU factorization of A are missing. Find them. $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 1 &1 &0\\cr a &-6 &1\\cr \\end{array}\\right] \\left[\\begin{array}{cc}-3 &-4\\cr 0 &b\\cr 0 &0\\cr \\end{array}\\right]=\\left[\\begin{array}{cc}-3 &c\\cr-3 &-2\\cr 12 &4\\cr \\end{array}\\right]$ $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v3": [ "-4", "2", "-4" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0234", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "LU factorization", "level": "3", "keywords": [], "problem_v1": "Use the given LU factorization to find all solutions to $\\textrm{A}\\textbf{x}=\\textbf{b}$ $\\textrm{A}=\\left[\\begin{array}{cccc} 1 &0 &0 &0\\cr 5 &1 &0 &0\\cr 2 &2 &1 &0\\cr 4 &-4 &-3 &1 \\end{array}\\right] \\left[\\begin{array}{ccc} 1 &1 &-2\\cr 0 &3 &-4\\cr 0 &0 &-1\\cr 0 &0 &0 \\end{array}\\right]$, $\\textbf{b}=\\left[\\begin{array}{c} 11\\cr 80\\cr 79\\cr-77 \\end{array}\\right]$ $\\textbf{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-2", "-1", "-7" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Use the given LU factorization to find all solutions to $\\textrm{A}\\textbf{x}=\\textbf{b}$ $\\textrm{A}=\\left[\\begin{array}{cccc} 1 &0 &0 &0\\cr-8 &1 &0 &0\\cr 8 &-7 &1 &0\\cr-3 &8 &-3 &1 \\end{array}\\right] \\left[\\begin{array}{ccc}-6 &-3 &1\\cr 0 &-8 &3\\cr 0 &0 &-1\\cr 0 &0 &0 \\end{array}\\right]$, $\\textbf{b}=\\left[\\begin{array}{c}-24\\cr 222\\cr-396\\cr 294 \\end{array}\\right]$ $\\textbf{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "6", "-6", "-6" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Use the given LU factorization to find all solutions to $\\textrm{A}\\textbf{x}=\\textbf{b}$ $\\textrm{A}=\\left[\\begin{array}{cccc} 1 &0 &0 &0\\cr-4 &1 &0 &0\\cr 2 &-4 &1 &0\\cr 1 &-6 &-3 &1 \\end{array}\\right] \\left[\\begin{array}{ccc} 6 &8 &7\\cr 0 &-6 &-4\\cr 0 &0 &-5\\cr 0 &0 &0 \\end{array}\\right]$, $\\textbf{b}=\\left[\\begin{array}{c} 17\\cr-110\\cr 157\\cr 404 \\end{array}\\right]$ $\\textbf{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-9", "1", "9" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0235", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "LU factorization", "level": "3", "keywords": [], "problem_v1": "If L and U are invertible, then $(LU)^{−1}$=$U^{−1}L^{−1}$. Find $A^{−1}$ from the given LU factorization: $A=LU=\\left[\\begin{array}{cc} 2 &8\\cr 10 &42 \\end{array}\\right]=\\left[\\begin{array}{cc} 1 &0\\cr 5 &1 \\end{array}\\right]\\left[\\begin{array}{cc} 2 &8\\cr 0 &2 \\end{array}\\right]$\n$A^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "0.5", "-2", "0", "0.5", "1", "0", "-5", "1", "10.5", "-2", "-2.5", "0.5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "If L and U are invertible, then $(LU)^{−1}$=$U^{−1}L^{−1}$. Find $A^{−1}$ from the given LU factorization: $A=LU=\\left[\\begin{array}{cc} 8 &-24\\cr-64 &185 \\end{array}\\right]=\\left[\\begin{array}{cc} 1 &0\\cr-8 &1 \\end{array}\\right]\\left[\\begin{array}{cc} 8 &-24\\cr 0 &-7 \\end{array}\\right]$\n$A^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "0.125", "-0.428571", "0", "-0.142857", "1", "0", "8", "1", "-3.30357", "-0.428571", "-1.14286", "-0.142857" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "If L and U are invertible, then $(LU)^{−1}$=$U^{−1}L^{−1}$. Find $A^{−1}$ from the given LU factorization: $A=LU=\\left[\\begin{array}{cc} 2 &2\\cr-8 &-12 \\end{array}\\right]=\\left[\\begin{array}{cc} 1 &0\\cr-4 &1 \\end{array}\\right]\\left[\\begin{array}{cc} 2 &2\\cr 0 &-4 \\end{array}\\right]$\n$A^{-1}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "0.5", "0.25", "0", "-0.25", "1", "0", "4", "1", "1.5", "0.25", "-1", "-0.25" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0237", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "LU factorization", "level": "3", "keywords": [], "problem_v1": "Find the $LU$ factorization of A=\\left[\\begin{array}{cc} 1 &1\\cr 3 &5 \\end{array}\\right] and use it to solve the system \\left[\\begin{array}{cc} 1 &1\\cr 3 &5 \\end{array}\\right] \\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]=\\left[\\begin{array}{c}-4\\cr-16 \\end{array}\\right].\n$A=LU=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "1", "0", "3", "1", "1", "1", "0", "2", "-2", "-2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the $LU$ factorization of A=\\left[\\begin{array}{cc} 5 &-4\\cr-25 &18 \\end{array}\\right] and use it to solve the system \\left[\\begin{array}{cc} 5 &-4\\cr-25 &18 \\end{array}\\right] \\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]=\\left[\\begin{array}{c} 33\\cr-161 \\end{array}\\right].\n$A=LU=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "1", "0", "-5", "1", "5", "-4", "0", "-2", "5", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the $LU$ factorization of A=\\left[\\begin{array}{cc} 1 &-2\\cr-2 &5 \\end{array}\\right] and use it to solve the system \\left[\\begin{array}{cc} 1 &-2\\cr-2 &5 \\end{array}\\right] \\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]=\\left[\\begin{array}{c} 1\\cr-4 \\end{array}\\right].\n$A=LU=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "1", "0", "-2", "1", "1", "-2", "0", "1", "-3", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0238", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "LU factorization", "level": "3", "keywords": [], "problem_v1": "Find the $LDU$ factorization of A=\\left[\\begin{array}{cc} 1 &2\\cr 3 &7 \\end{array}\\right]. That is, write $A=LDU$ where $L$ is a lower triangular matrix with ones on the diagonal, $D$ is a diagonal matrix, and $U$ is an upper triangular matrix with ones on the diagonal.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "0", "3", "1", "1", "0", "0", "1", "1", "2", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the $LDU$ factorization of A=\\left[\\begin{array}{cc} 5 &-10\\cr-25 &46 \\end{array}\\right]. That is, write $A=LDU$ where $L$ is a lower triangular matrix with ones on the diagonal, $D$ is a diagonal matrix, and $U$ is an upper triangular matrix with ones on the diagonal.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "-5", "1", "5", "0", "0", "-4", "1", "-2", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the $LDU$ factorization of A=\\left[\\begin{array}{cc} 1 &1\\cr-2 &-4 \\end{array}\\right]. That is, write $A=LDU$ where $L$ is a lower triangular matrix with ones on the diagonal, $D$ is a diagonal matrix, and $U$ is an upper triangular matrix with ones on the diagonal.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "0", "-2", "1", "1", "0", "0", "-2", "1", "1", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0239", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "QR factorization", "level": "3", "keywords": [], "problem_v1": "Find the $QR$ factorization of the given matrix.\n$\\left[\\begin{array}{ccc} 2 &-1 &1\\cr 2 &3 &-1\\cr 2 &-1 &-3\\cr 2 &3 &-5 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0.5", "-0.5", "0.5", "0.5", "0.5", "0.5", "0.5", "-0.5", "-0.5", "0.5", "0.5", "-0.5", "4", "2", "-4", "0", "4", "-2", "0", "0", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the $QR$ factorization of the given matrix.\n$\\left[\\begin{array}{ccc}-2 &5 &2\\cr 2 &-1 &2\\cr 2 &-1 &4\\cr 2 &-5 &0 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-0.5", "0.5", "0.5", "0.5", "0.5", "-0.5", "0.5", "0.5", "0.5", "0.5", "-0.5", "0.5", "4", "-6", "2", "0", "4", "4", "0", "0", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the $QR$ factorization of the given matrix.\n$\\left[\\begin{array}{ccc} 3 &-1 &-2\\cr 3 &-3 &0\\cr-3 &3 &-4\\cr 3 &-1 &-6 \\end{array}\\right]=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array} \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "0.5", "0.5", "0.5", "0.5", "-0.5", "-0.5", "-0.5", "0.5", "-0.5", "0.5", "0.5", "-0.5", "6", "-4", "-2", "0", "2", "-6", "0", "0", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0240", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "QR factorization", "level": "3", "keywords": [ "orthogonal' 'factorization" ], "problem_v1": "All vectors and subspaces are in ${\\mathbb R}^n$.\nCheck the true statements below: [ANS] A\\. In a $QR$ factorization, say $A=QR$ (when $A$ has linearly independent columns), the columns of $Q$ form an orthonormal basis for the column space of $A$. B\\. If $W=Span \\{x_{1},x_{2},x_{3}\\}$ with $\\{x_{1},x_{2},x_{3}\\}$ linearly independent, and if $\\{v_{1},v_{2},v_{3}\\}$ is an orthogonal set in $W$, then $\\{v_{1},v_{2},v_{3}\\}$ is an orthogonal basis for $W$. C\\. If $x$ is not in a subspace $W$, then $x-{\\rm proj}_W(x)$ is not zero.", "answer_v1": [ "AC" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C" ] ], "problem_v2": "All vectors and subspaces are in ${\\mathbb R}^n$.\nCheck the true statements below: [ANS] A\\. If $x$ is not in a subspace $W$, then $x-{\\rm proj}_W(x)$ is not zero. B\\. In a $QR$ factorization, say $A=QR$ (when $A$ has linearly independent columns), the columns of $Q$ form an orthonormal basis for the column space of $A$. C\\. If $W=Span \\{x_{1},x_{2},x_{3}\\}$ with $\\{x_{1},x_{2},x_{3}\\}$ linearly independent, and if $\\{v_{1},v_{2},v_{3}\\}$ is an orthogonal set in $W$, then $\\{v_{1},v_{2},v_{3}\\}$ is an orthogonal basis for $W$.", "answer_v2": [ "AB" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C" ] ], "problem_v3": "All vectors and subspaces are in ${\\mathbb R}^n$.\nCheck the true statements below: [ANS] A\\. If $W=Span \\{x_{1},x_{2},x_{3}\\}$ with $\\{x_{1},x_{2},x_{3}\\}$ linearly independent, and if $\\{v_{1},v_{2},v_{3}\\}$ is an orthogonal set in $W$, then $\\{v_{1},v_{2},v_{3}\\}$ is an orthogonal basis for $W$. B\\. If $x$ is not in a subspace $W$, then $x-{\\rm proj}_W(x)$ is not zero. C\\. In a $QR$ factorization, say $A=QR$ (when $A$ has linearly independent columns), the columns of $Q$ form an orthonormal basis for the column space of $A$.", "answer_v3": [ "BC" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C" ] ] }, { "id": "Linear_algebra_0241", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Singular value decomposition", "level": "3", "keywords": [], "problem_v1": "Let A=\\left[\\begin{array}{cc}-2 &14\\cr 14 &2\\cr-2 &14\\cr 14 &2 \\end{array}\\right]. A singular value decomposition of $A$ is A=\\left[\\begin{array}{cccc}-0.5 &0.5 &0.5 &-0.5\\cr 0.5 &0.5 &-0.5 &-0.5\\cr-0.5 &0.5 &-0.5 &0.5\\cr 0.5 &0.5 &0.5 &0.5 \\end{array}\\right] \\left[\\begin{array}{cc} 20 &0\\cr 0 &20\\cr 0 &0\\cr 0 &0 \\end{array}\\right] \\left[\\begin{array}{cc} 0.8 &-0.6\\cr 0.6 &0.8 \\end{array}\\right]. Find the least-squares solution of the linear system $A\\vec{x}=\\vec{b}$ where \\vec{b}=\\left[\\begin{array}{c}-1\\cr-1\\cr-1\\cr-4 \\end{array}\\right]. $\\vec{x}^*=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-0.165", "-0.095" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{cc} 15 &-5\\cr 15 &-5\\cr-9 &13\\cr-9 &13 \\end{array}\\right]. A singular value decomposition of $A$ is A=\\left[\\begin{array}{cccc} 0.5 &0.5 &0.5 &0.5\\cr 0.5 &0.5 &-0.5 &-0.5\\cr-0.5 &0.5 &0.5 &-0.5\\cr-0.5 &0.5 &-0.5 &0.5 \\end{array}\\right] \\left[\\begin{array}{cc} 30 &0\\cr 0 &10\\cr 0 &0\\cr 0 &0 \\end{array}\\right] \\left[\\begin{array}{cc} 0.8 &-0.6\\cr 0.6 &0.8 \\end{array}\\right]. Find the least-squares solution of the linear system $A\\vec{x}=\\vec{b}$ where \\vec{b}=\\left[\\begin{array}{c} 3\\cr-3\\cr-3\\cr-3 \\end{array}\\right]. $\\vec{x}^*=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-0.1", "-0.3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{cc}-19 &8\\cr-13 &16\\cr-13 &16\\cr-19 &8 \\end{array}\\right]. A singular value decomposition of $A$ is A=\\left[\\begin{array}{cccc} 0.5 &-0.5 &-0.5 &0.5\\cr 0.5 &0.5 &-0.5 &-0.5\\cr 0.5 &0.5 &0.5 &0.5\\cr 0.5 &-0.5 &0.5 &-0.5 \\end{array}\\right] \\left[\\begin{array}{cc} 40 &0\\cr 0 &10\\cr 0 &0\\cr 0 &0 \\end{array}\\right] \\left[\\begin{array}{cc}-0.8 &0.6\\cr 0.6 &0.8 \\end{array}\\right]. Find the least-squares solution of the linear system $A\\vec{x}=\\vec{b}$ where \\vec{b}=\\left[\\begin{array}{c}-5\\cr 1\\cr 5\\cr 3 \\end{array}\\right]. $\\vec{x}^*=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "0.2", "0.35" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0242", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Singular value decomposition", "level": "3", "keywords": [], "problem_v1": "Find the singular values $\\sigma_1 \\ge \\sigma_2$ of A=\\left[\\begin{array}{cc} 1 &1\\cr-3 &3 \\end{array}\\right]. $\\sigma_1=$ [ANS], $\\sigma_2=$ [ANS].\nFind unit vectors $\\vec{v}_1$ and $\\vec{v}_2$ such that $\\| A\\vec{v}_1 \\|=\\sigma_1$ and $\\| A\\vec{v}_2 \\|=\\sigma_2$.\n$\\vec{v}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\n$\\vec{v}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-0.707107", "0.707107", "-0.707107", "-0.707107" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the singular values $\\sigma_1 \\ge \\sigma_2$ of A=\\left[\\begin{array}{cc}-2 &1\\cr 1 &-2 \\end{array}\\right]. $\\sigma_1=$ [ANS], $\\sigma_2=$ [ANS].\nFind unit vectors $\\vec{v}_1$ and $\\vec{v}_2$ such that $\\| A\\vec{v}_1 \\|=\\sigma_1$ and $\\| A\\vec{v}_2 \\|=\\sigma_2$.\n$\\vec{v}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\n$\\vec{v}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-0.707107", "0.707107", "-0.707107", "-0.707107" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the singular values $\\sigma_1 \\ge \\sigma_2$ of A=\\left[\\begin{array}{cc}-1 &-2\\cr-2 &-1 \\end{array}\\right]. $\\sigma_1=$ [ANS], $\\sigma_2=$ [ANS].\nFind unit vectors $\\vec{v}_1$ and $\\vec{v}_2$ such that $\\| A\\vec{v}_1 \\|=\\sigma_1$ and $\\| A\\vec{v}_2 \\|=\\sigma_2$.\n$\\vec{v}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\n$\\vec{v}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "0.707107", "0.707107", "0.707107", "-0.707107" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0243", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Jordan form", "level": "3", "keywords": [ "linear algebra", "similar" ], "problem_v1": "Let A=\\left[\\begin{array}{ccc} 2 &0 &1\\cr-1 &1 &0\\cr-1 &0 &0 \\end{array}\\right]. Find the Jordan canonical form of $A$, where the blocks are ordered increasingly by eigenvalue and then by block size. $J=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "1", "0", "0", "1", "1", "0", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{ccc}-3 &2 &1\\cr 0 &-2 &0\\cr-1 &1 &-1 \\end{array}\\right]. Find the Jordan canonical form of $A$, where the blocks are ordered increasingly by eigenvalue and then by block size. $J=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-2", "1", "0", "0", "-2", "1", "0", "0", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{ccc}-2 &-2 &-1\\cr-1 &-5 &-3\\cr 2 &7 &4 \\end{array}\\right]. Find the Jordan canonical form of $A$, where the blocks are ordered increasingly by eigenvalue and then by block size. $J=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-1", "1", "0", "0", "-1", "1", "0", "0", "-1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0244", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Jordan form", "level": "2", "keywords": [ "" ], "problem_v1": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ where A=\\left[\\begin{array}{cccc}-1 &-1 &1 &-5\\cr-1 &0 &-1 &4\\cr 0 &0 &-1 &0\\cr 1 &1 &1 &2 \\end{array}\\right]. Find a matrix $B$ whose row space is smallest $L_A$-invariant subspace that contains the vector $\\left(1,-1,0,0\\right)$. $B=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "-1", "0", "0", "0", "-1", "0", "0", "1", "0", "0", "-1", "0", "0", "0", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ where A=\\left[\\begin{array}{cccc}-1 &-2 &0 &0\\cr-1 &3 &7 &-1\\cr 0 &-1 &-1 &0\\cr-2 &3 &8 &-1 \\end{array}\\right]. Find a matrix $B$ whose row space is smallest $L_A$-invariant subspace that contains the vector $\\left(0,0,0,1\\right)$. $B=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "0", "0", "0", "1", "0", "-1", "0", "-1", "2", "-2", "1", "-2", "0", "0", "0", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ where A=\\left[\\begin{array}{cccc} 15 &8 &-18 &4\\cr 0 &-1 &1 &0\\cr 9 &4 &-10 &3\\cr-4 &-3 &5 &0 \\end{array}\\right]. Find a matrix $B$ whose row space is smallest $L_A$-invariant subspace that contains the vector $\\left(0,2,1,1\\right)$. $B=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "0", "2", "1", "1", "2", "-1", "1", "-1", "0", "2", "1", "0", "0", "0", "0", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0245", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Jordan form", "level": "2", "keywords": [ "linear algebra", "transition matrix" ], "problem_v1": "Find the minimal polynomial $m(x)$ of $\\left[\\begin{array}{ccccc} 5 &1 &0 &0 &0\\cr 0 &5 &0 &0 &0\\cr 0 &0 &5 &0 &0\\cr 0 &0 &0 &2 &0\\cr 0 &0 &0 &0 &2 \\end{array}\\right]$. $m(x)=$ [ANS]", "answer_v1": [ "(x-5)*(x-5)*(x-2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the minimal polynomial $m(x)$ of $\\left[\\begin{array}{ccccc}-8 &1 &0 &0 &0\\cr 0 &-8 &0 &0 &0\\cr 0 &0 &-8 &0 &0\\cr 0 &0 &0 &8 &0\\cr 0 &0 &0 &0 &8 \\end{array}\\right]$. $m(x)=$ [ANS]", "answer_v2": [ "[x-(-8)]*[x-(-8)]*(x-8)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the minimal polynomial $m(x)$ of $\\left[\\begin{array}{ccccc}-4 &1 &0 &0 &0\\cr 0 &-4 &0 &0 &0\\cr 0 &0 &-4 &0 &0\\cr 0 &0 &0 &2 &0\\cr 0 &0 &0 &0 &2 \\end{array}\\right]$. $m(x)=$ [ANS]", "answer_v3": [ "[x-(-4)]*[x-(-4)]*(x-2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0246", "subject": "Linear_algebra", "topic": "Matrix factorizations", "subtopic": "Jordan form", "level": "4", "keywords": [], "problem_v1": "Let $\\lambda$ be an eigenvalue of the linear operator $L$ and define $L_\\lambda:=L-\\lambda I$. The following table lists the nullities of the powers of $L_\\lambda$.\n$\\begin{array}{cccccccc}\\hline k & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline \\text{nullity}(L_\\lambda^k) & 6 & 11 & 16 & 20 & 23 & 26 & 28 \\\\ \\hline \\end{array}$ Find the sizes of the Jordan blocks corresponding to $\\lambda$ of the Jordan form of the matrix of $L$ as a list of integers. Sizes: [ANS]", "answer_v1": [ "(1, 3, 4, 6, 7, 7)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Let $\\lambda$ be an eigenvalue of the linear operator $L$ and define $L_\\lambda:=L-\\lambda I$. The following table lists the nullities of the powers of $L_\\lambda$.\n$\\begin{array}{cccccc}\\hline k & 1 & 2 & 3 & 4 & 5 \\\\ \\hline \\text{nullity}(L_\\lambda^k) & 6 & 10 & 14 & 17 & 18 \\\\ \\hline \\end{array}$ Find the sizes of the Jordan blocks corresponding to $\\lambda$ of the Jordan form of the matrix of $L$ as a list of integers. Sizes: [ANS]", "answer_v2": [ "(1, 1, 3, 4, 4, 5)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Let $\\lambda$ be an eigenvalue of the linear operator $L$ and define $L_\\lambda:=L-\\lambda I$. The following table lists the nullities of the powers of $L_\\lambda$.\n$\\begin{array}{ccccc}\\hline k & 1 & 2 & 3 & 4 \\\\ \\hline \\text{nullity}(L_\\lambda^k) & 6 & 10 & 13 & 15 \\\\ \\hline \\end{array}$ Find the sizes of the Jordan blocks corresponding to $\\lambda$ of the Jordan form of the matrix of $L$ as a list of integers. Sizes: [ANS]", "answer_v3": [ "(1, 1, 2, 3, 4, 4)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0248", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Vectors", "level": "2", "keywords": [ "subspaces" ], "problem_v1": "Select the best statement. [ANS] A\\. The parallelogram rule for adding vectors is a visualization technique unsuited for computation. B\\. The parallelogram rule for adding vectors in a Euclidean space always works. C\\. The parallelogram rule for adding vectors only works with vectors in the first quadrant. D\\. The parallelogram rule for adding vectors only works if at least one of the vectors is in first quadrant. E\\. none of the above", "answer_v1": [ "B" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Select the best statement. [ANS] A\\. The parallelogram rule for adding vectors only works if at least one of the vectors is in first quadrant. B\\. The parallelogram rule for adding vectors only works with vectors in the first quadrant. C\\. The parallelogram rule for adding vectors is a visualization technique unsuited for computation. D\\. The parallelogram rule for adding vectors in a Euclidean space always works. E\\. none of the above", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Select the best statement. [ANS] A\\. The parallelogram rule for adding vectors in a Euclidean space always works. B\\. The parallelogram rule for adding vectors only works if at least one of the vectors is in first quadrant. C\\. The parallelogram rule for adding vectors only works with vectors in the first quadrant. D\\. The parallelogram rule for adding vectors is a visualization technique unsuited for computation. E\\. none of the above", "answer_v3": [ "A" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Linear_algebra_0249", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Vectors", "level": "2", "keywords": [ "Vector", "Operation" ], "problem_v1": "Let $x=(5, 2, 3)$ and $y=(5,-4,-4)$ be in $\\mathbb R^3$. Compute the following:\n$x+y$=([ANS], [ANS], [ANS])\n$2x=$ ([ANS], [ANS], [ANS])\n$x-y=$ ([ANS], [ANS], [ANS])\n2 $x+5 y$=([ANS], [ANS], [ANS])", "answer_v1": [ "10", "-2", "-1", "10", "4", "6", "0", "6", "7", "35", "-16", "-14" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $x=(-9, 9,-7)$ and $y=(-3, 9,-4)$ be in $\\mathbb R^3$. Compute the following:\n$x+y$=([ANS], [ANS], [ANS])\n$-7x=$ ([ANS], [ANS], [ANS])\n$x-y=$ ([ANS], [ANS], [ANS])-7 $x+4 y$=([ANS], [ANS], [ANS])", "answer_v2": [ "-12", "18", "-11", "63", "-63", "49", "-6", "0", "-3", "51", "-27", "33" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $x=(-4, 2,-5)$ and $y=(1,-6,-3)$ be in $\\mathbb R^3$. Compute the following:\n$x+y$=([ANS], [ANS], [ANS])\n$6x=$ ([ANS], [ANS], [ANS])\n$x-y=$ ([ANS], [ANS], [ANS])\n6 $x+10 y$=([ANS], [ANS], [ANS])", "answer_v3": [ "-3", "-4", "-8", "-24", "12", "-30", "-5", "8", "-2", "-14", "-48", "-60" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0250", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Vectors", "level": "3", "keywords": [ "Vector", "Operation" ], "problem_v1": "Let $x=(5, 2, 3)$ be in $\\mathbb R ^3$ and $a$ be in $\\mathbb R$. Then $a x$=([ANS], 10, [ANS]).", "answer_v1": [ "25", "15" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $x=(-9, 9,-7)$ be in $\\mathbb R ^3$ and $a$ be in $\\mathbb R$. Then $a x$=([ANS],-27, [ANS]).", "answer_v2": [ "27", "21" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $x=(-4, 2,-5)$ be in $\\mathbb R ^3$ and $a$ be in $\\mathbb R$. Then $a x$=([ANS],-12, [ANS]).", "answer_v3": [ "24", "30" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0251", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "2", "keywords": [ "subspaces" ], "problem_v1": "Let ${\\bf a}_1=\\left[{\\begin{matrix}3 \\cr 2 \\cr-3 \\cr\\end{matrix}}\\right]$, ${\\bf a}_2=\\left[{\\begin{matrix}-12 \\cr-6 \\cr 9 \\cr\\end{matrix}}\\right]$, and ${\\bf b}=\\left[{\\begin{matrix}-24 \\cr-10 \\cr 15 \\cr\\end{matrix}}\\right]$. Is ${\\bf b}$ a linear combination of ${\\bf a}_1$ and ${\\bf a}_2$? [ANS] A\\. Yes ${\\bf b}$ is a linear combination. B\\. ${\\bf b}$ is not a linear combination. C\\. We cannot tell if ${\\bf b}$ is a linear combination.\nEither fill in the coefficients of the vector equation, or enter \"NONE\" if no solution is possible. ${\\bf b}=$ [ANS] ${\\bf a}_1+$ [ANS] ${\\bf a}_2$", "answer_v1": [ "A", "4", "3" ], "answer_type_v1": [ "MCS", "NV", "NV" ], "options_v1": [ [ "A", "B", "C" ], [], [] ], "problem_v2": "Let ${\\bf a}_1=\\left[{\\begin{matrix}1 \\cr 3 \\cr-1 \\cr\\end{matrix}}\\right]$, ${\\bf a}_2=\\left[{\\begin{matrix}2 \\cr 4 \\cr 0 \\cr\\end{matrix}}\\right]$, and ${\\bf b}=\\left[{\\begin{matrix}-9 \\cr-17 \\cr-1 \\cr\\end{matrix}}\\right]$. Is ${\\bf b}$ a linear combination of ${\\bf a}_1$ and ${\\bf a}_2$? [ANS] A\\. Yes ${\\bf b}$ is a linear combination. B\\. ${\\bf b}$ is not a linear combination. C\\. We cannot tell if ${\\bf b}$ is a linear combination.\nEither fill in the coefficients of the vector equation, or enter \"NONE\" if no solution is possible. ${\\bf b}=$ [ANS] ${\\bf a}_1+$ [ANS] ${\\bf a}_2$", "answer_v2": [ "A", "1", "-5" ], "answer_type_v2": [ "MCS", "NV", "NV" ], "options_v2": [ [ "A", "B", "C" ], [], [] ], "problem_v3": "Let ${\\bf a}_1=\\left[{\\begin{matrix}-2 \\cr-2 \\cr 3 \\cr\\end{matrix}}\\right]$, ${\\bf a}_2=\\left[{\\begin{matrix}6 \\cr 8 \\cr-4 \\cr\\end{matrix}}\\right]$, and ${\\bf b}=\\left[{\\begin{matrix}-28 \\cr-36 \\cr 22 \\cr\\end{matrix}}\\right]$. Is ${\\bf b}$ a linear combination of ${\\bf a}_1$ and ${\\bf a}_2$? [ANS] A\\. ${\\bf b}$ is not a linear combination. B\\. Yes ${\\bf b}$ is a linear combination. C\\. We cannot tell if ${\\bf b}$ is a linear combination.\nEither fill in the coefficients of the vector equation, or enter \"NONE\" if no solution is possible. ${\\bf b}=$ [ANS] ${\\bf a}_1+$ [ANS] ${\\bf a}_2$", "answer_v3": [ "B", "2", "-4" ], "answer_type_v3": [ "MCS", "NV", "NV" ], "options_v3": [ [ "A", "B", "C" ], [], [] ] }, { "id": "Linear_algebra_0252", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "3", "keywords": [ "linear algebra", "vectors" ], "problem_v1": "Determine the center of mass for the vectors ${\\bf u}_1=(7, 2, 2)$ with weight $4$ kg. ${\\bf u}_2=(4, 3,-3)$ with weight $2$ kg. ${\\bf u}_3=(6,-2, 5)$ with weight $4$ kg. The center of mass is at ([ANS], [ANS], [ANS]).", "answer_v1": [ "6", "0.6", "2.2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Determine the center of mass for the vectors ${\\bf u}_1=(1, 8,-7)$ with weight $4$ kg. ${\\bf u}_2=(-3, 9,-3)$ with weight $3$ kg. ${\\bf u}_3=(-2, 1, 1)$ with weight $5$ kg. The center of mass is at ([ANS], [ANS], [ANS]).", "answer_v2": [ "-1.25", "5.33333333333333", "-2.66666666666667" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Determine the center of mass for the vectors ${\\bf u}_1=(3, 2,-4)$ with weight $2$ kg. ${\\bf u}_2=(1, 2,-3)$ with weight $2$ kg. ${\\bf u}_3=(8, 7, 2)$ with weight $1$ kg. The center of mass is at ([ANS], [ANS], [ANS]).", "answer_v3": [ "3.2", "3", "-2.4" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0253", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "2", "keywords": [], "problem_v1": "Let $\\vec{x}=\\left[\\begin{array}{c} 3\\cr 1 \\end{array}\\right]$ and $\\vec{y}=\\left[\\begin{array}{c} 2\\cr 3 \\end{array}\\right]$. Find the vector $\\vec{v}=3 \\vec{x}-6 \\vec{y}$ and its additive inverse.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $-\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-3", "-15", "3", "15" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $\\vec{x}=\\left[\\begin{array}{c}-5\\cr 6 \\end{array}\\right]$ and $\\vec{y}=\\left[\\begin{array}{c}-5\\cr-2 \\end{array}\\right]$. Find the vector $\\vec{v}=7 \\vec{x}-6 \\vec{y}$ and its additive inverse.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $-\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-5", "54", "5", "-54" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $\\vec{x}=\\left[\\begin{array}{c}-2\\cr 1 \\end{array}\\right]$ and $\\vec{y}=\\left[\\begin{array}{c}-3\\cr 1 \\end{array}\\right]$. Find the vector $\\vec{v}=3 \\vec{x}-5 \\vec{y}$ and its additive inverse.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $-\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "9", "-2", "-9", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0254", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "2", "keywords": [], "problem_v1": "Let $\\vec{x}=\\left[\\begin{array}{c} 3\\cr 1\\cr 2 \\end{array}\\right]$ and $y=\\left[\\begin{array}{c} 0\\cr-3\\cr-2 \\end{array}\\right]$. Find the vectors $\\vec{v}=5 \\vec{x}$, $\\vec{u}=\\vec{x}+\\vec{y}$, and $\\vec{w}=5 \\vec{x}+\\vec{y}$.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\vec{w}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "15", "5", "10", "3", "-2", "0", "15", "2", "8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $\\vec{x}=\\left[\\begin{array}{c}-5\\cr 0\\cr-5 \\end{array}\\right]$ and $y=\\left[\\begin{array}{c}-2\\cr 6\\cr-2 \\end{array}\\right]$. Find the vectors $\\vec{v}=3 \\vec{x}$, $\\vec{u}=\\vec{x}+\\vec{y}$, and $\\vec{w}=3 \\vec{x}+\\vec{y}$.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\vec{w}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-15", "0", "-15", "-7", "6", "-7", "-17", "6", "-17" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $\\vec{x}=\\left[\\begin{array}{c}-2\\cr 1\\cr-3 \\end{array}\\right]$ and $y=\\left[\\begin{array}{c} 1\\cr 0\\cr-2 \\end{array}\\right]$. Find the vectors $\\vec{v}=7 \\vec{x}$, $\\vec{u}=\\vec{x}+\\vec{y}$, and $\\vec{w}=7 \\vec{x}+\\vec{y}$.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\vec{w}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-14", "7", "-21", "-1", "1", "-5", "-13", "7", "-23" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0255", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "3", "keywords": [], "problem_v1": "Express the vector $\\vec{v}=\\left[\\begin{array}{c}-14\\cr-9 \\end{array}\\right]$ as a linear combination of $\\vec{x}=\\left[\\begin{array}{c} 4\\cr 1 \\end{array}\\right]$ and $\\vec{y}=\\left[\\begin{array}{c} 1\\cr 3 \\end{array}\\right]$.\n$\\vec{v}=$ [ANS] $\\vec{x}+$ [ANS] $\\vec{y}$.", "answer_v1": [ "-3", "-2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Express the vector $\\vec{v}=\\left[\\begin{array}{c}-26\\cr 34 \\end{array}\\right]$ as a linear combination of $\\vec{x}=\\left[\\begin{array}{c}-6\\cr 5 \\end{array}\\right]$ and $\\vec{y}=\\left[\\begin{array}{c}-5\\cr-2 \\end{array}\\right]$.\n$\\vec{v}=$ [ANS] $\\vec{x}+$ [ANS] $\\vec{y}$.", "answer_v2": [ "6", "-2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Express the vector $\\vec{v}=\\left[\\begin{array}{c} 14\\cr-6 \\end{array}\\right]$ as a linear combination of $\\vec{x}=\\left[\\begin{array}{c}-2\\cr 1 \\end{array}\\right]$ and $\\vec{y}=\\left[\\begin{array}{c}-3\\cr 1 \\end{array}\\right]$.\n$\\vec{v}=$ [ANS] $\\vec{x}+$ [ANS] $\\vec{y}$.", "answer_v3": [ "-4", "-2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0256", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "2", "keywords": [ "linear algebra", "vector space", "subspaces" ], "problem_v1": "Let \\vec{w}=\\left<2,3,-2\\right>, \\ \\vec{v}_1=\\left<1,1,-1\\right>, \\ \\vec{v}_2=\\left<-1,0,1\\right>. If possible, express $\\vec{w}$ as a linear combination of the vectors $\\vec{v}_1$ and $\\vec{v}_2$. Otherwise, enter DNE. For example, the answer $\\vec{w}=4 \\vec{v}_1+5 \\vec{v}_2$ would be entered $\\verb!4v1+5v2!$.\n$\\vec{w}=$ [ANS]", "answer_v1": [ "3*v1+1*v2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let \\vec{w}=\\left<20,5,-10\\right>, \\ \\vec{v}_1=\\left<-2,-1,1\\right>, \\ \\vec{v}_2=\\left<2,0,-1\\right>. If possible, express $\\vec{w}$ as a linear combination of the vectors $\\vec{v}_1$ and $\\vec{v}_2$. Otherwise, enter DNE. For example, the answer $\\vec{w}=4 \\vec{v}_1+5 \\vec{v}_2$ would be entered $\\verb!4v1+5v2!$.\n$\\vec{w}=$ [ANS]", "answer_v2": [ "-5*v1+5*v2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let \\vec{w}=\\left<3,-2,3\\right>, \\ \\vec{v}_1=\\left<-1,1,-1\\right>, \\ \\vec{v}_2=\\left<1,0,1\\right>. If possible, express $\\vec{w}$ as a linear combination of the vectors $\\vec{v}_1$ and $\\vec{v}_2$. Otherwise, enter DNE. For example, the answer $\\vec{w}=4 \\vec{v}_1+5 \\vec{v}_2$ would be entered $\\verb!4v1+5v2!$.\n$\\vec{w}=$ [ANS]", "answer_v3": [ "-2*v1+1*v2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0259", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "2", "keywords": [ "vector' 'line' 'parametric" ], "problem_v1": "Let $L$ be the line $y=6x+2$ in $\\mathbb{R}^2$. Find vectors $\\vec{a}$ and $\\vec{b}$ so that $\\vec{v}=\\vec{a}+t \\vec{b}$ is a parametric equation for $L$.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "0", "2", "1", "6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $L$ be the line $y=-\\left(9x+7\\right)$ in $\\mathbb{R}^2$. Find vectors $\\vec{a}$ and $\\vec{b}$ so that $\\vec{v}=\\vec{a}+t \\vec{b}$ is a parametric equation for $L$.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "0", "-7", "1", "-9" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $L$ be the line $y=-\\left(6x+4\\right)$ in $\\mathbb{R}^2$. Find vectors $\\vec{a}$ and $\\vec{b}$ so that $\\vec{v}=\\vec{a}+t \\vec{b}$ is a parametric equation for $L$.\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "0", "-4", "1", "-6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0260", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "", "keywords": [ "vector' 'line" ], "problem_v1": "Find the vector equation of a line $L$ going through the point $P=(3,1)$ and parallel to the line generated by multiples of the vector $v=\\left[\\begin{array}{c} 1\\cr 2 \\end{array}\\right]$.\n$L(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "3", "1", "1", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the vector equation of a line $L$ going through the point $P=(-5,6)$ and parallel to the line generated by multiples of the vector $v=\\left[\\begin{array}{c}-4\\cr-2 \\end{array}\\right]$.\n$L(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-5", "6", "-4", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the vector equation of a line $L$ going through the point $P=(-4,-2)$ and parallel to the line generated by multiples of the vector $v=\\left[\\begin{array}{c} 3\\cr 5 \\end{array}\\right]$.\n$L(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $+t$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "-4", "-2", "3", "5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0261", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Linear combinations", "level": "2", "keywords": [ "vector' 'line" ], "problem_v1": "Let $L$ be the line defined by the vector equation $\\vec{v}=\\left[\\begin{array}{c} 3\\cr 1 \\end{array}\\right]+t \\left[\\begin{array}{c} 2\\cr 4 \\end{array}\\right].$ Find the equation $y=mx+b$ for $L$.\n$y=$ [ANS]", "answer_v1": [ "2*x-5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $L$ be the line defined by the vector equation $\\vec{v}=\\left[\\begin{array}{c}-5\\cr 5 \\end{array}\\right]+t \\left[\\begin{array}{c}-5\\cr-2 \\end{array}\\right].$ Find the equation $y=mx+b$ for $L$.\n$y=$ [ANS]", "answer_v2": [ "0.4*x+7" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $L$ be the line defined by the vector equation $\\vec{v}=\\left[\\begin{array}{c}-2\\cr 1 \\end{array}\\right]+t \\left[\\begin{array}{c}-3\\cr-4 \\end{array}\\right].$ Find the equation $y=mx+b$ for $L$.\n$y=$ [ANS]", "answer_v3": [ "1.33333*x+3.66667" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0262", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Span", "level": "3", "keywords": [ "subspaces" ], "problem_v1": "Let $A=\\left[{\\begin{matrix}4 & 16 & 16 \\cr-2 &-12 & 0 \\cr-3 &-14 &-8 \\cr \\end{matrix}}\\right]$. We want to determine if the columns of $A$ span ${\\mathbb R}^3$. To check this we add [ANS] times the first row to the second. We then add [ANS] times the first row to the third. We then add [ANS] times the new second row to the new third row. We conclude that [ANS] A\\. The columns of $A$ do not span ${\\mathbb R}^3$. B\\. The columns of $A$ do span ${\\mathbb R}^3$. C\\. We cannot tell if the columns of $A$ span ${\\mathbb R}^3$.", "answer_v1": [ "0.5", "0.75", "-0.5", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C" ] ], "problem_v2": "Let $A=\\left[{\\begin{matrix}-2 &-2 & 2 \\cr-3 &-7 & 23 \\cr-5 &-6 & 10 \\cr \\end{matrix}}\\right]$. We want to determine if the columns of $A$ span ${\\mathbb R}^3$. To check this we add [ANS] times the first row to the second. We then add [ANS] times the first row to the third. We then add [ANS] times the new second row to the new third row. We conclude that [ANS] A\\. The columns of $A$ do not span ${\\mathbb R}^3$. B\\. The columns of $A$ do span ${\\mathbb R}^3$. C\\. We cannot tell if the columns of $A$ span ${\\mathbb R}^3$.", "answer_v2": [ "-1.5", "-2.5", "-0.25", "A" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C" ] ], "problem_v3": "Let $A=\\left[{\\begin{matrix}5 & 10 & 10 \\cr-5 &-12 &-6 \\cr 1 & 7 &-8 \\cr \\end{matrix}}\\right]$. We want to determine if the columns of $A$ span ${\\mathbb R}^3$. To check this we add [ANS] times the first row to the second. We then add [ANS] times the first row to the third. We then add [ANS] times the new second row to the new third row. We conclude that [ANS] A\\. The columns of $A$ do not span ${\\mathbb R}^3$. B\\. The columns of $A$ do span ${\\mathbb R}^3$. C\\. We cannot tell if the columns of $A$ span ${\\mathbb R}^3$.", "answer_v3": [ "1", "-0.2", "2.5", "A" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C" ] ] }, { "id": "Linear_algebra_0263", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Span", "level": "3", "keywords": [ "subspaces" ], "problem_v1": "Suppose a matrix $A$ has $n$ rows and $m$ columns. Select the best statement. (The best condition should work with any positive integer $n$.) [ANS] A\\. If $n 0\\}$ F\\. $\\{(2, y, z) \\ | \\ y, z$ arbitrary numbers $\\}$", "answer_v2": [ "ABD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of the following sets are subspaces of ${\\mathbb R}^3$? [ANS] A\\. $\\{(x, y, z) \\ | \\-4x+7 y=0, 3x+6 z=0\\}$ B\\. $\\{(x,y,z) \\ | \\ x+y+z=0\\}$ C\\. $\\{(x,y,z) \\ | \\ x, y, z > 0\\}$ D\\. $\\{(x, x+4, x-7) \\ | \\ x$ arbitrary number $\\}$ E\\. $\\{(6x, 2x,-8x) \\ | \\ x$ arbitrary number $\\}$ F\\. $\\{(x,y,z) \\ | \\-2x+8 y+9 z=-6\\}$", "answer_v3": [ "ABE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0290", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Basis and dimension", "level": "3", "keywords": [ "linear algebra", "subspaces", "basis" ], "problem_v1": "Find a basis of the given subspace by deleting linearly dependent vectors. span of $\\Bigg \\lbrace \\left[\\begin{array}{c}0\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}5\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}5\\\\-1\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\5\\\\5\\\\\\end{array}\\right] \\Bigg \\rbrace$ A basis is $\\Bigg \\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg \\rbrace$", "answer_v1": [ "5", "-1", "0", "1", "5", "5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find a basis of the given subspace by deleting linearly dependent vectors. span of $\\Bigg \\lbrace \\left[\\begin{array}{c}0\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}2\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}7\\\\-1\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\7\\\\7\\\\\\end{array}\\right] \\Bigg \\rbrace$ A basis is $\\Bigg \\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg \\rbrace$", "answer_v2": [ "7", "-1", "0", "1", "7", "7" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find a basis of the given subspace by deleting linearly dependent vectors. span of $\\Bigg \\lbrace \\left[\\begin{array}{c}0\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}3\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}5\\\\-1\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\5\\\\5\\\\\\end{array}\\right] \\Bigg \\rbrace$ A basis is $\\Bigg \\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg \\rbrace$", "answer_v3": [ "5", "-1", "0", "1", "5", "5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0291", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Basis and dimension", "level": "3", "keywords": [ "linear algebra", "subspaces", "basis", "nullity" ], "problem_v1": "Suppose that $S_1$ and $S_2$ are subspaces of $R^{6}$, with $dim(S_1)=m_1$ and $dim(S_2)=m_2$. If $S_1$ and $S_2$ have only $\\bf 0$ in common, then the maximum value of $m_1+m_2$ is [ANS]", "answer_v1": [ "6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $S_1$ and $S_2$ are subspaces of $R^{3}$, with $dim(S_1)=m_1$ and $dim(S_2)=m_2$. If $S_1$ and $S_2$ have only $\\bf 0$ in common, then the maximum value of $m_1+m_2$ is [ANS]", "answer_v2": [ "3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $S_1$ and $S_2$ are subspaces of $R^{4}$, with $dim(S_1)=m_1$ and $dim(S_2)=m_2$. If $S_1$ and $S_2$ have only $\\bf 0$ in common, then the maximum value of $m_1+m_2$ is [ANS]", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0292", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Basis and dimension", "level": "3", "keywords": [ "linear algebra", "subspaces", "basis" ], "problem_v1": "Suppose a subspace is spanned by the set $S$ of vectors shown. Find a subset of $\\mathbf S$ that forms a basis for the subspace, using the method of transforming a matrix to echelon form, where the columns of the matrix represent vectors spanning the subspace. $S=\\Bigg \\lbrace$ $\\left[\\begin{array}{c}1\\\\-3\\\\2\\\\\\end{array}\\right]$, $\\left[\\begin{array}{c}3\\\\-10\\\\8\\\\\\end{array}\\right]$, $\\left[\\begin{array}{c}-3\\\\10\\\\-8\\\\\\end{array}\\right]$ $\\Bigg \\rbrace$ A basis is $\\Bigg \\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg \\rbrace$ What is the dimension of the subspace? [ANS]", "answer_v1": [ "3", "-10", "8" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose a subspace is spanned by the set $S$ of vectors shown. Find a subset of $\\mathbf S$ that forms a basis for the subspace, using the method of transforming a matrix to echelon form, where the columns of the matrix represent vectors spanning the subspace. $S=\\Bigg \\lbrace$ $\\left[\\begin{array}{c}-2\\\\8\\\\-10\\\\\\end{array}\\right]$, $\\left[\\begin{array}{c}3\\\\-13\\\\20\\\\\\end{array}\\right]$, $\\left[\\begin{array}{c}-1\\\\5\\\\-10\\\\\\end{array}\\right]$ $\\Bigg \\rbrace$ A basis is $\\Bigg \\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg \\rbrace$ What is the dimension of the subspace? [ANS]", "answer_v2": [ "3", "-13", "20" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose a subspace is spanned by the set $S$ of vectors shown. Find a subset of $\\mathbf S$ that forms a basis for the subspace, using the method of transforming a matrix to echelon form, where the columns of the matrix represent vectors spanning the subspace. $S=\\Bigg \\lbrace$ $\\left[\\begin{array}{c}-1\\\\3\\\\-4\\\\\\end{array}\\right]$, $\\left[\\begin{array}{c}3\\\\-10\\\\16\\\\\\end{array}\\right]$, $\\left[\\begin{array}{c}-4\\\\13\\\\-20\\\\\\end{array}\\right]$ $\\Bigg \\rbrace$ A basis is $\\Bigg \\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg \\rbrace$ What is the dimension of the subspace? [ANS]", "answer_v3": [ "3", "-10", "16" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0293", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Basis and dimension", "level": "", "keywords": [], "problem_v1": "Find a linearly independent set of vectors that spans the same subspace of ${\\mathbb R}^4$ as that spanned by the vectors \\left[\\begin{array}{c} 1\\cr 0\\cr-1\\cr-1 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c}-5\\cr-4\\cr-1\\cr 5 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c} 1\\cr 2\\cr 0\\cr-2 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c} 2\\cr 2\\cr 1\\cr-2 \\end{array}\\right]. A linearly independent spanning set for the subspace is:\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v1": [ "2", "2", "1", "-2", "1", "0", "-1", "-1", "1", "0", "1", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find a linearly independent set of vectors that spans the same subspace of ${\\mathbb R}^4$ as that spanned by the vectors \\left[\\begin{array}{c}-3\\cr-1\\cr-2\\cr 1 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c} 0\\cr 3\\cr-1\\cr 2 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c} 6\\cr 2\\cr 7\\cr 0 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c} 6\\cr-1\\cr 5\\cr-4 \\end{array}\\right]. A linearly independent spanning set for the subspace is:\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v2": [ "-3", "-1", "-2", "1", "0", "3", "-1", "2", "0", "0", "-3", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find a linearly independent set of vectors that spans the same subspace of ${\\mathbb R}^4$ as that spanned by the vectors \\left[\\begin{array}{c}-3\\cr-2\\cr 1\\cr-1 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c} 1\\cr 0\\cr 3\\cr-2 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c} 0\\cr-2\\cr 2\\cr 1 \\end{array}\\right], \\ \\ \\ \\ \\left[\\begin{array}{c}-1\\cr-2\\cr 3\\cr-1 \\end{array}\\right]. A linearly independent spanning set for the subspace is:\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v3": [ "-1", "-2", "3", "-1", "1", "0", "3", "-2", "-2", "0", "-2", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0294", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Basis and dimension", "level": "", "keywords": [], "problem_v1": "The vectors \\vec{v}_1=\\left[\\begin{array}{c} 6\\cr-3\\cr 0 \\end{array}\\right], \\ \\ \\ \\ \\vec{v}_2=\\left[\\begin{array}{c}-3\\cr 6\\cr-3 \\end{array}\\right], \\ \\ \\ \\ \\vec{v}_3=\\left[\\begin{array}{c} 3\\cr 12\\cr k\\cr \\end{array}\\right] form a basis for ${\\mathbb R}^3$ if and only if $k\\ne$ [ANS].", "answer_v1": [ "-9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The vectors \\vec{v}_1=\\left[\\begin{array}{c} 1\\cr-1\\cr 0 \\end{array}\\right], \\ \\ \\ \\ \\vec{v}_2=\\left[\\begin{array}{c}-6\\cr 3\\cr 7 \\end{array}\\right], \\ \\ \\ \\ \\vec{v}_3=\\left[\\begin{array}{c}-4\\cr 1\\cr k\\cr \\end{array}\\right] form a basis for ${\\mathbb R}^3$ if and only if $k\\ne$ [ANS].", "answer_v2": [ "7" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The vectors \\vec{v}_1=\\left[\\begin{array}{c} 3\\cr-3\\cr 0 \\end{array}\\right], \\ \\ \\ \\ \\vec{v}_2=\\left[\\begin{array}{c}-6\\cr 4\\cr-4 \\end{array}\\right], \\ \\ \\ \\ \\vec{v}_3=\\left[\\begin{array}{c}-18\\cr 10\\cr k\\cr \\end{array}\\right] form a basis for ${\\mathbb R}^3$ if and only if $k\\ne$ [ANS].", "answer_v3": [ "-16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0298", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Basis and dimension", "level": "", "keywords": [ "bases' 'basis' 'matrix' 'linearly independent" ], "problem_v1": "Let $W_{1}$ be the set: $\\Bigg\\lbrace \\left[\\begin{array}{c}1\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\1\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\1\\\\1\\\\\\end{array}\\right] \\Bigg\\rbrace$ Determine if $W_{1}$ is a basis for ${\\mathbb R}^3$ and check the correct answer(s) below. [ANS] A\\. $W_{1}$ is not a basis because it is linearly dependent. B\\. $W_{1}$ is not a basis because it does not span ${\\mathbb R}^3$. C\\. $W_{1}$ is a basis.\nLet $W_{2}$ be the set: $\\Bigg\\lbrace \\left[\\begin{array}{c}1\\\\0\\\\1\\\\\\end{array}\\right], \\left[\\begin{array}{c}0\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}0\\\\1\\\\0\\\\\\end{array}\\right] \\Bigg\\rbrace$ Determine if $W_{2}$ is a basis for ${\\mathbb R}^3$ and check the correct answer(s) below. [ANS] A\\. $W_{2}$ is not a basis because it is linearly dependent. B\\. $W_{2}$ is a basis. C\\. $W_{2}$ is not a basis because it does not span ${\\mathbb R}^3$.", "answer_v1": [ "C", "AC" ], "answer_type_v1": [ "MCS", "MCM" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Let $W_{1}$ be the set: $\\Bigg\\lbrace \\left[\\begin{array}{c}1\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\1\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\1\\\\1\\\\\\end{array}\\right] \\Bigg\\rbrace$ Determine if $W_{1}$ is a basis for ${\\mathbb R}^3$ and check the correct answer(s) below. [ANS] A\\. $W_{1}$ is a basis. B\\. $W_{1}$ is not a basis because it does not span ${\\mathbb R}^3$. C\\. $W_{1}$ is not a basis because it is linearly dependent.\nLet $W_{2}$ be the set: $\\Bigg\\lbrace \\left[\\begin{array}{c}1\\\\0\\\\1\\\\\\end{array}\\right], \\left[\\begin{array}{c}0\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}0\\\\1\\\\0\\\\\\end{array}\\right] \\Bigg\\rbrace$ Determine if $W_{2}$ is a basis for ${\\mathbb R}^3$ and check the correct answer(s) below. [ANS] A\\. $W_{2}$ is not a basis because it is linearly dependent. B\\. $W_{2}$ is a basis. C\\. $W_{2}$ is not a basis because it does not span ${\\mathbb R}^3$.", "answer_v2": [ "A", "AC" ], "answer_type_v2": [ "MCS", "MCM" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Let $W_{1}$ be the set: $\\Bigg\\lbrace \\left[\\begin{array}{c}1\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\1\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}1\\\\1\\\\1\\\\\\end{array}\\right] \\Bigg\\rbrace$ Determine if $W_{1}$ is a basis for ${\\mathbb R}^3$ and check the correct answer(s) below. [ANS] A\\. $W_{1}$ is not a basis because it is linearly dependent. B\\. $W_{1}$ is a basis. C\\. $W_{1}$ is not a basis because it does not span ${\\mathbb R}^3$.\nLet $W_{2}$ be the set: $\\Bigg\\lbrace \\left[\\begin{array}{c}1\\\\0\\\\1\\\\\\end{array}\\right], \\left[\\begin{array}{c}0\\\\0\\\\0\\\\\\end{array}\\right], \\left[\\begin{array}{c}0\\\\1\\\\0\\\\\\end{array}\\right] \\Bigg\\rbrace$ Determine if $W_{2}$ is a basis for ${\\mathbb R}^3$ and check the correct answer(s) below. [ANS] A\\. $W_{2}$ is not a basis because it does not span ${\\mathbb R}^3$. B\\. $W_{2}$ is not a basis because it is linearly dependent. C\\. $W_{2}$ is a basis.", "answer_v3": [ "B", "AB" ], "answer_type_v3": [ "MCS", "MCM" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Linear_algebra_0299", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "3", "keywords": [ "linear algebra", "vector space", "subspace", "kernel" ], "problem_v1": "Find the null space for $A=\\left[\\begin{matrix} 7 & 5 \\cr 6 & 4 \\cr \\end{matrix}\\right].$ What is null $(A)$? [ANS] A\\. span $\\left\\{\\left[\\begin{matrix} 7 \\cr 5 \\cr \\end{matrix}\\right]\\right\\}$ B\\. ${\\mathbb R}^2$ C\\. span $\\left\\{\\left[\\begin{matrix} 6 \\cr 7 \\cr \\end{matrix}\\right]\\right\\}$ D\\. span $\\left\\{\\left[\\begin{matrix}-6 \\cr 7 \\cr \\end{matrix}\\right]\\right\\}$ E\\. span $\\left\\{\\left[\\begin{matrix} 7 \\cr 6 \\cr \\end{matrix}\\right]\\right\\}$ F\\. $\\left\\{\\left[\\begin{matrix} 0 \\cr 0 \\cr \\end{matrix}\\right]\\right\\}$ G\\. span $\\left\\{\\left[\\begin{matrix}-5 \\cr 7 \\cr \\end{matrix}\\right]\\right\\}$ H\\. none of the above.", "answer_v1": [ "F" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "Find the null space for $A=\\left[\\begin{matrix} 1 & 8 \\cr 2 &-3 \\cr \\end{matrix}\\right].$ What is null $(A)$? [ANS] A\\. span $\\left\\{\\left[\\begin{matrix} 1 \\cr 2 \\cr \\end{matrix}\\right]\\right\\}$ B\\. span $\\left\\{\\left[\\begin{matrix} 1 \\cr 8 \\cr \\end{matrix}\\right]\\right\\}$ C\\. span $\\left\\{\\left[\\begin{matrix}-8 \\cr 1 \\cr \\end{matrix}\\right]\\right\\}$ D\\. ${\\mathbb R}^2$ E\\. $\\left\\{\\left[\\begin{matrix} 0 \\cr 0 \\cr \\end{matrix}\\right]\\right\\}$ F\\. span $\\left\\{\\left[\\begin{matrix} 2 \\cr 1 \\cr \\end{matrix}\\right]\\right\\}$ G\\. span $\\left\\{\\left[\\begin{matrix}-2 \\cr 1 \\cr \\end{matrix}\\right]\\right\\}$ H\\. none of the above.", "answer_v2": [ "E" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "Find the null space for $A=\\left[\\begin{matrix} 3 & 5 \\cr 3 & 0 \\cr \\end{matrix}\\right].$ What is null $(A)$? [ANS] A\\. $\\left\\{\\left[\\begin{matrix} 0 \\cr 0 \\cr \\end{matrix}\\right]\\right\\}$ B\\. ${\\mathbb R}^2$ C\\. span $\\left\\{\\left[\\begin{matrix}-3 \\cr 3 \\cr \\end{matrix}\\right]\\right\\}$ D\\. span $\\left\\{\\left[\\begin{matrix} 3 \\cr 5 \\cr \\end{matrix}\\right]\\right\\}$ E\\. span $\\left\\{\\left[\\begin{matrix}-5 \\cr 3 \\cr \\end{matrix}\\right]\\right\\}$ F\\. span $\\left\\{\\left[\\begin{matrix} 3 \\cr 3 \\cr \\end{matrix}\\right]\\right\\}$ G\\. none of the above.", "answer_v3": [ "A" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Linear_algebra_0300", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "4", "keywords": [ "linear algebra", "subspaces", "basis", "nullity" ], "problem_v1": "Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix A is given to make your work easier. $A=\\left[\\begin{array}{ccc} 4 &19 &3\\cr 3 &1 &0\\cr-5 &2 &5\\cr 3 &5 &-2\\cr-5 &11 &6 \\end{array}\\right]$ $\\sim$ $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &1\\cr 0 &0 &0\\cr 0 &0 &0 \\end{array}\\right]$ Basis for the column space of $A$ is $\\Bigg \\lbrace$ \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$, \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$, \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$ $\\Bigg \\rbrace$ Basis for the row space of $A$ is $\\Big \\lbrace$ \\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$,\n\\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$,\n\\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$ $\\Big \\rbrace$ Note that since the only solution to $A{\\bf x}={\\bf 0}$ is the zero vector, there is no basis for the null space of $A$.", "answer_v1": [ "19", "1", "2", "5", "11", "3", "0", "5", "-2", "6", "0", "1", "0", "0", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix A is given to make your work easier. $A=\\left[\\begin{array}{ccc} 3 &26 &5\\cr 2 &1 &0\\cr-5 &-1 &5\\cr 2 &7 &-3\\cr-5 &10 &6 \\end{array}\\right]$ $\\sim$ $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &1\\cr 0 &0 &0\\cr 0 &0 &0 \\end{array}\\right]$ Basis for the column space of $A$ is $\\Bigg \\lbrace$ \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$, \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$, \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$ $\\Bigg \\rbrace$ Basis for the row space of $A$ is $\\Big \\lbrace$ \\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$,\n\\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$,\n\\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$ $\\Big \\rbrace$ Note that since the only solution to $A{\\bf x}={\\bf 0}$ is the zero vector, there is no basis for the null space of $A$.", "answer_v2": [ "26", "1", "-1", "7", "10", "5", "0", "5", "-3", "6", "0", "1", "0", "0", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix A is given to make your work easier. $A=\\left[\\begin{array}{ccc} 4 &22 &3\\cr 3 &1 &0\\cr-3 &-2 &5\\cr 2 &5 &-3\\cr-3 &12 &6 \\end{array}\\right]$ $\\sim$ $\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &1\\cr 0 &0 &0\\cr 0 &0 &0 \\end{array}\\right]$ Basis for the column space of $A$ is $\\Bigg \\lbrace$ \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$, \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$, \\left[\\Rule{0pt}{6em}{0pt}\\right. $\\left[\\Rule{0pt}{6em}{0pt}\\right.$ [ANS] [ANS] [ANS] [ANS] [ANS] \\left]\\Rule{0pt}{6em}{0pt}\\right. $\\left]\\Rule{0pt}{6em}{0pt}\\right.$ $\\Bigg \\rbrace$ Basis for the row space of $A$ is $\\Big \\lbrace$ \\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$,\n\\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$,\n\\left<\\Rule{0pt}{1.2em}{0pt}\\right. $\\left<\\Rule{0pt}{1.2em}{0pt}\\right.$ [ANS],, [ANS],, [ANS] \\left>\\Rule{0pt}{1.2em}{0pt}\\right. $\\left>\\Rule{0pt}{1.2em}{0pt}\\right.$ $\\Big \\rbrace$ Note that since the only solution to $A{\\bf x}={\\bf 0}$ is the zero vector, there is no basis for the null space of $A$.", "answer_v3": [ "22", "1", "-2", "5", "12", "3", "0", "5", "-3", "6", "0", "1", "0", "0", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0301", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "3", "keywords": [ "linear algebra", "vector space", "subspace", "kernel" ], "problem_v1": "If $A$ is an $7\\times 6$ matrix, then consider whether the set null($A$) is a subspace of ${\\mathbb R}^{7}$. Select true or false for each statement.\n[ANS] 1. This set is a subspace of ${\\mathbb R}^7$ [ANS] 2. This set contains the zero vector and is closed under vector addition and scalar multiplication. [ANS] 3. This set is a subset of the domain. [ANS] 4. This set is a subset of the codomain", "answer_v1": [ "FALSE", "TRUE", "TRUE", "False" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v2": "If $A$ is an $2\\times 8$ matrix, then consider whether the set null($A$) is a subspace of ${\\mathbb R}^{2}$. Select true or false for each statement.\n[ANS] 1. This set is a subset of the domain. [ANS] 2. This set contains the zero vector and is closed under vector addition and scalar multiplication. [ANS] 3. This set is a subspace of ${\\mathbb R}^2$ [ANS] 4. This set is a subset of the codomain", "answer_v2": [ "TRUE", "TRUE", "FALSE", "False" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v3": "If $A$ is an $4\\times 6$ matrix, then consider whether the set null($A$) is a subspace of ${\\mathbb R}^{4}$. Select true or false for each statement.\n[ANS] 1. This set is a subset of the domain. [ANS] 2. This set is a subspace of ${\\mathbb R}^4$ [ANS] 3. This set contains the zero vector and is closed under vector addition and scalar multiplication. [ANS] 4. This set is a subset of the codomain", "answer_v3": [ "TRUE", "FALSE", "TRUE", "False" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ] }, { "id": "Linear_algebra_0302", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "3", "keywords": [ "linear algebra", "subspaces", "basis", "row space", "column space", "null space" ], "problem_v1": "Suppose that $A$ is a $5 \\times 8$ matrix that has an echelon form with one zero row. Find the dimension of the row space of $A$, the dimension of the column space of $A$, and the dimension of the null space of $A$.\nThe dimension of the row space of $A$ is [ANS]. The dimension of the column space of $A$ is [ANS]. The dimension of the null space of $A$ is [ANS].", "answer_v1": [ "4", "4", "4" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that $A$ is a $3 \\times 9$ matrix that has an echelon form with no zero rows. Find the dimension of the row space of $A$, the dimension of the column space of $A$, and the dimension of the null space of $A$.\nThe dimension of the row space of $A$ is [ANS]. The dimension of the column space of $A$ is [ANS]. The dimension of the null space of $A$ is [ANS].", "answer_v2": [ "3", "3", "6" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that $A$ is a $3 \\times 8$ matrix that has an echelon form with no zero rows. Find the dimension of the row space of $A$, the dimension of the column space of $A$, and the dimension of the null space of $A$.\nThe dimension of the row space of $A$ is [ANS]. The dimension of the column space of $A$ is [ANS]. The dimension of the null space of $A$ is [ANS].", "answer_v3": [ "3", "3", "5" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0303", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "3", "keywords": [ "linear algebra", "subspaces", "basis", "row space", "column space" ], "problem_v1": "Suppose that $A$ is a $13 \\times 8$ matrix. If the dimension of the column space of $A$ is $5$, then the dimension of the row space of $A$ is [ANS].", "answer_v1": [ "5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $A$ is an $11 \\times 9$ matrix. If the dimension of the column space of $A$ is $3$, then the dimension of the row space of $A$ is [ANS].", "answer_v2": [ "3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $A$ is an $11 \\times 8$ matrix. If the dimension of the column space of $A$ is $4$, then the dimension of the row space of $A$ is [ANS].", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0304", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "3", "keywords": [ "linear algebra", "subspaces", "rank" ], "problem_v1": "Suppose that $A$ is an $n \\times m$ matrix of rank $5,$ the nullity of $A$ is $3,$ and the column space of $A$ is a subspace of $\\mathbf R ^ {8}$. Find the dimensions of $A$.\n$A$ has [ANS] rows and [ANS] columns.", "answer_v1": [ "8", "8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that $A$ is an $n \\times m$ matrix of rank $3,$ the nullity of $A$ is $4,$ and the column space of $A$ is a subspace of $\\mathbf R ^ {4}$. Find the dimensions of $A$.\n$A$ has [ANS] rows and [ANS] columns.", "answer_v2": [ "4", "7" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that $A$ is an $n \\times m$ matrix of rank $3,$ the nullity of $A$ is $3,$ and the column space of $A$ is a subspace of $\\mathbf R ^ {4}$. Find the dimensions of $A$.\n$A$ has [ANS] rows and [ANS] columns.", "answer_v3": [ "4", "6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0305", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "3", "keywords": [ "linear algebra", "subspaces", "rank" ], "problem_v1": "Suppose that $A$ is a 4 $\\times$ 14 matrix, and that $T(x)=Ax$. If $T$ is onto, then what is the dimension of the null space of $A$?\nThe nullity($A$)=[ANS]", "answer_v1": [ "10" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $A$ is a 5 $\\times$ 11 matrix, and that $T(x)=Ax$. If $T$ is onto, then what is the dimension of the null space of $A$?\nThe nullity($A$)=[ANS]", "answer_v2": [ "6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $A$ is a 4 $\\times$ 12 matrix, and that $T(x)=Ax$. If $T$ is onto, then what is the dimension of the null space of $A$?\nThe nullity($A$)=[ANS]", "answer_v3": [ "8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0306", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "4", "keywords": [ "linear algebra", "subspaces", "basis", "nullity" ], "problem_v1": "Find a basis for the null space of matrix A. A=$\\left[\\begin{array}{ccc} 6 &4 &0\\cr 1 &2 &3 \\end{array}\\right]$ Basis is $\\Bigg \\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg \\rbrace$", "answer_v1": [ "1.5", "-2.25", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find a basis for the null space of matrix A. A=$\\left[\\begin{array}{ccc} 3 &4 &0\\cr 1 &1 &1 \\end{array}\\right]$ Basis is $\\Bigg \\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg \\rbrace$", "answer_v2": [ "-4", "3", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Problem 1 2. ERROR caught by Translator while processing this problem", "answer_v3": [], "answer_type_v3": [], "options_v3": [] }, { "id": "Linear_algebra_0307", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "3", "keywords": [ "linear algebra", "subspaces", "rank" ], "problem_v1": "Suppose that $A$ is an $11 \\times 5$ matrix, and that $B$ is an equivalent matrix in echelon form. If the number of pivot columns of $B$ is $2$, find the nullity of $A$.\nThe nullity of $A$ is [ANS].", "answer_v1": [ "3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $A$ is an $8 \\times 6$ matrix, and that $B$ is an equivalent matrix in echelon form. If the number of pivot columns of $B$ is $1$, find the nullity of $A$.\nThe nullity of $A$ is [ANS].", "answer_v2": [ "5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $A$ is a $9 \\times 5$ matrix, and that $B$ is an equivalent matrix in echelon form. If the number of pivot columns of $B$ is $1$, find the nullity of $A$.\nThe nullity of $A$ is [ANS].", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0310", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "", "keywords": [ "matrix' 'null space" ], "problem_v1": "Let A=\\left[\\begin{array}{cccc}-16 &-9 &20 &-12\\cr-4 &3 &5 &4\\cr 0 &6 &0 &8 \\end{array}\\right]. Find a non-zero vector in the column space of $A$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-16", "-9", "20" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{cccc}-10 &-16 &15 &-20\\cr 4 &4 &-6 &5\\cr 0 &12 &0 &15 \\end{array}\\right]. Find a non-zero vector in the column space of $A$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-10", "-16", "15" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{cccc}-10 &-12 &15 &-16\\cr-2 &3 &3 &4\\cr 0 &9 &0 &12 \\end{array}\\right]. Find a non-zero vector in the column space of $A$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-10", "-12", "15" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0311", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Row, column, and null spaces", "level": "2", "keywords": [ "matrix' 'null space' 'column space" ], "problem_v1": "Let $W$ be the set of all vectors of the form \\begin{bmatrix} 3r-2s-t \\\\ r-2s \\\\ r+s+2t \\\\ 2r+s-3t \\end{bmatrix} with $r$, $s$ and $t$ real. Find a matrix $A$ such that $W=\\text{Col}(A)$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v1": [ "3", "-2", "-1", "1", "-2", "0", "1", "1", "2", "2", "1", "-3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $W$ be the set of all vectors of the form \\begin{bmatrix} 5s-5r+t \\\\ 5r-2s-5t \\\\ 2t-\\left(4r+3s\\right) \\\\-\\left(2r+2s+t\\right) \\end{bmatrix} with $r$, $s$ and $t$ real. Find a matrix $A$ such that $W=\\text{Col}(A)$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v2": [ "-5", "5", "1", "5", "-2", "-5", "-4", "-3", "2", "-2", "-2", "-1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $W$ be the set of all vectors of the form \\begin{bmatrix} 4t-\\left(2r+3s\\right) \\\\ r-2s-3t \\\\ 3s-2r-2t \\\\ r+5s-3t \\end{bmatrix} with $r$, $s$ and $t$ real. Find a matrix $A$ such that $W=\\text{Col}(A)$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}.", "answer_v3": [ "-2", "-3", "4", "1", "-2", "-3", "-2", "3", "-2", "1", "5", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0312", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [], "problem_v1": "Find the coordinate vector of $\\vec{x}=\\left[\\begin{array}{c} 2\\cr-2\\cr-2 \\end{array}\\right]$ with respect to the basis $B=\\left\\lbrace \\left[\\begin{array}{c} 1\\cr 7\\cr 6 \\end{array}\\right], \\left[\\begin{array}{c} 0\\cr 1\\cr-4 \\end{array}\\right], \\left[\\begin{array}{c} 0\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace$ for ${\\mathbb R}^3$.\n$[\\vec{x}]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "2", "-16", "-78" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the coordinate vector of $\\vec{x}=\\left[\\begin{array}{c}-2\\cr 5\\cr-2 \\end{array}\\right]$ with respect to the basis $B=\\left\\lbrace \\left[\\begin{array}{c} 1\\cr 2\\cr 8 \\end{array}\\right], \\left[\\begin{array}{c} 0\\cr 1\\cr-7 \\end{array}\\right], \\left[\\begin{array}{c} 0\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace$ for ${\\mathbb R}^3$.\n$[\\vec{x}]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-2", "9", "77" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the coordinate vector of $\\vec{x}=\\left[\\begin{array}{c} 1\\cr-3\\cr-2 \\end{array}\\right]$ with respect to the basis $B=\\left\\lbrace \\left[\\begin{array}{c} 1\\cr 4\\cr 6 \\end{array}\\right], \\left[\\begin{array}{c} 0\\cr 1\\cr-7 \\end{array}\\right], \\left[\\begin{array}{c} 0\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace$ for ${\\mathbb R}^3$.\n$[\\vec{x}]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "1", "-7", "-57" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0313", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [], "problem_v1": "Let $B$ be the basis of ${\\mathbb R}^2$ consisting of the vectors \\left\\lbrace \\left[\\begin{array}{c} 5\\cr 2 \\end{array}\\right], \\left[\\begin{array}{c} 1\\cr 3 \\end{array}\\right] \\right\\rbrace, and let $C$ be the basis consisting of \\left\\lbrace \\left[\\begin{array}{c}-2\\cr 3 \\end{array}\\right], \\left[\\begin{array}{c} 1\\cr-2 \\end{array}\\right] \\right\\rbrace.\nFind a matrix $P$ such that $[\\vec{x}]_C=P [\\vec{x}]_B$ for all $\\vec{x}$ in ${\\mathbb R}^2$.\n$P=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "-12", "-5", "-19", "-9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $B$ be the basis of ${\\mathbb R}^2$ consisting of the vectors \\left\\lbrace \\left[\\begin{array}{c} 3\\cr-2 \\end{array}\\right], \\left[\\begin{array}{c} 2\\cr 3 \\end{array}\\right] \\right\\rbrace, and let $C$ be the basis consisting of \\left\\lbrace \\left[\\begin{array}{c} 2\\cr-3 \\end{array}\\right], \\left[\\begin{array}{c}-1\\cr 2 \\end{array}\\right] \\right\\rbrace.\nFind a matrix $P$ such that $[\\vec{x}]_C=P [\\vec{x}]_B$ for all $\\vec{x}$ in ${\\mathbb R}^2$.\n$P=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "4", "7", "5", "12" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $B$ be the basis of ${\\mathbb R}^2$ consisting of the vectors \\left\\lbrace \\left[\\begin{array}{c} 3\\cr-2 \\end{array}\\right], \\left[\\begin{array}{c} 1\\cr 5 \\end{array}\\right] \\right\\rbrace, and let $C$ be the basis consisting of \\left\\lbrace \\left[\\begin{array}{c}-2\\cr-3 \\end{array}\\right], \\left[\\begin{array}{c}-1\\cr-2 \\end{array}\\right] \\right\\rbrace.\nFind a matrix $P$ such that $[\\vec{x}]_C=P [\\vec{x}]_B$ for all $\\vec{x}$ in ${\\mathbb R}^2$.\n$P=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-8", "3", "13", "-7" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0314", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [], "problem_v1": "Write the vector $\\left<-2,4,6\\right>$ as a linear combination of $\\vec{a}_1=\\left<3,1,1\\right>$, $\\vec{a}_2=\\left<2,-2,-2\\right>$ and $\\vec{a}_3=\\left<1,1,-1\\right>$. Express your answer in terms of the named vectors. Your answer should be in the form $4\\vec{a}_1+5 \\vec{a}_2+6 \\vec{a}_3$, which would be entered as $\\verb!4a1+5a2+6a3!$.\n$\\left<-2,4,6\\right>=$ [ANS]\nRepresent the vector $\\left<-2,4,6\\right>$ in terms of the ordered basis $\\mathcal{B}=\\lbrace \\left<3,1,1\\right>, \\left<2,-2,-2\\right>, \\left<1,1,-1\\right> \\rbrace$. Your answer should be a vector of the general form $\\verb+<1,2,3>+$.\n$\\lbrack \\left<-2,4,6\\right> \\rbrack_{\\mathcal{B}}=$ [ANS]", "answer_v1": [ "a1-2*a2-a3", "(1,-2,-1)" ], "answer_type_v1": [ "EX", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Write the vector $\\left<21,-13,13\\right>$ as a linear combination of $\\vec{a}_1=\\left<-5,5,-4\\right>$, $\\vec{a}_2=\\left<-2,5,-2\\right>$ and $\\vec{a}_3=\\left<-3,-2,1\\right>$. Express your answer in terms of the named vectors. Your answer should be in the form $4\\vec{a}_1+5 \\vec{a}_2+6 \\vec{a}_3$, which would be entered as $\\verb!4a1+5a2+6a3!$.\n$\\left<21,-13,13\\right>=$ [ANS]\nRepresent the vector $\\left<21,-13,13\\right>$ in terms of the ordered basis $\\mathcal{B}=\\lbrace \\left<-5,5,-4\\right>, \\left<-2,5,-2\\right>, \\left<-3,-2,1\\right> \\rbrace$. Your answer should be a vector of the general form $\\verb+<1,2,3>+$.\n$\\lbrack \\left<21,-13,13\\right> \\rbrack_{\\mathcal{B}}=$ [ANS]", "answer_v2": [ "a2-4*a1-a3", "(-4,1,-1)" ], "answer_type_v2": [ "EX", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Write the vector $\\left<11,9,-15\\right>$ as a linear combination of $\\vec{a}_1=\\left<-5,1,5\\right>$, $\\vec{a}_2=\\left<3,1,-4\\right>$ and $\\vec{a}_3=\\left<-2,2,2\\right>$. Express your answer in terms of the named vectors. Your answer should be in the form $4\\vec{a}_1+5 \\vec{a}_2+6 \\vec{a}_3$, which would be entered as $\\verb!4a1+5a2+6a3!$.\n$\\left<11,9,-15\\right>=$ [ANS]\nRepresent the vector $\\left<11,9,-15\\right>$ in terms of the ordered basis $\\mathcal{B}=\\lbrace \\left<-5,1,5\\right>, \\left<3,1,-4\\right>, \\left<-2,2,2\\right> \\rbrace$. Your answer should be a vector of the general form $\\verb+<1,2,3>+$.\n$\\lbrack \\left<11,9,-15\\right> \\rbrack_{\\mathcal{B}}=$ [ANS]", "answer_v3": [ "4*a2-a1+3*a3", "(-1,4,3)" ], "answer_type_v3": [ "EX", "OL" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0315", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [], "problem_v1": "Consider the following two ordered bases of $\\mathbb{R}^{3}$: \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace \\left<1,1,1\\right>, \\left<1,0,1\\right>, \\left<1,-1,0\\right> \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace \\left<-1,-1,1\\right>, \\left<1,2,-1\\right>, \\left<-2,-2,1\\right> \\rbrace. \\end{array}\nFind the change of basis matrix from the basis $\\mathcal{B}$ to the basis $\\mathcal{C}$.\n$\\lbrack id \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nFind the change of basis matrix from the basis $\\mathcal{C}$ to the basis $\\mathcal{B}$.\n$\\lbrack id \\rbrack_{\\mathcal{C}}^{\\mathcal{B}}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "3", "2", "-1", "0", "-1", "-2", "-2", "-2", "-1", "-3", "4", "-5", "4", "-5", "6", "-2", "2", "-3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the following two ordered bases of $\\mathbb{R}^{3}$: \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace \\left<-2,1,-1\\right>, \\left<2,0,1\\right>, \\left<-1,-1,0\\right> \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace \\left<0,-1,1\\right>, \\left<0,2,-1\\right>, \\left<-1,-1,0\\right> \\rbrace. \\end{array}\nFind the change of basis matrix from the basis $\\mathcal{B}$ to the basis $\\mathcal{C}$.\n$\\lbrack id \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nFind the change of basis matrix from the basis $\\mathcal{C}$ to the basis $\\mathcal{B}$.\n$\\lbrack id \\rbrack_{\\mathcal{C}}^{\\mathcal{B}}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "0", "2", "-1", "0", "2", "-2", "1", "1", "0", "0", "2", "-1", "0", "2", "-2", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the following two ordered bases of $\\mathbb{R}^{3}$: \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace \\left<-1,1,-1\\right>, \\left<-1,0,-1\\right>, \\left<0,0,-1\\right> \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace \\left<2,-1,-1\\right>, \\left<-2,0,1\\right>, \\left<-1,0,1\\right> \\rbrace. \\end{array}\nFind the change of basis matrix from the basis $\\mathcal{B}$ to the basis $\\mathcal{C}$.\n$\\lbrack id \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nFind the change of basis matrix from the basis $\\mathcal{C}$ to the basis $\\mathcal{B}$.\n$\\lbrack id \\rbrack_{\\mathcal{C}}^{\\mathcal{B}}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-1", "0", "0", "1", "2", "1", "-3", "-3", "-2", "-1", "0", "0", "-1", "2", "1", "3", "-3", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0316", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [], "problem_v1": "Find the representation of $\\left<4,-4,-3\\right>$ in each of the following ordered bases.\nRepresent the vector $\\left<4,-4,-3\\right>$ in terms of the ordered basis $\\mathcal{B}=\\lbrace \\vec{i}, \\vec{j}, \\vec{k} \\rbrace$.\n$\\lbrack \\left<4,-4,-3\\right> \\rbrack_{\\mathcal{B}}=$ [ANS]\nRepresent the vector $\\left<4,-4,-3\\right>$ in terms of the ordered basis $\\mathcal{C}=\\lbrace \\vec{e}_3, \\vec{e}_1, \\vec{e}_2 \\rbrace$.\n$\\lbrack \\left<4,-4,-3\\right> \\rbrack_{\\mathcal{C}}=$ [ANS]\nRepresent the vector $\\left<4,-4,-3\\right>$ in terms of the ordered basis $\\mathcal{D}=\\lbrace-\\vec{e}_2,-\\vec{e}_1, \\vec{e}_3 \\rbrace$.\n$\\lbrack \\left<4,-4,-3\\right> \\rbrack_{\\mathcal{D}}=$ [ANS]", "answer_v1": [ "(4,-4,-3)", "(-3,4,-4)", "(4,-4,-3)" ], "answer_type_v1": [ "OL", "OL", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the representation of $\\left<-8,8,-7\\right>$ in each of the following ordered bases.\nRepresent the vector $\\left<-8,8,-7\\right>$ in terms of the ordered basis $\\mathcal{B}=\\lbrace \\vec{i}, \\vec{j}, \\vec{k} \\rbrace$.\n$\\lbrack \\left<-8,8,-7\\right> \\rbrack_{\\mathcal{B}}=$ [ANS]\nRepresent the vector $\\left<-8,8,-7\\right>$ in terms of the ordered basis $\\mathcal{C}=\\lbrace \\vec{e}_3, \\vec{e}_1, \\vec{e}_2 \\rbrace$.\n$\\lbrack \\left<-8,8,-7\\right> \\rbrack_{\\mathcal{C}}=$ [ANS]\nRepresent the vector $\\left<-8,8,-7\\right>$ in terms of the ordered basis $\\mathcal{D}=\\lbrace-\\vec{e}_2,-\\vec{e}_1, \\vec{e}_3 \\rbrace$.\n$\\lbrack \\left<-8,8,-7\\right> \\rbrack_{\\mathcal{D}}=$ [ANS]", "answer_v2": [ "(-8,8,-7)", "(-7,-8,8)", "(-8,8,-7)" ], "answer_type_v2": [ "OL", "OL", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the representation of $\\left<1,-6,-3\\right>$ in each of the following ordered bases.\nRepresent the vector $\\left<1,-6,-3\\right>$ in terms of the ordered basis $\\mathcal{B}=\\lbrace \\vec{i}, \\vec{j}, \\vec{k} \\rbrace$.\n$\\lbrack \\left<1,-6,-3\\right> \\rbrack_{\\mathcal{B}}=$ [ANS]\nRepresent the vector $\\left<1,-6,-3\\right>$ in terms of the ordered basis $\\mathcal{C}=\\lbrace \\vec{e}_3, \\vec{e}_1, \\vec{e}_2 \\rbrace$.\n$\\lbrack \\left<1,-6,-3\\right> \\rbrack_{\\mathcal{C}}=$ [ANS]\nRepresent the vector $\\left<1,-6,-3\\right>$ in terms of the ordered basis $\\mathcal{D}=\\lbrace-\\vec{e}_2,-\\vec{e}_1, \\vec{e}_3 \\rbrace$.\n$\\lbrack \\left<1,-6,-3\\right> \\rbrack_{\\mathcal{D}}=$ [ANS]", "answer_v3": [ "(1,-6,-3)", "(-3,1,-6)", "(6,-1,-3)" ], "answer_type_v3": [ "OL", "OL", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0317", "subject": "Linear_algebra", "topic": "Euclidean spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [ "linear algebra", "transition matrix" ], "problem_v1": "Consider the ordered bases $B=((3,7),(-1,-2))$ and $C=((-1,2),(1,4))$ for the vector space ${\\mathbb R}^2$. a. Find the transition matrix from $C$ to the standard ordered basis $E=((1,0),(0,1))$. $T_C^E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. Find the transition matrix from $B$ to $E$. $T_B^E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. Find the transition matrix from $E$ to $B$. $T_E^B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. Find the transition matrix from $C$ to $B$. $T_C^B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. Find the coordinates of $u=(3,-3)$ in the ordered basis $B$. Note that $[u]_B=T_E^B [u]_E$. $[u]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} f. Find the coordinates of $v$ in the ordered basis $B$ if the coordinate vector of $v$ in $C$ is $[v]_C=(1,-1)$. $[v]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-1", "1", "2", "4", "3", "-1", "7", "-2", "-2", "1", "-7", "3", "4", "2", "13", "5", "-9", "-30", "2", "8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the ordered bases $B=((7,9),(-4,-5))$ and $C=((-2,4),(0,2))$ for the vector space ${\\mathbb R}^2$. a. Find the transition matrix from $C$ to the standard ordered basis $E=((1,0),(0,1))$. $T_C^E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. Find the transition matrix from $B$ to $E$. $T_B^E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. Find the transition matrix from $E$ to $B$. $T_E^B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. Find the transition matrix from $C$ to $B$. $T_C^B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. Find the coordinates of $u=(-3,1)$ in the ordered basis $B$. Note that $[u]_B=T_E^B [u]_E$. $[u]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} f. Find the coordinates of $v$ in the ordered basis $B$ if the coordinate vector of $v$ in $C$ is $[v]_C=(1,-2)$. $[v]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-2", "0", "4", "2", "7", "-4", "9", "-5", "-5", "4", "-9", "7", "26", "8", "46", "14", "19", "34", "10", "18" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the ordered bases $B=((-1,-7),(0,-1))$ and $C=((-3,-1),(0,-4))$ for the vector space ${\\mathbb R}^2$. a. Find the transition matrix from $C$ to the standard ordered basis $E=((1,0),(0,1))$. $T_C^E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} b. Find the transition matrix from $B$ to $E$. $T_B^E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} c. Find the transition matrix from $E$ to $B$. $T_E^B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} d. Find the transition matrix from $C$ to $B$. $T_C^B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} e. Find the coordinates of $u=(-3,2)$ in the ordered basis $B$. Note that $[u]_B=T_E^B [u]_E$. $[u]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} f. Find the coordinates of $v$ in the ordered basis $B$ if the coordinate vector of $v$ in $C$ is $[v]_C=(-2,-1)$. $[v]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-3", "0", "-1", "-4", "-1", "0", "-7", "-1", "-1", "0", "7", "-1", "3", "0", "-20", "4", "3", "-23", "-6", "36" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0318", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Linear combinations", "level": "2", "keywords": [ "linear algebra", "vector space", "subspaces" ], "problem_v1": "If possible, write $4-3x-3x^{2}$ as a linear combination of $1-x-x^{2}, 1-x^{2}$ and $x^{2}$. Otherwise, enter DNE in all answer blanks.\n$4-3x-3x^{2}=$ [ANS] $(1-x-x^{2})+$ [ANS] $(1-x^{2})+$ [ANS] $(x^{2}).$", "answer_v1": [ "3", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If possible, write $10-23x+14x^{2}$ as a linear combination of $x-1-x^{2}, 1-2x+x^{2}$ and $2x-x^{2}$. Otherwise, enter DNE in all answer blanks.\n$10-23x+14x^{2}=$ [ANS] $(x-1-x^{2})+$ [ANS] $(1-2x+x^{2})+$ [ANS] $(2x-x^{2}).$", "answer_v2": [ "-5", "5", "-4" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If possible, write $2+4x+x^{2}$ as a linear combination of $-x-x^{2},-x^{2}$ and $-1-x$. Otherwise, enter DNE in all answer blanks.\n$2+4x+x^{2}=$ [ANS] $(-x-x^{2})+$ [ANS] $(-x^{2})+$ [ANS] $(-1-x).$", "answer_v3": [ "-2", "1", "-2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0319", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Span", "level": "3", "keywords": [ "linear algebra", "span" ], "problem_v1": "Consider the subspaces U=\\text{span} \\lbrace \\left[\\begin{array}{ccc} 2 &-2 &-2 \\end{array}\\right],\\left[\\begin{array}{ccc} 8 &0 &0 \\end{array}\\right] \\rbrace and W=\\text{span} \\lbrace \\left[\\begin{array}{ccc} 1 &1 &-1 \\end{array}\\right],\\left[\\begin{array}{ccc} 7 &3 &1 \\end{array}\\right] \\rbrace of $V={\\mathbb R}^{1\\times 3}$. Find a matrix $X\\in V$ such that $U\\cap W=\\text{span} \\lbrace X \\rbrace$.\n$X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "3", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the subspaces U=\\text{span} \\lbrace \\left[\\begin{array}{ccc}-2 &5 &-2 \\end{array}\\right],\\left[\\begin{array}{ccc}-12 &15 &-10 \\end{array}\\right] \\rbrace and W=\\text{span} \\lbrace \\left[\\begin{array}{ccc}-3 &-2 &1 \\end{array}\\right],\\left[\\begin{array}{ccc}-18 &13 &-11 \\end{array}\\right] \\rbrace of $V={\\mathbb R}^{1\\times 3}$. Find a matrix $X\\in V$ such that $U\\cap W=\\text{span} \\lbrace X \\rbrace$.\n$X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-5", "5", "-4" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the subspaces U=\\text{span} \\lbrace \\left[\\begin{array}{ccc} 1 &-3 &-2 \\end{array}\\right],\\left[\\begin{array}{ccc}-1 &-2 &-4 \\end{array}\\right] \\rbrace and W=\\text{span} \\lbrace \\left[\\begin{array}{ccc} 3 &5 &4 \\end{array}\\right],\\left[\\begin{array}{ccc} 1 &6 &2 \\end{array}\\right] \\rbrace of $V={\\mathbb R}^{1\\times 3}$. Find a matrix $X\\in V$ such that $U\\cap W=\\text{span} \\lbrace X \\rbrace$.\n$X=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-2", "1", "-2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0320", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Span", "level": "2", "keywords": [ "linear algebra", "span" ], "problem_v1": "Let $V$ be the vector space of symmetric $2\\times 2$ matrices and $W$ be the subspace W=\\text{span} \\lbrace \\left[\\begin{array}{cc} 3 &1\\cr 1 &1 \\end{array}\\right],\\left[\\begin{array}{cc} 2 &-2\\cr-2 &-2 \\end{array}\\right] \\rbrace. a. Find a nonzero element $X$ in $W$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\nb. Find an element $Y$ in $V$ that is not in $W$. $Y=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "5", "-1", "-1", "-1", "0", "1", "1", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $V$ be the vector space of symmetric $2\\times 2$ matrices and $W$ be the subspace W=\\text{span} \\lbrace \\left[\\begin{array}{cc}-5 &5\\cr 5 &-4 \\end{array}\\right],\\left[\\begin{array}{cc}-2 &5\\cr 5 &-2 \\end{array}\\right] \\rbrace. a. Find a nonzero element $X$ in $W$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\nb. Find an element $Y$ in $V$ that is not in $W$. $Y=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-7", "10", "10", "-6", "-2", "1", "1", "-1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $V$ be the vector space of symmetric $2\\times 2$ matrices and $W$ be the subspace W=\\text{span} \\lbrace \\left[\\begin{array}{cc}-2 &1\\cr 1 &-2 \\end{array}\\right],\\left[\\begin{array}{cc} 1 &-3\\cr-3 &-2 \\end{array}\\right] \\rbrace. a. Find a nonzero element $X$ in $W$. $X=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\nb. Find an element $Y$ in $V$ that is not in $W$. $Y=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-1", "-2", "-2", "-4", "2", "1", "1", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0321", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Span", "level": "2", "keywords": [ "linear algebra", "span" ], "problem_v1": "Let $V$ be the vector space ${\\mathbb P}_3[x]$ of polynomials in $x$ with degree less than 3 and $W$ be the subspace W=\\text{span} \\lbrace 5+2x+2x^{2},4-4x-3x^{2} \\rbrace. a. Find a nonzero polynomial $p(x)$ in $W$.\n$p(x)=$ [ANS]\nb. Find a polynomial $q(x)$ in $V\\setminus W$. $q(x)=$ [ANS]", "answer_v1": [ "9-2*x-x^2", "-x^2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $V$ be the vector space ${\\mathbb P}_3[x]$ of polynomials in $x$ with degree less than 3 and $W$ be the subspace W=\\text{span} \\lbrace 8x-8-7x^{2},8x-3-3x^{2} \\rbrace. a. Find a nonzero polynomial $p(x)$ in $W$.\n$p(x)=$ [ANS]\nb. Find a polynomial $q(x)$ in $V\\setminus W$. $q(x)=$ [ANS]", "answer_v2": [ "16*x-11-10*x^2", "-(2+x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $V$ be the vector space ${\\mathbb P}_3[x]$ of polynomials in $x$ with degree less than 3 and $W$ be the subspace W=\\text{span} \\lbrace 2x-4-4x^{2},1-6x-3x^{2} \\rbrace. a. Find a nonzero polynomial $p(x)$ in $W$.\n$p(x)=$ [ANS]\nb. Find a polynomial $q(x)$ in $V\\setminus W$. $q(x)=$ [ANS]", "answer_v3": [ "-(3+4*x+7*x^2)", "2+2*x+2*x^2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0322", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Span", "level": "3", "keywords": [ "vector' 'linear' 'combination' 'span" ], "problem_v1": "Let $x, y, z$ be (non-zero) vectors and suppose $w=-3x-3y+2z$. If $z=x+y$, then $w=$ [ANS] $x+$ [ANS] $y$. Using the calculation above, mark the statements below that must be true. [ANS] A\\. Span(x, y, z)=Span(w, x, y) B\\. Span(y, z)=Span(w, y, z) C\\. Span(w, z)=Span(y, z) D\\. Span(w, y, z)=Span(w, z) E\\. Span(w, x, z)=Span(w, x)", "answer_v1": [ "-1", "-1", "ABE" ], "answer_type_v1": [ "NV", "NV", "MCM" ], "options_v1": [ [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Let $x, y, z$ be (non-zero) vectors and suppose $w=24x-18y-4z$. If $z=4x-3y$, then $w=$ [ANS] $x+$ [ANS] $y$. Using the calculation above, mark the statements below that must be true. [ANS] A\\. Span(w, z)=Span(w, x, y) B\\. Span(x, y)=Span(w, x) C\\. Span(y, z)=Span(x, z) D\\. Span(w, x)=Span(w, z) E\\. Span(w, x, z)=Span(x, y, z)", "answer_v2": [ "8", "-6", "BCE" ], "answer_type_v2": [ "NV", "NV", "MCM" ], "options_v2": [ [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Let $x, y, z$ be (non-zero) vectors and suppose $w=x-2y-2z$. If $z=x-2y$, then $w=$ [ANS] $x+$ [ANS] $y$. Using the calculation above, mark the statements below that must be true. [ANS] A\\. Span(w, x, y)=Span(w, x) B\\. Span(w, y)=Span(x, y, z) C\\. Span(x, z)=Span(y, z) D\\. Span(w, z)=Span(y, z) E\\. Span(x, y)=Span(w, z)", "answer_v3": [ "-1", "2", "ABC" ], "answer_type_v3": [ "NV", "NV", "MCM" ], "options_v3": [ [], [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Linear_algebra_0323", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Linear independence", "level": "3", "keywords": [ "subspaces" ], "problem_v1": "Let ${\\bf u}_4$ be a linear combination of $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$. Select the best statement. [ANS] A\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ is never a linearly independent set of vectors. B\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ could be a linearly dependent or linearly dependent set of vectors depending on the vector space chosen. C\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is a linearly dependent set of vectors unless one of $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is the zero vector. D\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ is always a linearly independent set of vectors. E\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ could be a linearly dependent or linearly dependent set of vectors depending on the vectors chosen. F\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is never a linearly dependent set of vectors. G\\. none of the above", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Let ${\\bf u}_4$ be a linear combination of $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$. Select the best statement. [ANS] A\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is a linearly dependent set of vectors unless one of $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is the zero vector. B\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ is never a linearly independent set of vectors. C\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ could be a linearly dependent or linearly dependent set of vectors depending on the vector space chosen. D\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ is always a linearly independent set of vectors. E\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is never a linearly dependent set of vectors. F\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ could be a linearly dependent or linearly dependent set of vectors depending on the vectors chosen. G\\. none of the above", "answer_v2": [ "B" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Let ${\\bf u}_4$ be a linear combination of $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$. Select the best statement. [ANS] A\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ is always a linearly independent set of vectors. B\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ could be a linearly dependent or linearly dependent set of vectors depending on the vectors chosen. C\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3, {\\bf u}_4}\\rbrace$ is never a linearly independent set of vectors. D\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is never a linearly dependent set of vectors. E\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ could be a linearly dependent or linearly dependent set of vectors depending on the vector space chosen. F\\. $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is a linearly dependent set of vectors unless one of $\\lbrace{{\\bf u}_1, {\\bf u}_2, {\\bf u}_3}\\rbrace$ is the zero vector. G\\. none of the above", "answer_v3": [ "C" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Linear_algebra_0324", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Linear independence", "level": "2", "keywords": [], "problem_v1": "Determine whether the following pairs of functions are linearly independent or not.\n[ANS] 1. $f(\\theta)=12\\cos 3\\theta$ and $g(\\theta)=48\\cos^3\\theta-36\\cos \\theta$ [ANS] 2. $f(t)=t$ and $g(t)=|t|$ [ANS] 3. $f(x)=e^{12x}$ and $g(x)=e^{12(x-1)}$", "answer_v1": [ "LINEARLY DEPENDENT", "LINEARLY INDEPENDENT", "LINEARLY DEPENDENT" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "Linearly dependent" ], [ "Linearly dependent" ], [ "Linearly dependent" ] ], "problem_v2": "Determine whether the following pairs of functions are linearly independent or not.\n[ANS] 1. $f(t)=t$ and $g(t)=|t|$ [ANS] 2. $f(t)=t^2+19t$ and $g(t)=t^2-19t$ [ANS] 3. $f(x)=e^{19x}$ and $g(x)=e^{19(x-1)}$", "answer_v2": [ "LINEARLY INDEPENDENT", "LINEARLY INDEPENDENT", "LINEARLY DEPENDENT" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "Linearly dependent" ], [ "Linearly dependent" ], [ "Linearly dependent" ] ], "problem_v3": "Determine whether the following pairs of functions are linearly independent or not.\n[ANS] 1. $f(t)=t^2+13t$ and $g(t)=t^2-13t$ [ANS] 2. $f(\\theta)=13\\cos 3\\theta$ and $g(\\theta)=52\\cos^3\\theta-39\\cos \\theta$ [ANS] 3. $f(x)=e^{13x}$ and $g(x)=e^{13(x-1)}$", "answer_v3": [ "LINEARLY INDEPENDENT", "LINEARLY DEPENDENT", "LINEARLY DEPENDENT" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "Linearly dependent" ], [ "Linearly dependent" ], [ "Linearly dependent" ] ] }, { "id": "Linear_algebra_0325", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Linear independence", "level": "2", "keywords": [ "matrices' 'basis' 'linearly independent' 'dependent" ], "problem_v1": "Determine whether or not the following sets $S$ of $2\\times 2$ matrices are linearly independent.\n[ANS] 1. $S=\\left\\{\\begin{pmatrix}-1&-2\\cr 2&-1 \\end{pmatrix},\\, \\begin{pmatrix}-2&-1\\cr-1&2 \\end{pmatrix},\\, \\begin{pmatrix}-2&2\\cr-1&0 \\end{pmatrix} \\right\\}$ [ANS] 2. $S=\\left\\{\\begin{pmatrix} 2&1\\cr 2&3 \\end{pmatrix},\\, \\begin{pmatrix}-8&-4\\cr-8&-12 \\end{pmatrix} \\right\\}$ [ANS] 3. $S=\\left\\{\\begin{pmatrix} 2&1\\cr 2&3 \\end{pmatrix}, \\, \\begin{pmatrix}-8&-12\\cr-4&-12 \\end{pmatrix},\\, \\begin{pmatrix} 1&-3\\cr 9&10 \\end{pmatrix},\\,\\right.$ $\\left. \\begin{pmatrix} 1&2\\cr-4&-4 \\end{pmatrix},\\, \\begin{pmatrix} 17&-31\\cr \\pi & e^2 \\end{pmatrix} \\right\\}$ [ANS] 4. $S=\\left\\{\\begin{pmatrix} 2&1\\cr 2&3 \\end{pmatrix},\\, \\begin{pmatrix}-8&-12\\cr-4&-12 \\end{pmatrix} \\right\\}$", "answer_v1": [ "LINEARLY_INDEPENDENT", "LINEARLY_DEPENDENT", "LINEARLY_DEPENDENT", "Linearly_Independent" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ] ], "problem_v2": "Determine whether or not the following sets $S$ of $2\\times 2$ matrices are linearly independent.\n[ANS] 1. $S=\\left\\{\\begin{pmatrix}-4&6\\cr-5&-2 \\end{pmatrix},\\, \\begin{pmatrix} 8&-12\\cr 10&4 \\end{pmatrix} \\right\\}$ [ANS] 2. $S=\\left\\{\\begin{pmatrix}-4&6\\cr-5&-2 \\end{pmatrix}, \\, \\begin{pmatrix} 8&-2\\cr 22&4 \\end{pmatrix},\\, \\begin{pmatrix} 1&-3\\cr 9&10 \\end{pmatrix},\\,\\right.$ $\\left. \\begin{pmatrix} 6&-4\\cr-12&-2 \\end{pmatrix},\\, \\begin{pmatrix} 17&-31\\cr \\pi & e^2 \\end{pmatrix} \\right\\}$ [ANS] 3. $S=\\left\\{\\begin{pmatrix}-4&6\\cr-5&-2 \\end{pmatrix},\\, \\begin{pmatrix} 8&-2\\cr 22&4 \\end{pmatrix} \\right\\}$ [ANS] 4. $S=\\left\\{\\begin{pmatrix}-1&-4\\cr-2&1 \\end{pmatrix},\\, \\begin{pmatrix}-4&-1\\cr-1&-2 \\end{pmatrix},\\, \\begin{pmatrix}-4&-2\\cr-1&0 \\end{pmatrix} \\right\\}$", "answer_v2": [ "LINEARLY_DEPENDENT", "LINEARLY_DEPENDENT", "LINEARLY_INDEPENDENT", "Linearly_Independent" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ] ], "problem_v3": "Determine whether or not the following sets $S$ of $2\\times 2$ matrices are linearly independent.\n[ANS] 1. $S=\\left\\{\\begin{pmatrix}-2&2\\cr-3&1 \\end{pmatrix}, \\, \\begin{pmatrix} 10&5\\cr 25&-5 \\end{pmatrix},\\, \\begin{pmatrix} 1&-3\\cr 9&10 \\end{pmatrix},\\,\\right.$ $\\left. \\begin{pmatrix} 2&-2\\cr-10&-5 \\end{pmatrix},\\, \\begin{pmatrix} 17&-31\\cr \\pi & e^2 \\end{pmatrix} \\right\\}$ [ANS] 2. $S=\\left\\{\\begin{pmatrix}-2&2\\cr-3&1 \\end{pmatrix},\\, \\begin{pmatrix} 10&5\\cr 25&-5 \\end{pmatrix} \\right\\}$ [ANS] 3. $S=\\left\\{\\begin{pmatrix} 1&4\\cr 4&3 \\end{pmatrix},\\, \\begin{pmatrix} 4&1\\cr 1&4 \\end{pmatrix},\\, \\begin{pmatrix} 4&4\\cr 1&0 \\end{pmatrix} \\right\\}$ [ANS] 4. $S=\\left\\{\\begin{pmatrix}-2&2\\cr-3&1 \\end{pmatrix},\\, \\begin{pmatrix} 10&-10\\cr 15&-5 \\end{pmatrix} \\right\\}$", "answer_v3": [ "LINEARLY_DEPENDENT", "LINEARLY_INDEPENDENT", "LINEARLY_INDEPENDENT", "Linearly_Dependent" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ], [ "Linearly_Independent", "Linearly_Dependent" ] ] }, { "id": "Linear_algebra_0326", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Linear independence", "level": "3", "keywords": [ "vectors' 'basis' 'linearly independent' 'dependent" ], "problem_v1": "Let $S=\\left\\{r, u, d, x \\right\\}$ be a set of vectors.\nIf $x=4 r+3 u+4 d$, determine whether or not $S$ is linearly independent. [ANS] 1. Determine whether or not the four vectors listed above are linearly independent or linearly dependent.\nIf $S$ is dependent, enter a non-trivial linear relation below. Otherwise, enter 0's for the coefficients. [ANS] $r+$ [ANS] $u+$ [ANS] $d+$ [ANS] $x=0$.", "answer_v1": [ "LINEARLY_DEPENDENT", "4", "3", "4", "-1" ], "answer_type_v1": [ "MCS", "NV", "NV", "NV" ], "options_v1": [ [ "Linearly_Independent", "Linearly_Dependent" ], [], [], [] ], "problem_v2": "Let $S=\\left\\{r, u, d, x \\right\\}$ be a set of vectors.\nIf $x=r+5 u+d$, determine whether or not $S$ is linearly independent. [ANS] 1. Determine whether or not the four vectors listed above are linearly independent or linearly dependent.\nIf $S$ is dependent, enter a non-trivial linear relation below. Otherwise, enter 0's for the coefficients. [ANS] $r+$ [ANS] $u+$ [ANS] $d+$ [ANS] $x=0$.", "answer_v2": [ "LINEARLY_DEPENDENT", "1", "5", "1", "-1" ], "answer_type_v2": [ "MCS", "NV", "NV", "NV" ], "options_v2": [ [ "Linearly_Independent", "Linearly_Dependent" ], [], [], [] ], "problem_v3": "Let $S=\\left\\{r, u, d, x \\right\\}$ be a set of vectors.\nIf $x=2 r+4 u+2 d$, determine whether or not $S$ is linearly independent. [ANS] 1. Determine whether or not the four vectors listed above are linearly independent or linearly dependent.\nIf $S$ is dependent, enter a non-trivial linear relation below. Otherwise, enter 0's for the coefficients. [ANS] $r+$ [ANS] $u+$ [ANS] $d+$ [ANS] $x=0$.", "answer_v3": [ "LINEARLY_DEPENDENT", "2", "4", "2", "-1" ], "answer_type_v3": [ "MCS", "NV", "NV", "NV" ], "options_v3": [ [ "Linearly_Independent", "Linearly_Dependent" ], [], [], [] ] }, { "id": "Linear_algebra_0327", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Subspaces", "level": "4", "keywords": [], "problem_v1": "Which of the following subsets of ${\\mathbb R}^{3\\times 3}$ are subspaces of ${\\mathbb R}^{3\\times 3}$? [ANS] A\\. The $3\\times 3$ matrices whose entries are all greater than or equal to $0$ B\\. The $3\\times 3$ matrices with all zeros in the second row C\\. The $3\\times 3$ matrices with trace $0$ (the trace of a matrix is the sum of its diagonal entries) D\\. The diagonal $3\\times 3$ matrices E\\. The invertible $3\\times 3$ matrices F\\. The $3\\times 3$ matrices with determinant $0$", "answer_v1": [ "BCD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Which of the following subsets of ${\\mathbb R}^{3\\times 3}$ are subspaces of ${\\mathbb R}^{3\\times 3}$? [ANS] A\\. The diagonal $3\\times 3$ matrices B\\. The $3\\times 3$ matrices whose entries are all integers C\\. The $3\\times 3$ matrices whose entries are all greater than or equal to $0$ D\\. The $3\\times 3$ matrices $A$ such that the vector $\\left(\\begin{array}{c} 0 \\cr 8 \\cr 2 \\end{array} \\right)$ is in the kernel of $A$ E\\. The non-invertible $3\\times 3$ matrices F\\. The $3\\times 3$ matrices with trace $0$ (the trace of a matrix is the sum of its diagonal entries)", "answer_v2": [ "ADF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of the following subsets of ${\\mathbb R}^{3\\times 3}$ are subspaces of ${\\mathbb R}^{3\\times 3}$? [ANS] A\\. The $3\\times 3$ matrices with trace $0$ (the trace of a matrix is the sum of its diagonal entries) B\\. The $3\\times 3$ matrices with all zeros in the second row C\\. The $3\\times 3$ matrices whose entries are all integers D\\. The $3\\times 3$ matrices in reduced row-echelon form E\\. The $3\\times 3$ matrices with determinant $0$ F\\. The $3\\times 3$ matrices $A$ such that the vector $\\left(\\begin{array}{c} 2 \\cr 5 \\cr 1 \\end{array} \\right)$ is in the kernel of $A$", "answer_v3": [ "ABF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0328", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Subspaces", "level": "4", "keywords": [], "problem_v1": "Which of the following subsets of $P_2$ are subspaces of $P_2$? [ANS] A\\. $\\{p(t) \\ | \\ p(-t)=p(t)$ for all $t\\}$ B\\. $\\{p(t) \\ | \\ p'(t)$ is constant $\\}$ C\\. $\\{p(t) \\ | \\ p(5)=5\\}$ D\\. $\\{p(t) \\ | \\ p(6)=0\\}$ E\\. $\\{p(t) \\ | \\ p'(t)+7 p(t)+2=0\\}$ F\\. $\\{p(t) \\ | \\ p'(4)=p(7)\\}$", "answer_v1": [ "ABDF" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Which of the following subsets of $P_2$ are subspaces of $P_2$? [ANS] A\\. $\\{p(t) \\ | \\ p'(t)$ is constant $\\}$ B\\. $\\{p(t) \\ | \\ p(0)=0\\}$ C\\. $\\{p(t) \\ | \\ \\int_0^{7} p(t)dt=0\\}$ D\\. $\\{p(t) \\ | \\ p'(t)+7 p(t)+5=0\\}$ E\\. $\\{p(t) \\ | \\ p(2)=1\\}$ F\\. $\\{p(t) \\ | \\ p'(8)=p(4)\\}$", "answer_v2": [ "ABCF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of the following subsets of $P_2$ are subspaces of $P_2$? [ANS] A\\. $\\{p(t) \\ | \\ p'(t)+4 p(t)+3=0\\}$ B\\. $\\{p(t) \\ | \\ p'(t)$ is constant $\\}$ C\\. $\\{p(t) \\ | \\ p(2)=0\\}$ D\\. $\\{p(t) \\ | \\ p'(5)=p(6)\\}$ E\\. $\\{p(t) \\ | \\ p(1)=2\\}$ F\\. $\\{p(t) \\ | \\ \\int_0^{3} p(t)dt=0\\}$", "answer_v3": [ "BCDF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0329", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Subspaces", "level": "4", "keywords": [], "problem_v1": "Determine whether the given set $S$ is a subspace of the vector space $V$. [ANS] A\\. $V=P_3$, and $S$ is the subset of $P_3$ consisting of all polynomials of the form $p(x)=ax^3+bx.$ B\\. $V=M_n({\\mathbb R})$, and $S$ is the subset of all nonsingular matrices. C\\. $V={\\mathbb R}^n$, and $S$ is the set of solutions to the homogeneous linear system $Ax=0$ where $A$ is a fixed $m\\times n$ matrix. D\\. $V=M_n({\\mathbb R})$, and $S$ is the subset of all symmetric matrices E\\. $V$ is the vector space of all real-valued functions defined on the interval $(-\\infty, \\infty)$, and $S$ is the subset of $V$ consisting of those functions satisfying $f(0)=0.$ F\\. $V={\\mathbb R}^5$, and $S$ is the set of vectors $(x_1,x_2,x_3)$ in $V$ satisfying $x_1-8x_2+x_3=7.$ G\\. $V$ is the vector space of all real-valued functions defined on the interval $[a,b]$, and $S$ is the subset of $V$ consisting of those functions satisfying $f(a)=7.$", "answer_v1": [ "ACDE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Determine whether the given set $S$ is a subspace of the vector space $V$. [ANS] A\\. $V={\\mathbb R}^4$, and $S$ is the set of vectors of the form $(0, x_2, 3, x_4).$ B\\. $V=M_n({\\mathbb R})$, and $S$ is the subset of all $n\\times n$ matrices with det $(A)=0.$ C\\. $V=M_n({\\mathbb R})$, and $S$ is the subset of all upper triangular matrices. D\\. $V=C^1({\\mathbb R})$, and $S$ is the subset of $V$ consisting of those functions satisfying $f'(0)\\ge 0.$ E\\. $V=C^2(I)$, and $S$ is the subset of $V$ consisting of those functions satisfying the differential equation $y''-4y'+3y=0.$ F\\. $V$ is the vector space of all real-valued functions defined on the interval $[a,b]$, and $S$ is the subset of $V$ consisting of those functions satisfying $f(a)=f(b).$ G\\. $V={\\mathbb R}^n$, and $S$ is the set of solutions to the homogeneous linear system $Ax=0$ where $A$ is a fixed $m\\times n$ matrix.", "answer_v2": [ "CEFG" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Determine whether the given set $S$ is a subspace of the vector space $V$. [ANS] A\\. $V={\\mathbb R}^n$, and $S$ is the set of solutions to the homogeneous linear system $Ax=0$ where $A$ is a fixed $m\\times n$ matrix. B\\. $V=M_n({\\mathbb R})$, and $S$ is the subset of all upper triangular matrices. C\\. $V={\\mathbb R}^2$, and $S$ consists of all vectors $(x_1, x_2)$ satisfying $x_1^2-x_2^2=0.$ D\\. $V=P_3$, and $S$ is the subset of $P_3$ consisting of all polynomials of the form $p(x)=ax^3+bx.$ E\\. $V=M_n({\\mathbb R})$, and $S$ is the subset of all nonsingular matrices. F\\. $V=C^2(I)$, and $S$ is the subset of $V$ consisting of those functions satisfying the differential equation $y''-4y'+3y=0.$ G\\. $V=P_2$, and $S$ is the subset of $P_2$ consisting of all polynomials of the form $p(x)=x^2+c.$", "answer_v3": [ "ABDF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Linear_algebra_0330", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Basis and dimension", "level": "2", "keywords": [], "problem_v1": "Find the dimensions of the following linear spaces.\n(a) The space of all diagonal $7 \\times 7$ matrices [ANS]\n(b) $P_5$ [ANS]\n(c) The space of all upper triangular $2 \\times 2$ matrices [ANS]", "answer_v1": [ "7", "6", "3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the dimensions of the following linear spaces.\n(a) $P_3$ [ANS]\n(b) The space of all lower triangular $6 \\times 6$ matrices [ANS]\n(c) ${\\mathbb R}^{2 \\times 7}$ [ANS]", "answer_v2": [ "4", "21", "14" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the dimensions of the following linear spaces.\n(a) The real linear space ${\\mathbb C}^{\\,7}$ [ANS]\n(b) The space of all upper triangular $2 \\times 2$ matrices [ANS]\n(c) ${\\mathbb R}^{3 \\times 6}$ [ANS]", "answer_v3": [ "14", "3", "18" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0331", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Basis and dimension", "level": "2", "keywords": [], "problem_v1": "(a) If $S$ is the subspace of $M_{6} ({\\mathbb R})$ consisting of all upper triangular matrices, then $\\mbox{dim}S=$ [ANS]\n(b) If $S$ is the subspace of $M_{7} ({\\mathbb R})$ consisting of all skew-symmetric matrices, then $\\mbox{dim}S=$ [ANS]", "answer_v1": [ "21", "21" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) If $S$ is the subspace of $M_{3} ({\\mathbb R})$ consisting of all upper triangular matrices, then $\\mbox{dim}S=$ [ANS]\n(b) If $S$ is the subspace of $M_{8} ({\\mathbb R})$ consisting of all diagonal matrices, then $\\mbox{dim}S=$ [ANS]", "answer_v2": [ "6", "8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) If $S$ is the subspace of $M_{4} ({\\mathbb R})$ consisting of all upper triangular matrices, then $\\mbox{dim}S=$ [ANS]\n(b) If $S$ is the subspace of $M_{6} ({\\mathbb R})$ consisting of all diagonal matrices, then $\\mbox{dim}S=$ [ANS]", "answer_v3": [ "10", "6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0332", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Basis and dimension", "level": "3", "keywords": [ "linear algebra", "vector space", "subspaces" ], "problem_v1": "Let $\\mathcal{P}_2$ be the vector space of all polynomials of degree 2 or less, and let $H$ be the subspace spanned by $-\\left(6x^{2}+7x+8\\right), \\-\\left(7x^{2}+4x+8\\right)$ and $-\\left(2x^{2}+3x+3\\right)$.\nThe dimension of the subspace $H$ is [ANS].\nIs $\\lbrace-\\left(6x^{2}+7x+8\\right),-\\left(7x^{2}+4x+8\\right),-\\left(2x^{2}+3x+3\\right) \\rbrace$ a basis for $\\mathcal{P}_2$? [ANS] A basis for the subspace $H$ is $\\big\\lbrace$ [ANS] $\\big\\rbrace$. Enter a polynomial or a comma separated list of polynomials.", "answer_v1": [ "3", "basis for P_2", "(-(6*x^2+7*x+8), -(7*x^2+4*x+8), -(2*x^2+3*x+3))" ], "answer_type_v1": [ "NV", "MCS", "UOL" ], "options_v1": [ [], [ "basis for P_2", "not a basis for P_2" ], [] ], "problem_v2": "Let $\\mathcal{P}_2$ be the vector space of all polynomials of degree 2 or less, and let $H$ be the subspace spanned by $7x^{2}-2x+3, \\-\\left(x^{2}+x+1\\right)$ and $2x^{2}-x+1$.\nThe dimension of the subspace $H$ is [ANS].\nIs $\\lbrace 7x^{2}-2x+3,-\\left(x^{2}+x+1\\right), 2x^{2}-x+1 \\rbrace$ a basis for $\\mathcal{P}_2$? [ANS] A basis for the subspace $H$ is $\\big\\lbrace$ [ANS] $\\big\\rbrace$. Enter a polynomial or a comma separated list of polynomials.", "answer_v2": [ "3", "basis for P_2", "(7*x^2-2*x+3, -(x^2+x+1), 2*x^2-x+1)" ], "answer_type_v2": [ "NV", "MCS", "UOL" ], "options_v2": [ [], [ "basis for P_2", "not a basis for P_2" ], [] ], "problem_v3": "Let $\\mathcal{P}_2$ be the vector space of all polynomials of degree 2 or less, and let $H$ be the subspace spanned by $6x^{2}-5x+2, \\ x^{2}-4x-2$ and $2x^{2}-x+1$.\nThe dimension of the subspace $H$ is [ANS].\nIs $\\lbrace 6x^{2}-5x+2, x^{2}-4x-2, 2x^{2}-x+1 \\rbrace$ a basis for $\\mathcal{P}_2$? [ANS] A basis for the subspace $H$ is $\\big\\lbrace$ [ANS] $\\big\\rbrace$. Enter a polynomial or a comma separated list of polynomials.", "answer_v3": [ "3", "basis for P_2", "(6*x^2-5*x+2, x^2-4*x-2, 2*x^2-x+1)" ], "answer_type_v3": [ "NV", "MCS", "UOL" ], "options_v3": [ [], [ "basis for P_2", "not a basis for P_2" ], [] ] }, { "id": "Linear_algebra_0333", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Basis and dimension", "level": "2", "keywords": [ "linear algebra", "span" ], "problem_v1": "Find a basis $\\lbrace p(x), q(x) \\rbrace$ for the vector space $\\lbrace f(x)\\in{\\mathbb P}_3[x] \\mid f'(5)=f(1) \\rbrace$ where ${\\mathbb P}_3[x]$ is the vector space of polynomials in $x$ with degree less than 3. $p(x)=$ [ANS], $q(x)=$ [ANS]", "answer_v1": [ "9+x^2", "x" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find a basis $\\lbrace p(x), q(x) \\rbrace$ for the vector space $\\lbrace f(x)\\in{\\mathbb P}_3[x] \\mid f'(-8)=f(1) \\rbrace$ where ${\\mathbb P}_3[x]$ is the vector space of polynomials in $x$ with degree less than 3. $p(x)=$ [ANS], $q(x)=$ [ANS]", "answer_v2": [ "x^2-17", "x" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find a basis $\\lbrace p(x), q(x) \\rbrace$ for the vector space $\\lbrace f(x)\\in{\\mathbb P}_3[x] \\mid f'(-4)=f(1) \\rbrace$ where ${\\mathbb P}_3[x]$ is the vector space of polynomials in $x$ with degree less than 3. $p(x)=$ [ANS], $q(x)=$ [ANS]", "answer_v3": [ "x^2-9", "x" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0334", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [], "problem_v1": "The set $B=\\lbrace 2+x^{2}, \\ 8+2x+4x^{2}, \\-\\left(13+4x+7x^{2}\\right) \\rbrace$ is a basis for $P_2$. Find the coordinates of $p(x)=-\\left(14+4x+7x^{2}\\right)$ relative to this basis:\n$[p(x)]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "1", "-2", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The set $B=\\lbrace 4x^{2}-4, \\ 8x^{2}-\\left(8+x\\right), \\ 33+3x-32x^{2} \\rbrace$ is a basis for $P_2$. Find the coordinates of $p(x)=61+7x-60x^{2}$ relative to this basis:\n$[p(x)]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "1", "-4", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The set $B=\\lbrace x^{2}-2, \\ 3x^{2}-\\left(6+3x\\right), \\ 9x^{2}-\\left(16+6x\\right) \\rbrace$ is a basis for $P_2$. Find the coordinates of $p(x)=50+18x-27x^{2}$ relative to this basis:\n$[p(x)]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-3", "-2", "-2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0335", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [], "problem_v1": "The set B=\\left\\lbrace \\left[\\begin{array}{cc} 1 &1\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &1\\cr 0 &2 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &-2 \\end{array}\\right] \\right\\rbrace is a basis of the space of upper-triangular $2\\times 2$ matrices.\nFind the coordinates of $M=\\left[\\begin{array}{cc}-3 &1\\cr 0 &1 \\end{array}\\right]$ with respect to this basis.\n$[M]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-3", "4", "3.5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The set B=\\left\\lbrace \\left[\\begin{array}{cc}-1 &3\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &-1\\cr 0 &-1 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &2 \\end{array}\\right] \\right\\rbrace is a basis of the space of upper-triangular $2\\times 2$ matrices.\nFind the coordinates of $M=\\left[\\begin{array}{cc}-3 &-6\\cr 0 &-3 \\end{array}\\right]$ with respect to this basis.\n$[M]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "3", "15", "6" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The set B=\\left\\lbrace \\left[\\begin{array}{cc}-1 &1\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &-1\\cr 0 &-2 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &2 \\end{array}\\right] \\right\\rbrace is a basis of the space of upper-triangular $2\\times 2$ matrices.\nFind the coordinates of $M=\\left[\\begin{array}{cc} 8 &7\\cr 0 &-6 \\end{array}\\right]$ with respect to this basis.\n$[M]_B=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-8", "-15", "-18" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0336", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Coordinate vectors and change of basis", "level": "3", "keywords": [], "problem_v1": "Let $\\mathcal{P}_3$ be the vector space of all polynomials of degree $3$ or less in the variable $x$. Let \\begin{array}{lcl} p_1(x) &=& 1+x+x^{2}+x^{3}, \\\\ p_2(x) &=& 2+2x+2x^{2}+2x^{3}, \\\\ p_3(x) &=&-1-2x-x^{2}-x^{3}, \\\\ p_4(x) &=&-3-6x-2x^{2}-2x^{3} \\\\ \\end{array} and let $\\mathcal{C}=\\lbrace p_1(x), p_2(x), p_3(x), p_4(x) \\rbrace.$\nUse coordinate representations with respect to the basis $\\mathcal{B}=\\lbrace 1, x, x^2, x^3 \\rbrace$ to determine whether the set $\\mathcal{C}$ forms a basis for $\\mathcal{P}_3$. [ANS]\nFind a basis for $\\mathrm{span}(\\mathcal{C})$. Enter a polynomial or a comma separated list of polynomials. $\\big\\lbrace$ [ANS] $\\big\\rbrace$\nThe dimension of $\\mathrm{span}(\\mathcal{C})$ is [ANS].", "answer_v1": [ "not a basis for P_3", "(1+x+x^2+x^3, -1-2*x-x^2-x^3, -3-6*x-2*x^2-2*x^3)", "3" ], "answer_type_v1": [ "MCS", "UOL", "NV" ], "options_v1": [ [ "basis for P_3", "not a basis for P_3" ], [], [] ], "problem_v2": "Let $\\mathcal{P}_3$ be the vector space of all polynomials of degree $3$ or less in the variable $x$. Let \\begin{array}{lcl} p_1(x) &=&-2+x-x^{2}, \\\\ p_2(x) &=&-2+x-x^{2}, \\\\ p_3(x) &=&-2-x^{2}, \\\\ p_4(x) &=&-7+x-3x^{2} \\\\ \\end{array} and let $\\mathcal{C}=\\lbrace p_1(x), p_2(x), p_3(x), p_4(x) \\rbrace.$\nUse coordinate representations with respect to the basis $\\mathcal{B}=\\lbrace 1, x, x^2, x^3 \\rbrace$ to determine whether the set $\\mathcal{C}$ forms a basis for $\\mathcal{P}_3$. [ANS]\nFind a basis for $\\mathrm{span}(\\mathcal{C})$. Enter a polynomial or a comma separated list of polynomials. $\\big\\lbrace$ [ANS] $\\big\\rbrace$\nThe dimension of $\\mathrm{span}(\\mathcal{C})$ is [ANS].", "answer_v2": [ "not a basis for P_3", "(-2+x-x^2, -2-x^2, -7+x-3*x^2)", "3" ], "answer_type_v2": [ "MCS", "UOL", "NV" ], "options_v2": [ [ "basis for P_3", "not a basis for P_3" ], [], [] ], "problem_v3": "Let $\\mathcal{P}_3$ be the vector space of all polynomials of degree $3$ or less in the variable $x$. Let \\begin{array}{lcl} p_1(x) &=&-1+x-x^{2}, \\\\ p_2(x) &=&-3+3x-3x^{2}, \\\\ p_3(x) &=& 1-2x+x^{2}, \\\\ p_4(x) &=&-2-3x-x^{2} \\\\ \\end{array} and let $\\mathcal{C}=\\lbrace p_1(x), p_2(x), p_3(x), p_4(x) \\rbrace.$\nUse coordinate representations with respect to the basis $\\mathcal{B}=\\lbrace 1, x, x^2, x^3 \\rbrace$ to determine whether the set $\\mathcal{C}$ forms a basis for $\\mathcal{P}_3$. [ANS]\nFind a basis for $\\mathrm{span}(\\mathcal{C})$. Enter a polynomial or a comma separated list of polynomials. $\\big\\lbrace$ [ANS] $\\big\\rbrace$\nThe dimension of $\\mathrm{span}(\\mathcal{C})$ is [ANS].", "answer_v3": [ "not a basis for P_3", "(-1+x-x^2, 1-2*x+x^2, -2-3*x-x^2)", "3" ], "answer_type_v3": [ "MCS", "UOL", "NV" ], "options_v3": [ [ "basis for P_3", "not a basis for P_3" ], [], [] ] }, { "id": "Linear_algebra_0338", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Coordinate vectors and change of basis", "level": "2", "keywords": [], "problem_v1": "Let M=\\left[\\begin{array}{cc} 2 &2\\cr 1 &1 \\end{array}\\right] be a vector in the vector space $V$ of $2 \\times 2$ matrices with real number entries. Let \\mathcal{B}=\\left\\lbrace \\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right] \\right\\rbrace be an ordered basis for $V$.\nWrite $M$ as a linear combination of elements of $\\mathcal{B}$.\n$\\left[\\begin{array}{cc} 2 &2\\cr 1 &1 \\end{array}\\right]=$ [ANS] $\\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right].$\nLet $\\lbrack M \\rbrack_{\\mathcal{B}}$ denote the coordinate representation of $M$ relative to the basis $\\mathcal{B}$. Find the coordinate vector representation for $M$ relative to the basis $\\mathcal{B}$. $\\lbrack M \\rbrack_{\\mathcal{B}}=$ [ANS].", "answer_v1": [ "2", "1", "1", "2", "(2,1,1,2)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Let M=\\left[\\begin{array}{cc}-4 &-1\\cr-3 &4 \\end{array}\\right] be a vector in the vector space $V$ of $2 \\times 2$ matrices with real number entries. Let \\mathcal{B}=\\left\\lbrace \\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right] \\right\\rbrace be an ordered basis for $V$.\nWrite $M$ as a linear combination of elements of $\\mathcal{B}$.\n$\\left[\\begin{array}{cc}-4 &-1\\cr-3 &4 \\end{array}\\right]=$ [ANS] $\\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right].$\nLet $\\lbrack M \\rbrack_{\\mathcal{B}}$ denote the coordinate representation of $M$ relative to the basis $\\mathcal{B}$. Find the coordinate vector representation for $M$ relative to the basis $\\mathcal{B}$. $\\lbrack M \\rbrack_{\\mathcal{B}}=$ [ANS].", "answer_v2": [ "-4", "4", "-3", "-1", "(-4,4,-3,-1)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Let M=\\left[\\begin{array}{cc}-2 &-3\\cr-2 &1 \\end{array}\\right] be a vector in the vector space $V$ of $2 \\times 2$ matrices with real number entries. Let \\mathcal{B}=\\left\\lbrace \\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right] \\right\\rbrace be an ordered basis for $V$.\nWrite $M$ as a linear combination of elements of $\\mathcal{B}$.\n$\\left[\\begin{array}{cc}-2 &-3\\cr-2 &1 \\end{array}\\right]=$ [ANS] $\\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right]+$ [ANS] $\\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right].$\nLet $\\lbrack M \\rbrack_{\\mathcal{B}}$ denote the coordinate representation of $M$ relative to the basis $\\mathcal{B}$. Find the coordinate vector representation for $M$ relative to the basis $\\mathcal{B}$. $\\lbrack M \\rbrack_{\\mathcal{B}}=$ [ANS].", "answer_v3": [ "-2", "1", "-2", "-3", "(-2,1,-2,-3)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "OL" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0339", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Coordinate vectors and change of basis", "level": "2", "keywords": [], "problem_v1": "Let $\\mathcal{P}_2$ denote the vector space of all polynomials in the variable $x$ of degree less than or equal to $2$. Let $\\mathcal{B}=\\lbrace 1, x, x^2 \\rbrace$ be an ordered basis for $\\mathcal{P}_2$. Let $\\lbrack ~ \\rbrack_{\\mathcal{B}}: \\mathcal{P}_2 \\to \\mathbb{R}^3$ be the linear transformation determined by \\lbrack 1 \\rbrack_{\\mathcal{B}}=\\vec{e}_1, \\ \\ \\ \\lbrack x \\rbrack_{\\mathcal{B}}=\\vec{e}_2, \\ \\ \\ \\lbrack x^2 \\rbrack_{\\mathcal{B}}=\\vec{e}_3. Find the coordinate vector representation for each of the following polynomials.\n$\\lbrack 5 \\rbrack_{\\mathcal{B}}=$ [ANS].\n$\\lbrack 2+2x^{2} \\rbrack_{\\mathcal{B}}=$ [ANS].\n$\\lbrack 4x^{2}-4x-3 \\rbrack_{\\mathcal{B}}=$ [ANS].\nIs the linear transformation $\\lbrack ~ \\rbrack_{\\mathcal{B}}$ an isomorphism? [ANS]", "answer_v1": [ "(5,0,0)", "(2,0,2)", "(-3,-4,4)", "isomorphism" ], "answer_type_v1": [ "OL", "OL", "OL", "MCS" ], "options_v1": [ [], [], [], [ "isomorphism", "not an isomorphism" ] ], "problem_v2": "Let $\\mathcal{P}_2$ denote the vector space of all polynomials in the variable $x$ of degree less than or equal to $2$. Let $\\mathcal{B}=\\lbrace 1, x, x^2 \\rbrace$ be an ordered basis for $\\mathcal{P}_2$. Let $\\lbrack ~ \\rbrack_{\\mathcal{B}}: \\mathcal{P}_2 \\to \\mathbb{R}^3$ be the linear transformation determined by \\lbrack 1 \\rbrack_{\\mathcal{B}}=\\vec{e}_1, \\ \\ \\ \\lbrack x \\rbrack_{\\mathcal{B}}=\\vec{e}_2, \\ \\ \\ \\lbrack x^2 \\rbrack_{\\mathcal{B}}=\\vec{e}_3. Find the coordinate vector representation for each of the following polynomials.\n$\\lbrack-8 \\rbrack_{\\mathcal{B}}=$ [ANS].\n$\\lbrack 8-7x^{2} \\rbrack_{\\mathcal{B}}=$ [ANS].\n$\\lbrack-3x^{2}+8x-3 \\rbrack_{\\mathcal{B}}=$ [ANS].\nIs the linear transformation $\\lbrack ~ \\rbrack_{\\mathcal{B}}$ an isomorphism? [ANS]", "answer_v2": [ "(-8,0,0)", "(8,0,-7)", "(-3,8,-3)", "isomorphism" ], "answer_type_v2": [ "OL", "OL", "OL", "MCS" ], "options_v2": [ [], [], [], [ "isomorphism", "not an isomorphism" ] ], "problem_v3": "Let $\\mathcal{P}_2$ denote the vector space of all polynomials in the variable $x$ of degree less than or equal to $2$. Let $\\mathcal{B}=\\lbrace 1, x, x^2 \\rbrace$ be an ordered basis for $\\mathcal{P}_2$. Let $\\lbrack ~ \\rbrack_{\\mathcal{B}}: \\mathcal{P}_2 \\to \\mathbb{R}^3$ be the linear transformation determined by \\lbrack 1 \\rbrack_{\\mathcal{B}}=\\vec{e}_1, \\ \\ \\ \\lbrack x \\rbrack_{\\mathcal{B}}=\\vec{e}_2, \\ \\ \\ \\lbrack x^2 \\rbrack_{\\mathcal{B}}=\\vec{e}_3. Find the coordinate vector representation for each of the following polynomials.\n$\\lbrack-4 \\rbrack_{\\mathcal{B}}=$ [ANS].\n$\\lbrack 2-4x^{2} \\rbrack_{\\mathcal{B}}=$ [ANS].\n$\\lbrack x^{2}-6x-3 \\rbrack_{\\mathcal{B}}=$ [ANS].\nIs the linear transformation $\\lbrack ~ \\rbrack_{\\mathcal{B}}$ an isomorphism? [ANS]", "answer_v3": [ "(-4,0,0)", "(2,0,-4)", "(-3,-6,1)", "isomorphism" ], "answer_type_v3": [ "OL", "OL", "OL", "MCS" ], "options_v3": [ [], [], [], [ "isomorphism", "not an isomorphism" ] ] }, { "id": "Linear_algebra_0341", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Examples", "level": "2", "keywords": [ "vectors", "vector operations" ], "problem_v1": "Select the true statements about zero vectors. There may be more than one correct answer. [ANS] A\\. The zero vector in \\(\\mathcal{P}_2 \\) is \\(\\langle 0,0,0 \\rangle \\). B\\. The zero vector in \\(\\mathcal{F}(\\mathbb{R},\\mathbb{R}) \\) is \\(f(0)=0 \\). C\\. The zero vector in \\(M_{2,2}(\\mathbb{R}) \\) is \\(\\left(\\begin{array}{rr} 0 & 0 \\\\ 0 & 0 \\end{array} \\right) \\). D\\. The zero vector in \\(\\mathcal{F}(\\lbrace 1, 2, 3, 4 \\rbrace, \\mathbb{R}) \\) is the function \\(f: \\mathbb{R} \\to \\mathbb{R} \\) defined by \\(f(1)=0 \\), \\(f(2)=0 \\), \\(f(3)=0 \\), and \\(f(4)=0 \\). E\\. The zero vector in \\(\\mathcal{P}_5 \\) is \\(f(t)=0 \\) for all real numbers \\(t \\). F\\. The zero vector in \\(M_{2,2}(\\mathbb{R}) \\) is \\(\\langle 0, 0, 0, 0 \\rangle \\).\nSelect the true statements about vectors in vector spaces. There may be more than one correct answer. [ANS] A\\. The function \\(f(t)=\\ln(t) \\) is a vector in \\(\\mathcal{F}(\\mathbb{R},\\mathbb{R}) \\). B\\. The sum \\(\\langle 2,3 \\rangle+4 \\mathbf{e_1} \\) is a vector in \\(\\mathbb{R}^2 \\). C\\. The function \\(f(t)=e^t \\) is a vector in \\(\\mathcal{P}_{\\infty} \\). D\\. The function \\(f(t)=2+3t \\) is a vector in \\(\\mathcal{P}_5 \\). E\\. If \\(f(t) \\) is in \\(\\mathcal{P}_2 \\), then \\(f(t^3) \\) is in \\(\\mathcal{P}_2 \\). F\\. If \\(f(t) \\) is in \\(\\mathcal{P}_2 \\), then \\((f(t))^2 \\) is in \\(\\mathcal{P}_2 \\). G\\. The additive inverse of the vector \\(f(t)=4+5t+6t^2 \\) in \\(\\mathcal{P}_2 \\) is \\(f(-t) \\).", "answer_v1": [ "CDE", "BD" ], "answer_type_v1": [ "MCM", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Select the true statements about zero vectors. There may be more than one correct answer. [ANS] A\\. The zero vector in \\(M_{2,2}(\\mathbb{R}) \\) is \\(\\langle 0, 0, 0, 0 \\rangle \\). B\\. The zero vector in \\(\\mathcal{F}(\\mathbb{R},\\mathbb{R}) \\) is \\(f(0)=0 \\). C\\. The zero vector in \\(\\mathcal{P}_2 \\) is \\(\\langle 0,0,0 \\rangle \\). D\\. The zero vector in \\(M_{2,2}(\\mathbb{R}) \\) is \\(\\left(\\begin{array}{rr} 0 & 0 \\\\ 0 & 0 \\end{array} \\right) \\). E\\. The zero vector in \\(\\mathcal{F}(\\lbrace 1, 2, 3, 4 \\rbrace, \\mathbb{R}) \\) is the function \\(f: \\mathbb{R} \\to \\mathbb{R} \\) defined by \\(f(1)=0 \\), \\(f(2)=0 \\), \\(f(3)=0 \\), and \\(f(4)=0 \\). F\\. The zero vector in \\(\\mathcal{P}_5 \\) is \\(f(t)=0 \\) for all real numbers \\(t \\).\nSelect the true statements about vectors in vector spaces. There may be more than one correct answer. [ANS] A\\. If \\(f(t) \\) is in \\(\\mathcal{P}_2 \\), then \\((f(t))^2 \\) is in \\(\\mathcal{P}_2 \\). B\\. The function \\(f(t)=e^t \\) is a vector in \\(\\mathcal{P}_{\\infty} \\). C\\. The sum \\(\\langle 2,3 \\rangle+4 \\mathbf{e_1} \\) is a vector in \\(\\mathbb{R}^2 \\). D\\. The additive inverse of the vector \\(f(t)=4+5t+6t^2 \\) in \\(\\mathcal{P}_2 \\) is \\(f(-t) \\). E\\. The function \\(f(t)=\\ln(t) \\) is a vector in \\(\\mathcal{F}(\\mathbb{R},\\mathbb{R}) \\). F\\. The function \\(f(t)=2+3t \\) is a vector in \\(\\mathcal{P}_5 \\). G\\. If \\(f(t) \\) is in \\(\\mathcal{P}_2 \\), then \\(f(t^3) \\) is in \\(\\mathcal{P}_2 \\).", "answer_v2": [ "DEF", "CF" ], "answer_type_v2": [ "MCM", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Select the true statements about zero vectors. There may be more than one correct answer. [ANS] A\\. The zero vector in \\(M_{2,2}(\\mathbb{R}) \\) is \\(\\langle 0, 0, 0, 0 \\rangle \\). B\\. The zero vector in \\(M_{2,2}(\\mathbb{R}) \\) is \\(\\left(\\begin{array}{rr} 0 & 0 \\\\ 0 & 0 \\end{array} \\right) \\). C\\. The zero vector in \\(\\mathcal{F}(\\lbrace 1, 2, 3, 4 \\rbrace, \\mathbb{R}) \\) is the function \\(f: \\mathbb{R} \\to \\mathbb{R} \\) defined by \\(f(1)=0 \\), \\(f(2)=0 \\), \\(f(3)=0 \\), and \\(f(4)=0 \\). D\\. The zero vector in \\(\\mathcal{P}_2 \\) is \\(\\langle 0,0,0 \\rangle \\). E\\. The zero vector in \\(\\mathcal{P}_5 \\) is \\(f(t)=0 \\) for all real numbers \\(t \\). F\\. The zero vector in \\(\\mathcal{F}(\\mathbb{R},\\mathbb{R}) \\) is \\(f(0)=0 \\).\nSelect the true statements about vectors in vector spaces. There may be more than one correct answer. [ANS] A\\. The function \\(f(t)=2+3t \\) is a vector in \\(\\mathcal{P}_5 \\). B\\. The additive inverse of the vector \\(f(t)=4+5t+6t^2 \\) in \\(\\mathcal{P}_2 \\) is \\(f(-t) \\). C\\. If \\(f(t) \\) is in \\(\\mathcal{P}_2 \\), then \\(f(t^3) \\) is in \\(\\mathcal{P}_2 \\). D\\. If \\(f(t) \\) is in \\(\\mathcal{P}_2 \\), then \\((f(t))^2 \\) is in \\(\\mathcal{P}_2 \\). E\\. The function \\(f(t)=e^t \\) is a vector in \\(\\mathcal{P}_{\\infty} \\). F\\. The sum \\(\\langle 2,3 \\rangle+4 \\mathbf{e_1} \\) is a vector in \\(\\mathbb{R}^2 \\). G\\. The function \\(f(t)=\\ln(t) \\) is a vector in \\(\\mathcal{F}(\\mathbb{R},\\mathbb{R}) \\).", "answer_v3": [ "BCE", "AF" ], "answer_type_v3": [ "MCM", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Linear_algebra_0342", "subject": "Linear_algebra", "topic": "Abstract vector spaces", "subtopic": "Examples", "level": "4", "keywords": [ "vector space" ], "problem_v1": "Let $V={\\mathbb R}^2$. For $(u_1,u_2),(v_1,v_2) \\in V$ and $a\\in{\\mathbb R}$ define vector addition by $(u_1,u_2) \\boxplus (v_1,v_2):=(u_1+v_1-2,u_2+v_2-1)$ and scalar multiplication by $a \\boxdot (u_1,u_2):=(au_1-2a+2,au_2-a+1)$. It can be shown that $(V,\\boxplus,\\boxdot)$ is a vector space over the scalar field $\\mathbb R$. Find the following: the sum: $(-3,-2)\\boxplus (1,0)=$ ([ANS], [ANS]) the scalar multiple: $3\\boxdot (-3,-2)=$ ([ANS], [ANS]) the zero vector: $\\underline{0}_V=$ ([ANS], [ANS]) the additive inverse of $(x,y)$: $\\boxminus (x,y)=$ ([ANS], [ANS])", "answer_v1": [ "-4", "-3", "-13", "-8", "2", "1", "4-x", "2-y" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $V={\\mathbb R}^2$. For $(u_1,u_2),(v_1,v_2) \\in V$ and $a\\in{\\mathbb R}$ define vector addition by $(u_1,u_2) \\boxplus (v_1,v_2):=(u_1+v_1+3,u_2+v_2-3)$ and scalar multiplication by $a \\boxdot (u_1,u_2):=(au_1+3a-3,au_2-3a+3)$. It can be shown that $(V,\\boxplus,\\boxdot)$ is a vector space over the scalar field $\\mathbb R$. Find the following: the sum: $(1,-1)\\boxplus (-8,6)=$ ([ANS], [ANS]) the scalar multiple: $-6\\boxdot (1,-1)=$ ([ANS], [ANS]) the zero vector: $\\underline{0}_V=$ ([ANS], [ANS]) the additive inverse of $(x,y)$: $\\boxminus (x,y)=$ ([ANS], [ANS])", "answer_v2": [ "-4", "2", "-27", "27", "-3", "3", "-(6+x)", "6-y" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $V={\\mathbb R}^2$. For $(u_1,u_2),(v_1,v_2) \\in V$ and $a\\in{\\mathbb R}$ define vector addition by $(u_1,u_2) \\boxplus (v_1,v_2):=(u_1+v_1+1,u_2+v_2-1)$ and scalar multiplication by $a \\boxdot (u_1,u_2):=(au_1+a-1,au_2-a+1)$. It can be shown that $(V,\\boxplus,\\boxdot)$ is a vector space over the scalar field $\\mathbb R$. Find the following: the sum: $(-4,-3)\\boxplus (1,6)=$ ([ANS], [ANS]) the scalar multiple: $8\\boxdot (-4,-3)=$ ([ANS], [ANS]) the zero vector: $\\underline{0}_V=$ ([ANS], [ANS]) the additive inverse of $(x,y)$: $\\boxminus (x,y)=$ ([ANS], [ANS])", "answer_v3": [ "-2", "2", "-25", "-31", "-1", "1", "-(2+x)", "2-y" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0343", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "3", "keywords": [], "problem_v1": "Find the eigenvalues $\\lambda_1 < \\lambda_2 < \\lambda_3$ and associated unit eigenvectors $\\vec{u}_1, \\vec{u}_2, \\vec{u}_3$ of the symmetric matrix A=\\left[\\begin{array}{ccc}-1 &3 &-2\\cr 3 &-1 &-2\\cr-2 &-2 &4 \\end{array}\\right]. The eigenvalue $\\lambda_1=$ [ANS] has associated unit eigenvector $\\vec{u}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $\\lambda_2=$ [ANS] has associated unit eigenvector $\\vec{u}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $\\lambda_3=$ [ANS] has associated unit eigenvector $\\vec{u}_3=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}. Note: The eigenvectors above form an orthonormal eigenbasis for $A$.", "answer_v1": [ "-0.707107", "0.707107", "0", "0.57735", "0.57735", "0.57735", "0.408248", "0.408248", "-0.816497" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the eigenvalues $\\lambda_1 < \\lambda_2 < \\lambda_3$ and associated unit eigenvectors $\\vec{u}_1, \\vec{u}_2, \\vec{u}_3$ of the symmetric matrix A=\\left[\\begin{array}{ccc}-3 &3 &3\\cr 3 &0 &0\\cr 3 &0 &0 \\end{array}\\right]. The eigenvalue $\\lambda_1=$ [ANS] has associated unit eigenvector $\\vec{u}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $\\lambda_2=$ [ANS] has associated unit eigenvector $\\vec{u}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $\\lambda_3=$ [ANS] has associated unit eigenvector $\\vec{u}_3=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}. Note: The eigenvectors above form an orthonormal eigenbasis for $A$.", "answer_v2": [ "-0.816497", "0.408248", "0.408248", "0", "-0.707107", "0.707107", "0.57735", "0.57735", "0.57735" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the eigenvalues $\\lambda_1 < \\lambda_2 < \\lambda_3$ and associated unit eigenvectors $\\vec{u}_1, \\vec{u}_2, \\vec{u}_3$ of the symmetric matrix A=\\left[\\begin{array}{ccc}-4 &2 &2\\cr 2 &3 &-5\\cr 2 &-5 &3 \\end{array}\\right]. The eigenvalue $\\lambda_1=$ [ANS] has associated unit eigenvector $\\vec{u}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $\\lambda_2=$ [ANS] has associated unit eigenvector $\\vec{u}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $\\lambda_3=$ [ANS] has associated unit eigenvector $\\vec{u}_3=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}. Note: The eigenvectors above form an orthonormal eigenbasis for $A$.", "answer_v3": [ "-0.816497", "0.408248", "0.408248", "0.57735", "0.57735", "0.57735", "0", "-0.707107", "0.707107" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0344", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "3", "keywords": [], "problem_v1": "The matrix A=\\left[\\begin{array}{ccc} 0 &2 &-2\\cr 2 &0 &-2\\cr 2 &2 &-4 \\end{array}\\right] has two real eigenvalues, one of multiplicity $1$ and one of multiplicity $2$. Find the eigenvalues and a basis for each eigenspace.\nThe eigenvalue $\\lambda_1$ is [ANS] and a basis for its associated eigenspace is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$\nThe eigenvalue $\\lambda_2$ is [ANS] and a basis for its associated eigenspace is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v1": [ "1", "1", "1", "1", "0", "1", "1", "-1", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "The matrix A=\\left[\\begin{array}{ccc}-10 &10 &20\\cr 5 &-5 &-10\\cr-5 &5 &10 \\end{array}\\right] has two real eigenvalues, one of multiplicity $1$ and one of multiplicity $2$. Find the eigenvalues and a basis for each eigenspace.\nThe eigenvalue $\\lambda_1$ is [ANS] and a basis for its associated eigenspace is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$\nThe eigenvalue $\\lambda_2$ is [ANS] and a basis for its associated eigenspace is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v2": [ "-2", "1", "-1", "2", "0", "1", "-1", "-1", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "The matrix A=\\left[\\begin{array}{ccc} 0 &3 &0\\cr 0 &-3 &0\\cr 0 &3 &0 \\end{array}\\right] has two real eigenvalues, one of multiplicity $1$ and one of multiplicity $2$. Find the eigenvalues and a basis for each eigenspace.\nThe eigenvalue $\\lambda_1$ is [ANS] and a basis for its associated eigenspace is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$\nThe eigenvalue $\\lambda_2$ is [ANS] and a basis for its associated eigenspace is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v3": [ "-1", "1", "-1", "-1", "0", "-1", "0", "0", "-1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0345", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "3", "keywords": [], "problem_v1": "Find the eigenvalues and eigenvectors of the matrix \\left[\\begin{array}{ccc} 0 &-4 &0\\cr-8 &-4 &8\\cr-4 &-4 &4 \\end{array}\\right]. From smallest to largest, the eigenvalues are $\\lambda_1 < \\lambda_2 < \\lambda_3$ where\n$\\lambda_1=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\lambda_2=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\lambda_3=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "1", "1", "1", "1", "0", "1", "1", "-1", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the eigenvalues and eigenvectors of the matrix \\left[\\begin{array}{ccc} 2 &2 &-3\\cr 7 &-3 &-7\\cr-2 &2 &1 \\end{array}\\right]. From smallest to largest, the eigenvalues are $\\lambda_1 < \\lambda_2 < \\lambda_3$ where\n$\\lambda_1=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\lambda_2=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\lambda_3=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-1", "1", "-1", "1", "0", "1", "-1", "-1", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the eigenvalues and eigenvectors of the matrix \\left[\\begin{array}{ccc}-1 &1 &0\\cr 0 &-2 &0\\cr-4 &1 &3 \\end{array}\\right]. From smallest to largest, the eigenvalues are $\\lambda_1 < \\lambda_2 < \\lambda_3$ where\n$\\lambda_1=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\lambda_2=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\lambda_3=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "-1", "1", "-1", "-1", "0", "-1", "0", "0", "-1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0346", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "3", "keywords": [], "problem_v1": "Find the eigenvalues of the matrix A=\\left[\\begin{array}{cc} 0 &0\\cr-4 &4 \\end{array}\\right] The eigenvalues are [ANS]. (Enter your answers as a comma separated list. The list you enter should have repeated items if there are eigenvalues with multiplicity greater than one.)", "answer_v1": [ "(0, 4)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find the eigenvalues of the matrix A=\\left[\\begin{array}{cc}-46 &-120\\cr 15 &39 \\end{array}\\right] The eigenvalues are [ANS]. (Enter your answers as a comma separated list. The list you enter should have repeated items if there are eigenvalues with multiplicity greater than one.)", "answer_v2": [ "(-6, -1)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find the eigenvalues of the matrix A=\\left[\\begin{array}{cc}-4 &0\\cr-7 &3 \\end{array}\\right] The eigenvalues are [ANS]. (Enter your answers as a comma separated list. The list you enter should have repeated items if there are eigenvalues with multiplicity greater than one.)", "answer_v3": [ "(-4, 3)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0347", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "4", "keywords": [], "problem_v1": "The matrix A=\\left[\\begin{array}{ccc} 1 &1 &0\\cr 0 &1 &1\\cr k &0 &0\\cr \\end{array}\\right] has three distinct real eigenvalues if and only if [ANS] $< k <$ [ANS].", "answer_v1": [ "0", "0.148148" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The matrix A=\\left[\\begin{array}{ccc}-2 &-1 &0\\cr 1 &1 &-1\\cr k &0 &0\\cr \\end{array}\\right] has three distinct real eigenvalues if and only if [ANS] $< k <$ [ANS].", "answer_v2": [ "-0.185185", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The matrix A=\\left[\\begin{array}{ccc}-1 &1 &0\\cr 5 &0 &1\\cr k &0 &0\\cr \\end{array}\\right] has three distinct real eigenvalues if and only if [ANS] $< k <$ [ANS].", "answer_v3": [ "-3", "6.48148" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0348", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "2", "keywords": [], "problem_v1": "Find the three distinct real eigenvalues of the matrix B=\\left[\\begin{array}{ccc} 7 &1 &1\\cr 0 &-3 &-2\\cr 0 &0 &2 \\end{array}\\right]. The eigenvalues are [ANS]. (Enter your answers as a comma separated list.)", "answer_v1": [ "(7, -3, 2)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find the three distinct real eigenvalues of the matrix B=\\left[\\begin{array}{ccc} 5 &-6 &-3\\cr 0 &6 &1\\cr 0 &0 &-9 \\end{array}\\right]. The eigenvalues are [ANS]. (Enter your answers as a comma separated list.)", "answer_v2": [ "(5, 6, -9)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find the three distinct real eigenvalues of the matrix B=\\left[\\begin{array}{ccc} 1 &6 &8\\cr 0 &-7 &7\\cr 0 &0 &4 \\end{array}\\right]. The eigenvalues are [ANS]. (Enter your answers as a comma separated list.)", "answer_v3": [ "(1, -7, 4)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0349", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "2", "keywords": [], "problem_v1": "Given that $\\vec{v}_1=\\left[\\begin{array}{c} 1\\cr-1 \\end{array}\\right]$ and $\\vec{v}_2=\\left[\\begin{array}{c} 2\\cr-1 \\end{array}\\right]$ are eigenvectors of the matrix A=\\left[\\begin{array}{cc}-4 &-8\\cr 4 &8 \\end{array}\\right] determine the corresponding eigenvalues.\n$\\lambda_1=$ [ANS]. $\\lambda_2=$ [ANS].", "answer_v1": [ "4", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Given that $\\vec{v}_1=\\left[\\begin{array}{c}-1\\cr-2 \\end{array}\\right]$ and $\\vec{v}_2=\\left[\\begin{array}{c} 0\\cr-1 \\end{array}\\right]$ are eigenvectors of the matrix A=\\left[\\begin{array}{cc}-1 &0\\cr 10 &-6 \\end{array}\\right] determine the corresponding eigenvalues.\n$\\lambda_1=$ [ANS]. $\\lambda_2=$ [ANS].", "answer_v2": [ "-1", "-6" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Given that $\\vec{v}_1=\\left[\\begin{array}{c} 1\\cr-1 \\end{array}\\right]$ and $\\vec{v}_2=\\left[\\begin{array}{c} 2\\cr-1 \\end{array}\\right]$ are eigenvectors of the matrix A=\\left[\\begin{array}{cc} 10 &14\\cr-7 &-11 \\end{array}\\right] determine the corresponding eigenvalues.\n$\\lambda_1=$ [ANS]. $\\lambda_2=$ [ANS].", "answer_v3": [ "-4", "3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0350", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "3", "keywords": [], "problem_v1": "For which value of $k$ does the matrix A=\\left[\\begin{array}{cc} 4 &k\\cr 2 &2\\cr \\end{array}\\right] have one real eigenvalue of multiplicity $2$?\n$k=$ [ANS].", "answer_v1": [ "-(4-2)^2/(4*2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For which value of $k$ does the matrix A=\\left[\\begin{array}{cc}-7 &k\\cr 8 &-6\\cr \\end{array}\\right] have one real eigenvalue of multiplicity $2$?\n$k=$ [ANS].", "answer_v2": [ "-(-7--6)^2/(4*8)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For which value of $k$ does the matrix A=\\left[\\begin{array}{cc}-3 &k\\cr 2 &-4\\cr \\end{array}\\right] have one real eigenvalue of multiplicity $2$?\n$k=$ [ANS].", "answer_v3": [ "-(-3--4)^2/(4*2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0351", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "2", "keywords": [], "problem_v1": "Find the characteristic polynomial of the matrix A=\\left[\\begin{array}{ccc} 3 &1 &0\\cr 0 &1 &2\\cr-2 &-2 &0 \\end{array}\\right]. $p(x)=$ [ANS].", "answer_v1": [ "x^3-4*x^2+7*x-8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the characteristic polynomial of the matrix A=\\left[\\begin{array}{ccc}-5 &5 &0\\cr 0 &-4 &-2\\cr 5 &-2 &0 \\end{array}\\right]. $p(x)=$ [ANS].", "answer_v2": [ "x^3+9*x^2+16*x+30" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the characteristic polynomial of the matrix A=\\left[\\begin{array}{ccc}-2 &1 &0\\cr 0 &-2 &1\\cr-3 &-2 &0 \\end{array}\\right]. $p(x)=$ [ANS].", "answer_v3": [ "x^3+4*x^2+6*x+7" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0352", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "3", "keywords": [], "problem_v1": "Let $C^{\\infty}(\\mathbb{R})$ be the vector space of \"smooth\" functions, i.e., real-valued functions $f(x)$ in the variable $x$ that have infinitely many derivatives at all points $x \\in \\mathbb{R}$.\nLet $D: C^{\\infty}(\\mathbb{R}) \\to C^{\\infty}(\\mathbb{R})$ and $D^2: C^{\\infty}(\\mathbb{R}) \\to C^{\\infty}(\\mathbb{R})$ be the linear transformations defined by the first derivative $D(f(x))=f'(x)$ and the second derivative $D^2(f(x))=f''(x)$.\nDetermine whether the smooth function $g(x)=8e^{-2x}$ is an eigenvector of $D$. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue=[ANS]\nDetermine whether the smooth function $h(x)=\\cos\\!\\left(6x\\right)$ is an eigenvector of $D^2$. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue=[ANS]", "answer_v1": [ "-2", "-1*6^2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $C^{\\infty}(\\mathbb{R})$ be the vector space of \"smooth\" functions, i.e., real-valued functions $f(x)$ in the variable $x$ that have infinitely many derivatives at all points $x \\in \\mathbb{R}$.\nLet $D: C^{\\infty}(\\mathbb{R}) \\to C^{\\infty}(\\mathbb{R})$ and $D^2: C^{\\infty}(\\mathbb{R}) \\to C^{\\infty}(\\mathbb{R})$ be the linear transformations defined by the first derivative $D(f(x))=f'(x)$ and the second derivative $D^2(f(x))=f''(x)$.\nDetermine whether the smooth function $g(x)=2e^{-1x}$ is an eigenvector of $D$. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue=[ANS]\nDetermine whether the smooth function $h(x)=\\sin\\!\\left(3x\\right)$ is an eigenvector of $D^2$. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue=[ANS]", "answer_v2": [ "-1", "-1*3^2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $C^{\\infty}(\\mathbb{R})$ be the vector space of \"smooth\" functions, i.e., real-valued functions $f(x)$ in the variable $x$ that have infinitely many derivatives at all points $x \\in \\mathbb{R}$.\nLet $D: C^{\\infty}(\\mathbb{R}) \\to C^{\\infty}(\\mathbb{R})$ and $D^2: C^{\\infty}(\\mathbb{R}) \\to C^{\\infty}(\\mathbb{R})$ be the linear transformations defined by the first derivative $D(f(x))=f'(x)$ and the second derivative $D^2(f(x))=f''(x)$.\nDetermine whether the smooth function $g(x)=4e^{-2x}$ is an eigenvector of $D$. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue=[ANS]\nDetermine whether the smooth function $h(x)=\\cos\\!\\left(4x\\right)$ is an eigenvector of $D^2$. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue=[ANS]", "answer_v3": [ "-2", "-1*4^2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0354", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "2", "keywords": [ "linear algebra", "transition matrix" ], "problem_v1": "Find the characteristic polynomial $f(x)$ of $\\left[\\begin{array}{ccc}-5 &-8 &-1\\cr 4 &7 &1\\cr-8 &-10 &0 \\end{array}\\right]$. $f(x)=$ [ANS]", "answer_v1": [ "(2-x)*(-1-x)*(1-x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the characteristic polynomial $f(x)$ of $\\left[\\begin{array}{ccc}-3 &0 &0\\cr 8 &1 &0\\cr 17 &6 &-2 \\end{array}\\right]$. $f(x)=$ [ANS]", "answer_v2": [ "(-2-x)*(-3-x)*(1-x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the characteristic polynomial $f(x)$ of $\\left[\\begin{array}{ccc}-1 &-4 &-2\\cr 4 &7 &2\\cr-4 &-8 &-3 \\end{array}\\right]$. $f(x)=$ [ANS]", "answer_v3": [ "(-1-x)*(3-x)*(1-x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0355", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "2", "keywords": [ "eigenvalues' 'eigenvectors" ], "problem_v1": "Determine if $\\lambda$ is an eigenvalue of the matrix $A$.\n[ANS] 1. $A=\\left[\\begin{array}{cc}-31 &18\\cr-60 &35 \\end{array}\\right]$ and $\\lambda=1$ [ANS] 2. $A=\\left[\\begin{array}{cc} 17 &12\\cr-18 &-13 \\end{array}\\right]$ and $\\lambda=5$ [ANS] 3. $A=\\left[\\begin{array}{cc}-7 &4\\cr-8 &5 \\end{array}\\right]$ and $\\lambda=-3$", "answer_v1": [ "NO", "YES", "YES" ], "answer_type_v1": [ "TF", "TF", "TF" ], "options_v1": [ [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ] ], "problem_v2": "Determine if $\\lambda$ is an eigenvalue of the matrix $A$.\n[ANS] 1. $A=\\left[\\begin{array}{cc}-13 &-6\\cr 36 &17 \\end{array}\\right]$ and $\\lambda=-5$ [ANS] 2. $A=\\left[\\begin{array}{cc}-31 &-16\\cr 48 &25 \\end{array}\\right]$ and $\\lambda=1$ [ANS] 3. $A=\\left[\\begin{array}{cc}-23 &-24\\cr 16 &17 \\end{array}\\right]$ and $\\lambda=-3$", "answer_v2": [ "NO", "YES", "NO" ], "answer_type_v2": [ "TF", "TF", "TF" ], "options_v2": [ [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ] ], "problem_v3": "Determine if $\\lambda$ is an eigenvalue of the matrix $A$.\n[ANS] 1. $A=\\left[\\begin{array}{cc}-3 &2\\cr 0 &-5 \\end{array}\\right]$ and $\\lambda=-5$ [ANS] 2. $A=\\left[\\begin{array}{cc} 8 &-6\\cr 12 &-10 \\end{array}\\right]$ and $\\lambda=8$ [ANS] 3. $A=\\left[\\begin{array}{cc} 8 &-12\\cr 6 &-10 \\end{array}\\right]$ and $\\lambda=-4$", "answer_v3": [ "YES", "NO", "YES" ], "answer_type_v3": [ "TF", "TF", "TF" ], "options_v3": [ [ "Yes", "No" ], [ "Yes", "No" ], [ "Yes", "No" ] ] }, { "id": "Linear_algebra_0356", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Computing eigenvalues and eigenvectors", "level": "3", "keywords": [ "eigenvalues' 'eigenvectors" ], "problem_v1": "The matrix A=\\left[\\begin{array}{ccc} 13 &0 &18\\cr 18 &-5 &18\\cr-12 &0 &-17 \\end{array}\\right] has $\\lambda=-5$ as an eigenvalue with multiplicity $2$ and $\\lambda=1$ as an eigenvalue with multiplicity $1$. Give one associated eigenvector for each of the eigenvalues. The eigenvalue $-5$ has associated eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}. The eigenvalue $1$ has associated eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "-1", "-1", "1", "3", "3", "-2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The matrix A=\\left[\\begin{array}{ccc}-11 &0 &7\\cr-7 &-4 &7\\cr-14 &0 &10 \\end{array}\\right] has $\\lambda=-4$ as an eigenvalue with multiplicity $2$ and $\\lambda=3$ as an eigenvalue with multiplicity $1$. Give one associated eigenvector for each of the eigenvalues. The eigenvalue $-4$ has associated eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}. The eigenvalue $3$ has associated eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-1", "-1", "-1", "1", "1", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The matrix A=\\left[\\begin{array}{ccc} 22 &0 &-27\\cr-9 &-5 &9\\cr 18 &0 &-23 \\end{array}\\right] has $\\lambda=-5$ as an eigenvalue with multiplicity $2$ and $\\lambda=4$ as an eigenvalue with multiplicity $1$. Give one associated eigenvector for each of the eigenvalues. The eigenvalue $-5$ has associated eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}. The eigenvalue $4$ has associated eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "-1", "1", "-1", "-3", "1", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0357", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Properties", "level": "3", "keywords": [], "problem_v1": "Suppose that the trace of a $2\\times 2$ matrix $A$ is ${\\rm tr}(A)=9$ and the determinant is ${\\rm det}(A)=14$. Find the eigenvalues of $A$.\nThe eigenvalues of $A$ are [ANS]. (Enter your answers as a comma separated list.)", "answer_v1": [ "(2, 7)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Suppose that the trace of a $2\\times 2$ matrix $A$ is ${\\rm tr}(A)=-9$ and the determinant is ${\\rm det}(A)=8$. Find the eigenvalues of $A$.\nThe eigenvalues of $A$ are [ANS]. (Enter your answers as a comma separated list.)", "answer_v2": [ "(-8, -1)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Suppose that the trace of a $2\\times 2$ matrix $A$ is ${\\rm tr}(A)=-3$ and the determinant is ${\\rm det}(A)=-18$. Find the eigenvalues of $A$.\nThe eigenvalues of $A$ are [ANS]. (Enter your answers as a comma separated list.)", "answer_v3": [ "(-6, 3)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0358", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Properties", "level": "3", "keywords": [], "problem_v1": "If $\\vec{v}_1=\\left[\\begin{array}{c} 3\\cr 1 \\end{array}\\right]$ and $\\vec{v}_2=\\left[\\begin{array}{c} 1\\cr 2 \\end{array}\\right]$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\\lambda_1=-2$ and $\\lambda_2=6$, respectively,\nthen $A(\\vec{v}_1+\\vec{v}_2)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\nand $A(3 \\vec{v}_1)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0", "10", "-18", "-6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "If $\\vec{v}_1=\\left[\\begin{array}{c}-5\\cr 5 \\end{array}\\right]$ and $\\vec{v}_2=\\left[\\begin{array}{c}-4\\cr-2 \\end{array}\\right]$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\\lambda_1=5$ and $\\lambda_2=-2$, respectively,\nthen $A(\\vec{v}_1+\\vec{v}_2)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\nand $A(-2 \\vec{v}_1)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-17", "29", "50", "-50" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "If $\\vec{v}_1=\\left[\\begin{array}{c}-2\\cr 1 \\end{array}\\right]$ and $\\vec{v}_2=\\left[\\begin{array}{c}-3\\cr-2 \\end{array}\\right]$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\\lambda_1=3$ and $\\lambda_2=5$, respectively,\nthen $A(\\vec{v}_1+\\vec{v}_2)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\nand $A(-3 \\vec{v}_1)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-21", "-7", "18", "-9" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0359", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Properties", "level": "4", "keywords": [], "problem_v1": "Supppose $A$ is an invertible $n\\times n$ matrix and $\\vec{v}$ is an eigenvector of $A$ with associated eigenvalue $7$. Convince yourself that $\\vec{v}$ is an eigenvector of the following matrices, and find the associated eigenvalues.\nThe matrix $A^{6}$ has an eigenvalue [ANS].\nThe matrix $A^{-1}$ has an eigenvalue [ANS].\nThe matrix $A-7 I_n$ has an eigenvalue [ANS].\nThe matrix $4 A$ has an eigenvalue [ANS].", "answer_v1": [ "7^6", "1/7", "7+-7", "4*7" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Supppose $A$ is an invertible $n\\times n$ matrix and $\\vec{v}$ is an eigenvector of $A$ with associated eigenvalue $2$. Convince yourself that $\\vec{v}$ is an eigenvector of the following matrices, and find the associated eigenvalues.\nThe matrix $A^{3}$ has an eigenvalue [ANS].\nThe matrix $A^{-1}$ has an eigenvalue [ANS].\nThe matrix $A+4 I_n$ has an eigenvalue [ANS].\nThe matrix $-4 A$ has an eigenvalue [ANS].", "answer_v2": [ "2^3", "1/2", "2+4", "-4*2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Supppose $A$ is an invertible $n\\times n$ matrix and $\\vec{v}$ is an eigenvector of $A$ with associated eigenvalue $4$. Convince yourself that $\\vec{v}$ is an eigenvector of the following matrices, and find the associated eigenvalues.\nThe matrix $A^{4}$ has an eigenvalue [ANS].\nThe matrix $A^{-1}$ has an eigenvalue [ANS].\nThe matrix $A-6 I_n$ has an eigenvalue [ANS].\nThe matrix $4 A$ has an eigenvalue [ANS].", "answer_v3": [ "4^4", "1/4", "4+-6", "4*4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0360", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Properties", "level": "4", "keywords": [], "problem_v1": "Suppose a $3\\times 3$ matrix $A$ has only two distinct eigenvalues. Suppose that ${\\rm tr}(A)=0$ and ${\\rm det}(A)=-54$. Find the eigenvalues of $A$ with their algebraic multiplicities.\nThe smaller eigenvalue=[ANS] has multiplicity [ANS], and the larger eigenvalue=[ANS] has multiplicity [ANS].", "answer_v1": [ "-6", "1", "3", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose a $3\\times 3$ matrix $A$ has only two distinct eigenvalues. Suppose that ${\\rm tr}(A)=0$ and ${\\rm det}(A)=16$. Find the eigenvalues of $A$ with their algebraic multiplicities.\nThe smaller eigenvalue=[ANS] has multiplicity [ANS], and the larger eigenvalue=[ANS] has multiplicity [ANS].", "answer_v2": [ "-2", "2", "4", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose a $3\\times 3$ matrix $A$ has only two distinct eigenvalues. Suppose that ${\\rm tr}(A)=-1$ and ${\\rm det}(A)=45$. Find the eigenvalues of $A$ with their algebraic multiplicities.\nThe smaller eigenvalue=[ANS] has multiplicity [ANS], and the larger eigenvalue=[ANS] has multiplicity [ANS].", "answer_v3": [ "-3", "2", "5", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0361", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Properties", "level": "4", "keywords": [ "linear algebra", "matrix", "characteristic polynomial" ], "problem_v1": "Let $M=\\left[\\begin{array}{cc} 3 &1\\cr 2 &3 \\end{array}\\right]$. Find $c_1$ and $c_2$ such that $M^2+c_1 M+c_2 I_2=0$, where $I_2$ is the identity $2\\times 2$ matrix.\n$c_1=$ [ANS], $c_2=$ [ANS].", "answer_v1": [ "-6", "7" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $M=\\left[\\begin{array}{cc}-5 &6\\cr-5 &-2 \\end{array}\\right]$. Find $c_1$ and $c_2$ such that $M^2+c_1 M+c_2 I_2=0$, where $I_2$ is the identity $2\\times 2$ matrix.\n$c_1=$ [ANS], $c_2=$ [ANS].", "answer_v2": [ "7", "40" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $M=\\left[\\begin{array}{cc}-2 &1\\cr-3 &1 \\end{array}\\right]$. Find $c_1$ and $c_2$ such that $M^2+c_1 M+c_2 I_2=0$, where $I_2$ is the identity $2\\times 2$ matrix.\n$c_1=$ [ANS], $c_2=$ [ANS].", "answer_v3": [ "1", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0362", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Properties", "level": "3", "keywords": [ "determinant' 'characteristic polynomial' 'eigenvalues" ], "problem_v1": "Let $A$ be a $2\\times2$ matrix with trace 11 and determinant 30. Find the eigenvalues $\\lambda_1$, $\\lambda_2$ of $A$.[ANS]", "answer_v1": [ "(5,6)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Let $A$ be a $2\\times2$ matrix with trace 0 and determinant-64. Find the eigenvalues $\\lambda_1$, $\\lambda_2$ of $A$.[ANS]", "answer_v2": [ "(-8,8)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Let $A$ be a $2\\times2$ matrix with trace-2 and determinant-8. Find the eigenvalues $\\lambda_1$, $\\lambda_2$ of $A$.[ANS]", "answer_v3": [ "(-4,2)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0363", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Properties", "level": "4", "keywords": [ "eigenvalues' 'eigenvectors" ], "problem_v1": "$A$ is an $n\\times n$ matrix.\nCheck the true statements below: [ANS] A\\. A steady-state vector for a stochastic matrix is actually an eigenvector. B\\. The eigenvalues of a matrix are on its main diagonal. C\\. If $v_{1}$ and $v_{2}$ are linearly independent eigenvectors, then they correspond to distinct eigenvalues. D\\. If $Ax=\\lambda x$ for some vector $x$, then $x$ is an eigenvector of $A$. E\\. An eigenspace of $A$ is just a null space of a certain matrix.", "answer_v1": [ "AE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "$A$ is an $n\\times n$ matrix.\nCheck the true statements below: [ANS] A\\. An eigenspace of $A$ is just a null space of a certain matrix. B\\. The eigenvalues of a matrix are on its main diagonal. C\\. If $v_{1}$ and $v_{2}$ are linearly independent eigenvectors, then they correspond to distinct eigenvalues. D\\. A steady-state vector for a stochastic matrix is actually an eigenvector. E\\. If $Ax=\\lambda x$ for some vector $x$, then $x$ is an eigenvector of $A$.", "answer_v2": [ "AD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "$A$ is an $n\\times n$ matrix.\nCheck the true statements below: [ANS] A\\. If $Ax=\\lambda x$ for some vector $x$, then $x$ is an eigenvector of $A$. B\\. An eigenspace of $A$ is just a null space of a certain matrix. C\\. The eigenvalues of a matrix are on its main diagonal. D\\. If $v_{1}$ and $v_{2}$ are linearly independent eigenvectors, then they correspond to distinct eigenvalues. E\\. A steady-state vector for a stochastic matrix is actually an eigenvector.", "answer_v3": [ "BE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Linear_algebra_0364", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Properties", "level": "4", "keywords": [ "eigenvalues' 'eigenvectors" ], "problem_v1": "$A$ is an $n\\times n$ matrix.\nCheck the true statements below: [ANS] A\\. A matrix $A$ is not invertible if and only if 0 is an eigenvalue of $A$. B\\. If $Ax=\\lambda x$ for some vector $x$, then $\\lambda$ is an eigenvalue of $A$. C\\. To find the eigenvalues of $A$, reduce $A$ to echelon form. D\\. A number $c$ is an eigenvalue of $A$ if and only if the equation $(A-cI)x=0$ has a nontrivial solution $x$. E\\. Finding an eigenvector of $A$ might be difficult, but checking whether a given vector is in fact an eigenvector is easy.", "answer_v1": [ "ADE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "$A$ is an $n\\times n$ matrix.\nCheck the true statements below: [ANS] A\\. A matrix $A$ is not invertible if and only if 0 is an eigenvalue of $A$. B\\. If $Ax=\\lambda x$ for some vector $x$, then $\\lambda$ is an eigenvalue of $A$. C\\. A number $c$ is an eigenvalue of $A$ if and only if the equation $(A-cI)x=0$ has a nontrivial solution $x$. D\\. To find the eigenvalues of $A$, reduce $A$ to echelon form. E\\. Finding an eigenvector of $A$ might be difficult, but checking whether a given vector is in fact an eigenvector is easy.", "answer_v2": [ "ACE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "$A$ is an $n\\times n$ matrix.\nCheck the true statements below: [ANS] A\\. If $Ax=\\lambda x$ for some vector $x$, then $\\lambda$ is an eigenvalue of $A$. B\\. A matrix $A$ is not invertible if and only if 0 is an eigenvalue of $A$. C\\. A number $c$ is an eigenvalue of $A$ if and only if the equation $(A-cI)x=0$ has a nontrivial solution $x$. D\\. To find the eigenvalues of $A$, reduce $A$ to echelon form. E\\. Finding an eigenvector of $A$ might be difficult, but checking whether a given vector is in fact an eigenvector is easy.", "answer_v3": [ "BCE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Linear_algebra_0365", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Complex eigenvalues and eigenvectors", "level": "3", "keywords": [], "problem_v1": "Find all the eigenvalues (real and complex) of the matrix M=\\left[\\begin{array}{cccc} 7 &-5 &-18 &0\\cr-5 &7 &14 &0\\cr 5 &-5 &-12 &0\\cr-4 &0 &8 &-2 \\end{array}\\right]. The eigenvalues are [ANS]. (Enter your answers as a comma separated list.)", "answer_v1": [ "(2, -2, 4i, -4i)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all the eigenvalues (real and complex) of the matrix M=\\left[\\begin{array}{cccc} 5 &0 &-3 &1\\cr-13 &0 &4 &-5\\cr 16 &0 &-4 &8\\cr-2 &0 &3 &2 \\end{array}\\right]. The eigenvalues are [ANS]. (Enter your answers as a comma separated list.)", "answer_v2": [ "(0, 3, 2.82843i, -2.82843i)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all the eigenvalues (real and complex) of the matrix M=\\left[\\begin{array}{cccc}-10 &17 &11 &-5\\cr-4 &6 &1 &0\\cr 4 &-6 &-1 &0\\cr 15 &-23 &-13 &6 \\end{array}\\right]. The eigenvalues are [ANS]. (Enter your answers as a comma separated list.)", "answer_v3": [ "(0, 1, 4.3589i, -4.3589i)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0366", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Complex eigenvalues and eigenvectors", "level": "4", "keywords": [], "problem_v1": "Find all the values of $k$ for which the matrix \\left[\\begin{array}{ccc} 0 &1 &0\\cr 0 &0 &1\\cr 0 &k-7 &-k+8\\cr \\end{array}\\right] is not diagonalizable over ${\\mathbb C}$.\n$k=$ [ANS] (Enter your answers as a comma separated list.)", "answer_v1": [ "(7, 6)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all the values of $k$ for which the matrix \\left[\\begin{array}{ccc} 0 &1 &0\\cr 0 &0 &1\\cr 0 &k-1 &-k+2\\cr \\end{array}\\right] is not diagonalizable over ${\\mathbb C}$.\n$k=$ [ANS] (Enter your answers as a comma separated list.)", "answer_v2": [ "(1, 0)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all the values of $k$ for which the matrix \\left[\\begin{array}{ccc} 0 &1 &0\\cr 0 &0 &1\\cr 0 &k-3 &-k+4\\cr \\end{array}\\right] is not diagonalizable over ${\\mathbb C}$.\n$k=$ [ANS] (Enter your answers as a comma separated list.)", "answer_v3": [ "(3, 2)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0367", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Complex eigenvalues and eigenvectors", "level": "2", "keywords": [ "" ], "problem_v1": "Consider the multiplication operator $L_A:\\mathbb{C}^2\\to\\mathbb{C}^2$ where A=\\left[\\begin{array}{cc}-2 &1\\cr-2 &0 \\end{array}\\right]. Find a basis $B$ for a one dimensional $L_A$-invariant subspace. $B=\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $\\}$", "answer_v1": [ "1", "1+i" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Consider the multiplication operator $L_A:\\mathbb{C}^2\\to\\mathbb{C}^2$ where A=\\left[\\begin{array}{cc} 2 &3\\cr-6 &-4 \\end{array}\\right]. Find a basis $B$ for a one dimensional $L_A$-invariant subspace. $B=\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $\\}$", "answer_v2": [ "-i", "1+i" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Consider the multiplication operator $L_A:\\mathbb{C}^2\\to\\mathbb{C}^2$ where A=\\left[\\begin{array}{cc}-3 &2\\cr-1 &-1 \\end{array}\\right]. Find a basis $B$ for a one dimensional $L_A$-invariant subspace. $B=\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} $\\}$", "answer_v3": [ "-1+i", "-1" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0368", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Complex eigenvalues and eigenvectors", "level": "", "keywords": [], "problem_v1": "Find the eigenvalues and eigenvectors for $A=\\left[\\begin{array}{cc} 0 &-6\\cr 3 &-6 \\end{array}\\right]$.\nThe eigenvalue $a+b i=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $a-b i=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "1-i", "-i", "1+i", "i" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the eigenvalues and eigenvectors for $A=\\left[\\begin{array}{cc}-1 &-30\\cr 3 &17 \\end{array}\\right]$.\nThe eigenvalue $a+b i=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $a-b i=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-1-3i", "i", "-1+3i", "-i" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the eigenvalues and eigenvectors for $A=\\left[\\begin{array}{cc}-8 &6\\cr-3 &-2 \\end{array}\\right]$.\nThe eigenvalue $a+b i=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.\nThe eigenvalue $a-b i=$ [ANS] has an eigenvector \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "-1-i", "-i", "-1+i", "i" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0369", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Quadratic forms", "level": "2", "keywords": [ "linear algebra", "matrix", "quadratic form" ], "problem_v1": "If A=\\left[\\begin{array}{cc} 5 &2\\cr 2 &2 \\end{array}\\right] and $Q(\\vec{x})=\\vec{x}^T A \\vec{x},$ then\n$Q(\\vec{e}_1)=$ [ANS], $Q(\\vec{e}_2)=$ [ANS].", "answer_v1": [ "5", "2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If A=\\left[\\begin{array}{cc}-8 &8\\cr 8 &-7 \\end{array}\\right] and $Q(\\vec{x})=\\vec{x}^T A \\vec{x},$ then\n$Q(\\vec{e}_1)=$ [ANS], $Q(\\vec{e}_2)=$ [ANS].", "answer_v2": [ "-8", "-7" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If A=\\left[\\begin{array}{cc}-4 &2\\cr 2 &-4 \\end{array}\\right] and $Q(\\vec{x})=\\vec{x}^T A \\vec{x},$ then\n$Q(\\vec{e}_1)=$ [ANS], $Q(\\vec{e}_2)=$ [ANS].", "answer_v3": [ "-4", "-4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0370", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Quadratic forms", "level": "2", "keywords": [ "linear algebra", "matrix", "quadratic form" ], "problem_v1": "Write the matrix of the quadratic form Q(\\vec{x})=5x_{1}^{2}+2x_{2}^{2}+2x_{3}^{2}+4x_{1}x_{2}-4x_{1}x_{3}-3x_{2}x_{3}. $A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "5", "2", "-2", "2", "2", "-1.5", "-2", "-1.5", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Write the matrix of the quadratic form Q(\\vec{x})=8x_{2}^{2}-8x_{1}^{2}-7x_{3}^{2}-3x_{1}x_{2}+8x_{1}x_{3}-3x_{2}x_{3}. $A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-8", "-1.5", "4", "-1.5", "8", "-1.5", "4", "-1.5", "-7" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Write the matrix of the quadratic form Q(\\vec{x})=2x_{2}^{2}-4x_{1}^{2}-4x_{3}^{2}+x_{1}x_{2}-6x_{1}x_{3}-3x_{2}x_{3}. $A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-4", "0.5", "-3", "0.5", "2", "-1.5", "-3", "-1.5", "-4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0371", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Quadratic forms", "level": "4", "keywords": [ "linear algebra", "matrix", "quadratic form", "eigenvalue" ], "problem_v1": "The matrix A=\\left[\\begin{array}{ccc}-6 &0 &0\\cr 0 &-0.4 &-0.8\\cr 0 &-0.8 &-1.6 \\end{array}\\right] has three distinct eigenvalues, $\\lambda_1 < \\lambda_2 < \\lambda_3$. Enter the eigenvalues as a comma separated list. [ANS].\nClassify the quadratic form $Q(\\vec{x})=\\vec{x}^TA\\vec{x}:$ [ANS] A\\. $Q(\\vec{x})$ is negative semidefinite B\\. $Q(\\vec{x})$ is negative definite C\\. $Q(\\vec{x})$ is positive semidefinite D\\. $Q(\\vec{x})$ is indefinite E\\. $Q(\\vec{x})$ is positive definite", "answer_v1": [ "(-6, -2, 0)", "A" ], "answer_type_v1": [ "OL", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The matrix A=\\left[\\begin{array}{ccc} 2.1 &0 &0.3\\cr 0 &5 &0\\cr 0.3 &0 &2.9 \\end{array}\\right] has three distinct eigenvalues, $\\lambda_1 < \\lambda_2 < \\lambda_3$. Enter the eigenvalues as a comma separated list. [ANS].\nClassify the quadratic form $Q(\\vec{x})=\\vec{x}^TA\\vec{x}:$ [ANS] A\\. $Q(\\vec{x})$ is positive definite B\\. $Q(\\vec{x})$ is positive semidefinite C\\. $Q(\\vec{x})$ is negative definite D\\. $Q(\\vec{x})$ is indefinite E\\. $Q(\\vec{x})$ is negative semidefinite", "answer_v2": [ "(2, 3, 5)", "A" ], "answer_type_v2": [ "OL", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "The matrix A=\\left[\\begin{array}{ccc} 0 &0 &0\\cr 0 &5.6 &-0.8\\cr 0 &-0.8 &4.4 \\end{array}\\right] has three distinct eigenvalues, $\\lambda_1 < \\lambda_2 < \\lambda_3$. Enter the eigenvalues as a comma separated list. [ANS].\nClassify the quadratic form $Q(\\vec{x})=\\vec{x}^TA\\vec{x}:$ [ANS] A\\. $Q(\\vec{x})$ is positive definite B\\. $Q(\\vec{x})$ is indefinite C\\. $Q(\\vec{x})$ is negative definite D\\. $Q(\\vec{x})$ is positive semidefinite E\\. $Q(\\vec{x})$ is negative semidefinite", "answer_v3": [ "(0, 4, 6)", "D" ], "answer_type_v3": [ "OL", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Linear_algebra_0372", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Determine for which of the following matrices $M$ the zero state is a stable equilibrium of the dynamical system $x(t+1)=Mx(t)$. [ANS] A\\. $M=\\left\\lbrack \\begin{array}{rrr}-3 & 0 &-1 \\\\ 0 &-4 & 0 \\\\ 7 & 0 & 9 \\end{array} \\right\\rbrack$ B\\. $M=\\left\\lbrack \\begin{array}{rr} 0 &-0.3 \\\\ 2 & 0 \\end{array} \\right\\rbrack$ C\\. $M=\\left\\lbrack \\begin{array}{rrr}-0.1 &-0.1 &-0.1 \\\\-0.1 &-0.1 &-0.1 \\\\-0.1 &-0.1 &-0.1 \\end{array} \\right\\rbrack$ D\\. $M=\\left\\lbrack \\begin{array}{rr}-4 & 0 \\\\ 0 & 0.5 \\end{array} \\right\\rbrack$ E\\. $M=\\left\\lbrack \\begin{array}{rr} 0.5 & 0 \\\\ 0 & 0.6 \\end{array} \\right\\rbrack$ F\\. $M=\\left\\lbrack \\begin{array}{rr}-3 & 0.2 \\\\-0.2 &-3 \\end{array} \\right\\rbrack$", "answer_v1": [ "BCE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Determine for which of the following matrices $M$ the zero state is a stable equilibrium of the dynamical system $x(t+1)=Mx(t)$. [ANS] A\\. $M=\\left\\lbrack \\begin{array}{rr}-9 & 0 \\\\ 0 &-0.1 \\end{array} \\right\\rbrack$ B\\. $M=\\left\\lbrack \\begin{array}{rrr}-7 & 0 &-8 \\\\ 0 &-8 & 0 \\\\-4 & 0 & 0 \\end{array} \\right\\rbrack$ C\\. $M=\\left\\lbrack \\begin{array}{rrr}-0.2 &-0.2 &-0.2 \\\\-0.2 &-0.2 &-0.2 \\\\-0.2 &-0.2 &-0.2 \\end{array} \\right\\rbrack$ D\\. $M=\\left\\lbrack \\begin{array}{rr}-0.8 & 0 \\\\ 0 &-0.2 \\end{array} \\right\\rbrack$ E\\. $M=\\left\\lbrack \\begin{array}{rr} 0.1 & 0.3 \\\\-0.3 & 0.1 \\end{array} \\right\\rbrack$ F\\. $M=\\left\\lbrack \\begin{array}{rr} 0 &-1 \\\\ 2 & 0 \\end{array} \\right\\rbrack$", "answer_v2": [ "CDE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Determine for which of the following matrices $M$ the zero state is a stable equilibrium of the dynamical system $x(t+1)=Mx(t)$. [ANS] A\\. $M=\\left\\lbrack \\begin{array}{rrr} 0.7 & 0.7 & 0.7 \\\\ 0.7 & 0.7 & 0.7 \\\\ 0.7 & 0.7 & 0.7 \\end{array} \\right\\rbrack$ B\\. $M=\\left\\lbrack \\begin{array}{rr} 0 &-5 \\\\ 3 & 0 \\end{array} \\right\\rbrack$ C\\. $M=\\left\\lbrack \\begin{array}{rrr} 0.2 & 0 & 0.1 \\\\ 0 &-0.4 & 0 \\\\ 0.1 & 0 &-0.2 \\end{array} \\right\\rbrack$ D\\. $M=\\left\\lbrack \\begin{array}{rr}-0.4 & 0 \\\\ 0 & 0.3 \\end{array} \\right\\rbrack$ E\\. $M=\\left\\lbrack \\begin{array}{rr}-3 & 0 \\\\ 0 & 0.8 \\end{array} \\right\\rbrack$ F\\. $M=\\left\\lbrack \\begin{array}{rr}-0.3 & 0.2 \\\\-0.2 &-0.3 \\end{array} \\right\\rbrack$", "answer_v3": [ "CDF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0373", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Find the equilibrium state of the dynamical system\n\\begin{array}{l} x_1(t+1)=0.1x_1(t)-0.1x_2(t)-2, \\cr x_1(t+1)=2x_1(t)+0.9x_2(t)+1. \\end{array} $\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-1.03448", "-10.6897" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the equilibrium state of the dynamical system\n\\begin{array}{l} x_1(t+1)=-0.5x_1(t)+0.1x_2(t)-2, \\cr x_1(t+1)=-7.3x_1(t)-1.1x_2(t)-3. \\end{array} $\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-1.15979", "2.60309" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the equilibrium state of the dynamical system\n\\begin{array}{l} x_1(t+1)=-0.3x_1(t)-0.1x_2(t)-2, \\cr x_1(t+1)=0.500000000000001x_1(t)-0.5x_2(t)+3. \\end{array} $\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-1.65", "1.45" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0374", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Find real-valued closed formulas for the trajectory $x(t+1)=Ax(t)$, where A=\\left[\\begin{array}{cc}-0.6 &0.8\\cr-0.8 &-0.6 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{x}(0)=\\left[\\begin{array}{c} 1\\cr 0 \\end{array}\\right]. $\\vec{x}(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "cos(4.06889*t)*1-sin(4.06889*t)*0", "sin(4.06889*t)*1+cos(4.06889*t)*0" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find real-valued closed formulas for the trajectory $x(t+1)=Ax(t)$, where A=\\left[\\begin{array}{cc} 0.6 &0.8\\cr-0.8 &0.6 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{x}(0)=\\left[\\begin{array}{c} 0\\cr 1 \\end{array}\\right]. $\\vec{x}(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "cos(-0.927295*t)*0-sin(-0.927295*t)*1", "sin(-0.927295*t)*0+cos(-0.927295*t)*1" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find real-valued closed formulas for the trajectory $x(t+1)=Ax(t)$, where A=\\left[\\begin{array}{cc}-0.6 &-0.8\\cr 0.8 &-0.6 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{x}(0)=\\left[\\begin{array}{c} 0\\cr 1 \\end{array}\\right]. $\\vec{x}(t)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "cos(2.2143*t)*0-sin(2.2143*t)*1", "sin(2.2143*t)*0+cos(2.2143*t)*1" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0375", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Consider a species of bird that can be split into three age groupings: those aged 0-1 years, those aged 1-2 years, and those aged 2-3 years. The population is observed once a year. Given that the Leslie matrix is equal to L=\\left[\\begin{array}{ccc} 0 &3 &1\\cr 0.3 &0 &0\\cr 0 &0.5 &0 \\end{array}\\right] and the initial population distribution is $900$ of the first age group, $1800$ of the second age group, and $3200$ of the oldest age group, answer the following questions.\nThe initial population vector is $\\mathbf{x}_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nHow many birds aged 1-2 years are there after 10 years? [ANS]\nHow many birds aged 0-1 years are there after 20 years? [ANS]\nHow many birds are there after 30 years? [ANS]\nCalculate the dominant eigenvalue of the Leslie matrix good to 3 decimal places. $\\lambda_1\\=\\ $ [ANS]\nWhat is the long-term growth rate of this population of birds as a percent? growth rate=[ANS] (The growth rate is the percentage of growth over/under 100\\%.)\nAre the birds thriving, static, or going extinct? [ANS]", "answer_v1": [ "900", "1800", "3200" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider a species of bird that can be split into three age groupings: those aged 0-1 years, those aged 1-2 years, and those aged 2-3 years. The population is observed once a year. Given that the Leslie matrix is equal to L=\\left[\\begin{array}{ccc} 0 &2 &1.5\\cr 0.2 &0 &0\\cr 0 &0.4 &0 \\end{array}\\right] and the initial population distribution is $1200$ of the first age group, $1800$ of the second age group, and $2400$ of the oldest age group, answer the following questions.\nThe initial population vector is $\\mathbf{x}_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nHow many birds aged 1-2 years are there after 10 years? [ANS]\nHow many birds aged 0-1 years are there after 20 years? [ANS]\nHow many birds are there after 30 years? [ANS]\nCalculate the dominant eigenvalue of the Leslie matrix good to 3 decimal places. $\\lambda_1\\=\\ $ [ANS]\nWhat is the long-term growth rate of this population of birds as a percent? growth rate=[ANS] (The growth rate is the percentage of growth over/under 100\\%.)\nAre the birds thriving, static, or going extinct? [ANS]", "answer_v2": [ "1200", "1800", "2400" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider a species of bird that can be split into three age groupings: those aged 0-1 years, those aged 1-2 years, and those aged 2-3 years. The population is observed once a year. Given that the Leslie matrix is equal to L=\\left[\\begin{array}{ccc} 0 &2 &1\\cr 0.2 &0 &0\\cr 0 &0.5 &0 \\end{array}\\right] and the initial population distribution is $900$ of the first age group, $1800$ of the second age group, and $3600$ of the oldest age group, answer the following questions.\nThe initial population vector is $\\mathbf{x}_0=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nHow many birds aged 1-2 years are there after 10 years? [ANS]\nHow many birds aged 0-1 years are there after 20 years? [ANS]\nHow many birds are there after 30 years? [ANS]\nCalculate the dominant eigenvalue of the Leslie matrix good to 3 decimal places. $\\lambda_1\\=\\ $ [ANS]\nWhat is the long-term growth rate of this population of birds as a percent? growth rate=[ANS] (The growth rate is the percentage of growth over/under 100\\%.)\nAre the birds thriving, static, or going extinct? [ANS]", "answer_v3": [ "900", "1800", "3600" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0376", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Applications", "level": "4", "keywords": [], "problem_v1": "Consider this matrix that represents the discrete dynamical system $\\mathbf{x}_{k+1}=D\\mathbf{x}_k$:\nD=\\left[\\begin{array}{cc} 0.9 &0.3\\cr 0 &0.4 \\end{array}\\right].\nFind the eigenvalues of $D$ and write as a comma-separated list. [ANS]\nClassify the origin as an attractor, repeller, or saddle point? [ANS]", "answer_v1": [ "(0.4, 0.9)", "attractor" ], "answer_type_v1": [ "UOL", "MCS" ], "options_v1": [ [], [ "attractor", "repeller", "saddle" ] ], "problem_v2": "Consider this matrix that represents the discrete dynamical system $\\mathbf{x}_{k+1}=D\\mathbf{x}_k$:\nD=\\left[\\begin{array}{cc}-1.6 &1.6\\cr 0 &-1.4 \\end{array}\\right].\nFind the eigenvalues of $D$ and write as a comma-separated list. [ANS]\nClassify the origin as an attractor, repeller, or saddle point? [ANS]", "answer_v2": [ "(-1.6, -1.4)", "repeller" ], "answer_type_v2": [ "UOL", "MCS" ], "options_v2": [ [], [ "attractor", "repeller", "saddle" ] ], "problem_v3": "Consider this matrix that represents the discrete dynamical system $\\mathbf{x}_{k+1}=D\\mathbf{x}_k$:\nD=\\left[\\begin{array}{cc}-0.8 &0.4\\cr 0 &-0.9 \\end{array}\\right].\nFind the eigenvalues of $D$ and write as a comma-separated list. [ANS]\nClassify the origin as an attractor, repeller, or saddle point? [ANS]", "answer_v3": [ "(-0.9, -0.8)", "attractor" ], "answer_type_v3": [ "UOL", "MCS" ], "options_v3": [ [], [ "attractor", "repeller", "saddle" ] ] }, { "id": "Linear_algebra_0377", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Applications", "level": "4", "keywords": [], "problem_v1": "Let $M$ be a $2 \\times 2$ matrix with eigenvalues $\\lambda_1=0.4, \\ \\lambda_2=1$ with corresponding eigenvectors\n\\mathbf{v}_1=\\left[\\begin{array}{c} 1\\cr-1 \\end{array}\\right], \\mathbf{v}_2=\\left[\\begin{array}{c} 0\\cr 1 \\end{array}\\right]. Consider the difference equation\n\\mathbf{x}_{k+1}=M\\mathbf{x}_k with initial condition $\\mathbf{x}_0=\\left[\\begin{array}{c} 3\\cr 3 \\end{array}\\right]$. Write the initial condition as a linear combination of the eigenvectors of $M$.\nThat is, write $\\mathbf{x}_0=c_1 \\mathbf{v_1}+c_2 \\mathbf{v_2}$ $=$ [ANS] $\\mathbf{v_1}+$ [ANS] $\\mathbf{v_2}$ In general, $\\mathbf{x}_k=$ [ANS] $\\big($ [ANS] $\\big)^k \\ \\mathbf{v}_1+$ [ANS] $\\big($ [ANS] $\\big)^k \\ \\mathbf{v}_2$ Specifically, $\\mathbf{x}_{4}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} For large $k$, $\\mathbf{x}_{k} \\rightarrow$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.0768", "5.9232", "0", "6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $M$ be a $2 \\times 2$ matrix with eigenvalues $\\lambda_1=-0.8, \\ \\lambda_2=1$ with corresponding eigenvectors\n\\mathbf{v}_1=\\left[\\begin{array}{c} 2\\cr-2 \\end{array}\\right], \\mathbf{v}_2=\\left[\\begin{array}{c} 0\\cr-2 \\end{array}\\right]. Consider the difference equation\n\\mathbf{x}_{k+1}=M\\mathbf{x}_k with initial condition $\\mathbf{x}_0=\\left[\\begin{array}{c} 4\\cr 9 \\end{array}\\right]$. Write the initial condition as a linear combination of the eigenvectors of $M$.\nThat is, write $\\mathbf{x}_0=c_1 \\mathbf{v_1}+c_2 \\mathbf{v_2}$ $=$ [ANS] $\\mathbf{v_1}+$ [ANS] $\\mathbf{v_2}$ In general, $\\mathbf{x}_k=$ [ANS] $\\big($ [ANS] $\\big)^k \\ \\mathbf{v}_1+$ [ANS] $\\big($ [ANS] $\\big)^k \\ \\mathbf{v}_2$ Specifically, $\\mathbf{x}_{3}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} For large $k$, $\\mathbf{x}_{k} \\rightarrow$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-2.048", "15.048", "0", "13" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $M$ be a $2 \\times 2$ matrix with eigenvalues $\\lambda_1=-0.4, \\ \\lambda_2=1$ with corresponding eigenvectors\n\\mathbf{v}_1=\\left[\\begin{array}{c} 1\\cr-1 \\end{array}\\right], \\mathbf{v}_2=\\left[\\begin{array}{c} 0\\cr-1 \\end{array}\\right]. Consider the difference equation\n\\mathbf{x}_{k+1}=M\\mathbf{x}_k with initial condition $\\mathbf{x}_0=\\left[\\begin{array}{c} 5\\cr 2 \\end{array}\\right]$. Write the initial condition as a linear combination of the eigenvectors of $M$.\nThat is, write $\\mathbf{x}_0=c_1 \\mathbf{v_1}+c_2 \\mathbf{v_2}$ $=$ [ANS] $\\mathbf{v_1}+$ [ANS] $\\mathbf{v_2}$ In general, $\\mathbf{x}_k=$ [ANS] $\\big($ [ANS] $\\big)^k \\ \\mathbf{v}_1+$ [ANS] $\\big($ [ANS] $\\big)^k \\ \\mathbf{v}_2$ Specifically, $\\mathbf{x}_{3}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} For large $k$, $\\mathbf{x}_{k} \\rightarrow$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-0.32", "7.32", "0", "7" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0378", "subject": "Linear_algebra", "topic": "Eigenvalues and eigenvectors", "subtopic": "Applications", "level": "2", "keywords": [ "" ], "problem_v1": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ where A=\\left[\\begin{array}{cccc} 1 &1 &-2 &3\\cr-2 &-3 &0 &-1\\cr 0 &0 &3 &-1\\cr 0 &0 &-2 &-3 \\end{array}\\right]. Let $W$ be the subspace of $V=\\mathbb{R}^4$ generated by $\\lbrace \\left(1,-1,0,0\\right),\\left(0,-1,0,0\\right)\\}$. It is easy to see that $W$ is invariant under $L_A$. Define the linear operator $K$ on the quotient space $V/W$ by $K(u+W):=L_A(u)+W$. Find the characteristic polynomial $f(t)$ of $K$. $f(t)=$ [ANS]", "answer_v1": [ "-[(3+t)*(3-t)+2]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ where A=\\left[\\begin{array}{cccc} 5 &0 &-9 &0\\cr-4 &-1 &3 &-1\\cr 3 &0 &-5 &0\\cr-3 &-2 &-2 &-1 \\end{array}\\right]. Let $W$ be the subspace of $V=\\mathbb{R}^4$ generated by $\\lbrace \\left(0,0,0,1\\right),\\left(0,-1,0,-1\\right)\\}$. It is easy to see that $W$ is invariant under $L_A$. Define the linear operator $K$ on the quotient space $V/W$ by $K(u+W):=L_A(u)+W$. Find the characteristic polynomial $f(t)$ of $K$. $f(t)=$ [ANS]", "answer_v2": [ "3-(1-t)*(1+t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ where A=\\left[\\begin{array}{cccc}-1 &0 &1 &-5\\cr-34 &-24 &50 &4\\cr-15 &-10 &22 &-2\\cr-14 &-12 &22 &6 \\end{array}\\right]. Let $W$ be the subspace of $V=\\mathbb{R}^4$ generated by $\\lbrace \\left(0,2,1,1\\right),\\left(2,-1,1,-1\\right)\\}$. It is easy to see that $W$ is invariant under $L_A$. Define the linear operator $K$ on the quotient space $V/W$ by $K(u+W):=L_A(u)+W$. Find the characteristic polynomial $f(t)$ of $K$. $f(t)=$ [ANS]", "answer_v3": [ "1-(3-t)*(4+t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0379", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with dot products", "level": "2", "keywords": [], "problem_v1": "Let $\\left\\lbrace \\vec{e}_1, \\ \\vec{e}_2, \\ \\vec{e}_3, \\ \\vec{e}_4, \\ \\vec{e}_5, \\ \\vec{e}_6 \\right\\rbrace$ be the standard basis in ${\\mathbb R}^6$. Find the length of the vector $\\vec{x}=5\\vec{e}_1+4\\vec{e}_2-3\\vec{e}_3+4\\vec{e}_4-3\\vec{e}_5-4\\vec{e}_6$.\n$\\| \\vec{x} \\|=$ [ANS].", "answer_v1": [ "sqrt(91)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $\\left\\lbrace \\vec{e}_1, \\ \\vec{e}_2, \\ \\vec{e}_3, \\ \\vec{e}_4, \\ \\vec{e}_5, \\ \\vec{e}_6 \\right\\rbrace$ be the standard basis in ${\\mathbb R}^6$. Find the length of the vector $\\vec{x}=2\\vec{e}_1-2\\vec{e}_2-5\\vec{e}_3-2\\vec{e}_4-4\\vec{e}_5-4\\vec{e}_6$.\n$\\| \\vec{x} \\|=$ [ANS].", "answer_v2": [ "sqrt(69)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $\\left\\lbrace \\vec{e}_1, \\ \\vec{e}_2, \\ \\vec{e}_3, \\ \\vec{e}_4, \\ \\vec{e}_5, \\ \\vec{e}_6 \\right\\rbrace$ be the standard basis in ${\\mathbb R}^6$. Find the length of the vector $\\vec{x}=3\\vec{e}_1+3\\vec{e}_2-2\\vec{e}_3+5\\vec{e}_4-5\\vec{e}_5-3\\vec{e}_6$.\n$\\| \\vec{x} \\|=$ [ANS].", "answer_v3": [ "sqrt(81)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0380", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with dot products", "level": "2", "keywords": [], "problem_v1": "Find the angle $\\alpha$ between the vectors \\left[\\begin{array}{c} 3\\cr 1\\cr-2 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 1\\cr 2\\cr-2 \\end{array}\\right]. $\\alpha=$ [ANS].", "answer_v1": [ "acos(0.801784)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the angle $\\alpha$ between the vectors \\left[\\begin{array}{c}-5\\cr-4\\cr 5 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 5\\cr-2\\cr-2 \\end{array}\\right]. $\\alpha=$ [ANS].", "answer_v2": [ "acos(-0.578542)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the angle $\\alpha$ between the vectors \\left[\\begin{array}{c}-2\\cr-2\\cr-3 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 1\\cr 1\\cr-2 \\end{array}\\right]. $\\alpha=$ [ANS].", "answer_v3": [ "acos(0.19803)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0381", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with dot products", "level": "2", "keywords": [], "problem_v1": "Find the dot product of \\vec{x}=\\left[\\begin{array}{c}-7\\cr-5 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{y}=\\left[\\begin{array}{c} 4\\cr 8 \\end{array}\\right]. $\\vec{x}\\cdot \\vec{y}=$ [ANS].", "answer_v1": [ "-7*4+-5*8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the dot product of \\vec{x}=\\left[\\begin{array}{c} 0\\cr-9 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{y}=\\left[\\begin{array}{c}-2\\cr-4 \\end{array}\\right]. $\\vec{x}\\cdot \\vec{y}=$ [ANS].", "answer_v2": [ "0*-2+-9*-4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the dot product of \\vec{x}=\\left[\\begin{array}{c}-3\\cr-6 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{y}=\\left[\\begin{array}{c} 2\\cr 5 \\end{array}\\right]. $\\vec{x}\\cdot \\vec{y}=$ [ANS].", "answer_v3": [ "-3*2+-6*5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0382", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with dot products", "level": "2", "keywords": [], "problem_v1": "Find the length of the vector $\\vec{x}=\\left[\\begin{array}{c} 3\\cr 1\\cr 3 \\end{array}\\right].$ $\\| \\vec{x} \\|=$ [ANS].", "answer_v1": [ "sqrt(3^2+1^2+3^2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the vector $\\vec{x}=\\left[\\begin{array}{c}-5\\cr 5\\cr-5 \\end{array}\\right].$ $\\| \\vec{x} \\|=$ [ANS].", "answer_v2": [ "sqrt((-5)^2+5^2+(-5)^2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the vector $\\vec{x}=\\left[\\begin{array}{c}-2\\cr 1\\cr-3 \\end{array}\\right].$ $\\| \\vec{x} \\|=$ [ANS].", "answer_v3": [ "sqrt((-2)^2+1^2+(-3)^2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0383", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with dot products", "level": "2", "keywords": [], "problem_v1": "Find the norm of $\\vec{x}$ and the unit vector $\\vec{u}$ in the direction of $\\vec{x}$ if \\vec{x}=\\left[\\begin{array}{c} 3\\cr 1\\cr 1\\cr 2 \\end{array}\\right]. $\\| \\vec{x} \\|=$ [ANS],\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.774597", "0.258199", "0.258199", "0.516398" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the norm of $\\vec{x}$ and the unit vector $\\vec{u}$ in the direction of $\\vec{x}$ if \\vec{x}=\\left[\\begin{array}{c}-5\\cr 5\\cr-4\\cr-2 \\end{array}\\right]. $\\| \\vec{x} \\|=$ [ANS],\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-0.597614", "0.597614", "-0.478091", "-0.239046" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the norm of $\\vec{x}$ and the unit vector $\\vec{u}$ in the direction of $\\vec{x}$ if \\vec{x}=\\left[\\begin{array}{c}-2\\cr 1\\cr-2\\cr 1 \\end{array}\\right]. $\\| \\vec{x} \\|=$ [ANS],\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-0.632456", "0.316228", "-0.632456", "0.316228" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0384", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with dot products", "level": "2", "keywords": [ "Vector", "Dot Product" ], "problem_v1": "Suppose $\\vec{u}=\\left<3,1,1\\right>$, $\\vec{v}=\\left<2,-2,-2\\right>$ and $\\vec{w}=\\left<1,1,-1\\right>$. Then:\n$\\begin{array}{ccc}\\hline \\vec{u}\\cdot \\vec{v} &=& [ANS] \\\\ \\hline \\vec{u}\\cdot \\vec{w} &=& [ANS] \\\\ \\hline \\vec{v}\\cdot \\vec{w} &=& [ANS] \\\\ \\hline \\vec{v}\\cdot \\vec{v} &=& [ANS] \\\\ \\hline \\vec{u}\\cdot \\left(\\vec{v}+\\vec{w}\\right) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "2", "3", "2", "12", "5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Suppose $\\vec{u}=\\left<-5,5,-4\\right>$, $\\vec{v}=\\left<-2,5,-2\\right>$ and $\\vec{w}=\\left<-3,-2,1\\right>$. Then:\n$\\begin{array}{ccc}\\hline \\vec{u}\\cdot \\vec{v} &=& [ANS] \\\\ \\hline \\vec{u}\\cdot \\vec{w} &=& [ANS] \\\\ \\hline \\vec{v}\\cdot \\vec{w} &=& [ANS] \\\\ \\hline \\vec{v}\\cdot \\vec{v} &=& [ANS] \\\\ \\hline \\vec{u}\\cdot \\left(\\vec{v}+\\vec{w}\\right) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "43", "1", "-6", "33", "44" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Suppose $\\vec{u}=\\left<-2,1,-2\\right>$, $\\vec{v}=\\left<1,-3,-2\\right>$ and $\\vec{w}=\\left<3,5,4\\right>$. Then:\n$\\begin{array}{ccc}\\hline \\vec{u}\\cdot \\vec{v} &=& [ANS] \\\\ \\hline \\vec{u}\\cdot \\vec{w} &=& [ANS] \\\\ \\hline \\vec{v}\\cdot \\vec{w} &=& [ANS] \\\\ \\hline \\vec{v}\\cdot \\vec{v} &=& [ANS] \\\\ \\hline \\vec{u}\\cdot \\left(\\vec{v}+\\vec{w}\\right) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "-1", "-9", "-20", "14", "-10" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Linear_algebra_0385", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with dot products", "level": "2", "keywords": [ "Vector", "Angle", "Dot Product", "Cross Product" ], "problem_v1": "Suppose $\\vec{u}=\\left<4,1,1\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<3,-4,-3\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<1,3,2\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<2,2,-2\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<-2,0,8\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\end{array}$", "answer_v1": [ "an acute angle", "an acute angle", "an acute angle", "a right angle" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ] ], "problem_v2": "Suppose $\\vec{u}=\\left<1,5,-4\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<8,0,2\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<-2,5,-2\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<-3,2,3\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<1,-4,-1\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\end{array}$", "answer_v2": [ "a right angle", "an acute angle", "an obtuse angle", "an obtuse angle" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ] ], "problem_v3": "Suppose $\\vec{u}=\\left<2,1,-2\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<1,2,-2\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<2,-2,-3\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<4,0,4\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\\\ \\hline \\left<3,5,5\\right>makes & & [ANS]an acute anglewith \\vec{u} \\\\ \\hline \\end{array}$", "answer_v3": [ "an acute angle", "an acute angle", "a right angle", "an acute angle" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ], [ "an obtuse angle", "a right angle", "an acute angle" ] ] }, { "id": "Linear_algebra_0386", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [ "dot product", "inner product" ], "problem_v1": "If $f(x)$ and $g(x)$ are arbitrary polynomials of degree at most 2, then the mapping \\langle f,g\\rangle=f(-1)g(-1)+f(0)g(0)+f(2)g(2) defines an inner product in $P_2$. Use this inner product to find $\\langle f,g\\rangle$, $\\| f \\|$, $\\| g \\|$, and the angle $\\alpha_{f,g}$ between $f(x)$ and $g(x)$ for f(x)=3x^2+5x-4 \\ \\mbox{and} \\ g(x)=2x^2-4x+1. $\\langle f,g\\rangle=$ [ANS], $\\| f \\|=$ [ANS], $\\| g \\|=$ [ANS], $\\alpha_{f,g}=$ [ANS].", "answer_v1": [ "-28", "19.3907", "7.14143", "1.7744" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "If $f(x)$ and $g(x)$ are arbitrary polynomials of degree at most 2, then the mapping \\langle f,g\\rangle=f(-3)g(-3)+f(0)g(0)+f(3)g(3) defines an inner product in $P_2$. Use this inner product to find $\\langle f,g\\rangle$, $\\| f \\|$, $\\| g \\|$, and the angle $\\alpha_{f,g}$ between $f(x)$ and $g(x)$ for f(x)=2x^2+3x+8 \\ \\mbox{and} \\ g(x)=2x^2-6x-3. $\\langle f,g\\rangle=$ [ANS], $\\| f \\|=$ [ANS], $\\| g \\|=$ [ANS], $\\alpha_{f,g}=$ [ANS].", "answer_v2": [ "432", "39.724", "33.2716", "1.23782" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "If $f(x)$ and $g(x)$ are arbitrary polynomials of degree at most 2, then the mapping \\langle f,g\\rangle=f(-3)g(-3)+f(0)g(0)+f(2)g(2) defines an inner product in $P_2$. Use this inner product to find $\\langle f,g\\rangle$, $\\| f \\|$, $\\| g \\|$, and the angle $\\alpha_{f,g}$ between $f(x)$ and $g(x)$ for f(x)=2x^2+4x-6 \\ \\mbox{and} \\ g(x)=3x^2-2x+8. $\\langle f,g\\rangle=$ [ANS], $\\| f \\|=$ [ANS], $\\| g \\|=$ [ANS], $\\alpha_{f,g}=$ [ANS].", "answer_v3": [ "112", "11.6619", "44.7325", "1.35442" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0387", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "4", "keywords": [ "dot product", "inner product", "norm" ], "problem_v1": "Find the norm $\\| x \\|$ of $x=\\left(1,- \\frac{1}{8} , \\frac{1}{64} ,- \\frac{1}{512} , \\frac{1}{4096} , \\ldots, \\frac{1}{(-8) ^{n-1} }, \\ldots \\right)$ in $\\ell_2$.\n$\\|x\\|=$ [ANS].", "answer_v1": [ "sqrt(64/63)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the norm $\\| x \\|$ of $x=\\left(1,- \\frac{1}{2} , \\frac{1}{4} ,- \\frac{1}{8} , \\frac{1}{16} , \\ldots, \\frac{1}{(-2) ^{n-1} }, \\ldots \\right)$ in $\\ell_2$.\n$\\|x\\|=$ [ANS].", "answer_v2": [ "sqrt(4/3)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the norm $\\| x \\|$ of $x=\\left(1,- \\frac{1}{4} , \\frac{1}{16} ,- \\frac{1}{64} , \\frac{1}{256} , \\ldots, \\frac{1}{(-4) ^{n-1} }, \\ldots \\right)$ in $\\ell_2$.\n$\\|x\\|=$ [ANS].", "answer_v3": [ "sqrt(16/15)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0388", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [ "linear algebra", "inner product", "dot product", "norm", "angle", "function space" ], "problem_v1": "Use the inner product \\langle f,g \\rangle=\\int_0^1 f(x)g(x) \\, dx in the vector space $C^0 \\lbrack 0,1\\rbrack$ of continuous functions on the domain $\\lbrack 0, 1 \\rbrack$ to find $\\langle f,g\\rangle$, $\\| f \\|$, $\\| g\\|$, and the angle $\\alpha_{f,g}$ between $f(x)$ and $g(x)$ for f(x)=5x^2-6 \\ \\mbox{and} \\ g(x)=3x+5. $\\langle f,g\\rangle=$ [ANS], $\\|f\\|=$ [ANS], $\\|g\\|=$ [ANS], $\\alpha_{f,g}$ [ANS].", "answer_v1": [ "-26.9167", "4.58258", "6.55744", "2.68087" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Use the inner product \\langle f,g \\rangle=\\int_0^1 f(x)g(x) \\, dx in the vector space $C^0 \\lbrack 0,1\\rbrack$ of continuous functions on the domain $\\lbrack 0, 1 \\rbrack$ to find $\\langle f,g\\rangle$, $\\| f \\|$, $\\| g\\|$, and the angle $\\alpha_{f,g}$ between $f(x)$ and $g(x)$ for f(x)=-10x^2-9 \\ \\mbox{and} \\ g(x)=-6x-3. $\\langle f,g\\rangle=$ [ANS], $\\|f\\|=$ [ANS], $\\|g\\|=$ [ANS], $\\alpha_{f,g}$ [ANS].", "answer_v2": [ "79", "12.6886", "6.245", "0.0778728" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Use the inner product \\langle f,g \\rangle=\\int_0^1 f(x)g(x) \\, dx in the vector space $C^0 \\lbrack 0,1\\rbrack$ of continuous functions on the domain $\\lbrack 0, 1 \\rbrack$ to find $\\langle f,g\\rangle$, $\\| f \\|$, $\\| g\\|$, and the angle $\\alpha_{f,g}$ between $f(x)$ and $g(x)$ for f(x)=-5x^2-6 \\ \\mbox{and} \\ g(x)=-6x+1. $\\langle f,g\\rangle=$ [ANS], $\\|f\\|=$ [ANS], $\\|g\\|=$ [ANS], $\\alpha_{f,g}$ [ANS].", "answer_v3": [ "17.8333", "7.81025", "2.64575", "0.529588" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0389", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [], "problem_v1": "If $A$ and $B$ are arbitrary real $m\\times n$ matrices, then the mapping \\langle A,B \\rangle={\\rm trace}(A^T B) defines an inner product in ${\\mathbb R}^{m\\times n}$. Use this inner product to find $\\langle A,B \\rangle$, the norms $\\|A\\|$ and $\\|B\\|$, and the angle $\\alpha_{A,B}$ between $A$ and $B$ for A=\\left[\\begin{array}{cc} 2 &1\\cr 1 &2\\cr-1 &-1 \\end{array}\\right] \\ \\mbox{and} \\ B=\\left[\\begin{array}{cc} 1 &-1\\cr 1 &-2\\cr-1 &-1 \\end{array}\\right]. $\\langle A,B \\rangle=$ [ANS], $\\|A\\|=$ [ANS], $\\|B\\|=$ [ANS], $\\alpha_{A,B}=$ [ANS].", "answer_v1": [ "0", "3.4641", "3", "1.5708" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "If $A$ and $B$ are arbitrary real $m\\times n$ matrices, then the mapping \\langle A,B \\rangle={\\rm trace}(A^T B) defines an inner product in ${\\mathbb R}^{m\\times n}$. Use this inner product to find $\\langle A,B \\rangle$, the norms $\\|A\\|$ and $\\|B\\|$, and the angle $\\alpha_{A,B}$ between $A$ and $B$ for A=\\left[\\begin{array}{cc}-3 &3\\cr-2 &-1\\cr 3 &-1 \\end{array}\\right] \\ \\mbox{and} \\ B=\\left[\\begin{array}{cc}-2 &-1\\cr-3 &1\\cr 2 &-2 \\end{array}\\right]. $\\langle A,B \\rangle=$ [ANS], $\\|A\\|=$ [ANS], $\\|B\\|=$ [ANS], $\\alpha_{A,B}=$ [ANS].", "answer_v2": [ "16", "5.74456", "4.79583", "0.95113" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "If $A$ and $B$ are arbitrary real $m\\times n$ matrices, then the mapping \\langle A,B \\rangle={\\rm trace}(A^T B) defines an inner product in ${\\mathbb R}^{m\\times n}$. Use this inner product to find $\\langle A,B \\rangle$, the norms $\\|A\\|$ and $\\|B\\|$, and the angle $\\alpha_{A,B}$ between $A$ and $B$ for A=\\left[\\begin{array}{cc}-1 &1\\cr-2 &-2\\cr-1 &2 \\end{array}\\right] \\ \\mbox{and} \\ B=\\left[\\begin{array}{cc} 3 &3\\cr-2 &-1\\cr-2 &-3 \\end{array}\\right]. $\\langle A,B \\rangle=$ [ANS], $\\|A\\|=$ [ANS], $\\|B\\|=$ [ANS], $\\alpha_{A,B}=$ [ANS].", "answer_v3": [ "2", "3.87298", "6", "1.48462" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0390", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [ "orthogonal", "sturm", "liouville" ], "problem_v1": "On the interval $[-8,8]$ we know that $x$ and $x^2$ are orthogonal. Let $p=x+ax^2+bx^3$. Then\n$\\langle p,x \\rangle=$ [ANS]\n$\\langle p,x^2 \\rangle=$ [ANS]\nSo if we want p to be orthogonal to both $x$ and $x^2$ we have to solve the system of equations [ANS] $=0$ [ANS] $=0$\nWhich gives us $p=$ [ANS]", "answer_v1": [ "2/3*8^3+b*2/5*8^5", "a*2/5*8^5", "2/3*8^3+b*2/5*8^5", "a*2/5*8^5", "x+-0.0260417*x^3" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "On the interval $[-2,2]$ we know that $x$ and $x^2$ are orthogonal. Let $p=x+ax^2+bx^3$. Then\n$\\langle p,x \\rangle=$ [ANS]\n$\\langle p,x^2 \\rangle=$ [ANS]\nSo if we want p to be orthogonal to both $x$ and $x^2$ we have to solve the system of equations [ANS] $=0$ [ANS] $=0$\nWhich gives us $p=$ [ANS]", "answer_v2": [ "2/3*2^3+b*2/5*2^5", "a*2/5*2^5", "2/3*2^3+b*2/5*2^5", "a*2/5*2^5", "x+-0.416667*x^3" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "On the interval $[-4,4]$ we know that $x$ and $x^2$ are orthogonal. Let $p=x+ax^2+bx^3$. Then\n$\\langle p,x \\rangle=$ [ANS]\n$\\langle p,x^2 \\rangle=$ [ANS]\nSo if we want p to be orthogonal to both $x$ and $x^2$ we have to solve the system of equations [ANS] $=0$ [ANS] $=0$\nWhich gives us $p=$ [ANS]", "answer_v3": [ "2/3*4^3+b*2/5*4^5", "a*2/5*4^5", "2/3*4^3+b*2/5*4^5", "a*2/5*4^5", "x+-0.104167*x^3" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Linear_algebra_0391", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [ "orthogonal", "sturm", "liouville" ], "problem_v1": "On the interval $[-7,7]$\n$\\begin{array}{ccccccccc}\\hline \\langle x^{2},x \\rangle=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert x^{2}\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "-7", "7", "x^2*x", "0", "-7", "7", "x^2*x^2", "sqrt(2/5*7^5)" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "On the interval $[-1,1]$\n$\\begin{array}{ccccccccc}\\hline \\langle x^{2},x \\rangle=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert x^{2}\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "-1", "1", "x^2*x", "0", "-1", "1", "x^2*x^2", "sqrt(2/5*1^5)" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "On the interval $[-3,3]$\n$\\begin{array}{ccccccccc}\\hline \\langle x^{2},x \\rangle=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert x^{2}\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "-3", "3", "x^2*x", "0", "-3", "3", "x^2*x^2", "sqrt(2/5*3^5)" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0392", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [ "orthogonal", "sturm", "liouville" ], "problem_v1": "On the interval $(-\\infty,\\infty)$ with weight function $e^{-x^{2}}$\n$\\begin{array}{ccccccccc}\\hline \\langle 4x,x^{2}+2 \\rangle=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert 4x\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "-infinity", "infinity", "4*x*(x^2+2)*e^(-x^2)", "0", "-infinity", "infinity", "e^(-x^2)*4*x*4*x", "4*(pi/4)^{1/4}" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "On the interval $(-\\infty,\\infty)$ with weight function $e^{-x^{2}}$\n$\\begin{array}{ccccccccc}\\hline \\langle-6x,6x^{2}-5 \\rangle=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert-6x\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "-infinity", "infinity", "-6*x*(6*x^2-5)*e^(-x^2)", "0", "-infinity", "infinity", "e^(-x^2)*-6*x*-6*x", "-6*(pi/4)^(1/4)" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "On the interval $(-\\infty,\\infty)$ with weight function $e^{-x^{2}}$\n$\\begin{array}{ccccccccc}\\hline \\langle-3x,2x^{2}-3 \\rangle=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert-3x\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "-infinity", "infinity", "-3*x*(2*x^2-3)*e^(-x^2)", "0", "-infinity", "infinity", "e^(-x^2)*-3*x*-3*x", "3*(pi/4)^(1/4)" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0393", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [ "orthogonal", "sturm", "liouville" ], "problem_v1": "$0$ On the interval $[0,\\pi]$\n$\\begin{array}{ccccccc}\\hline \\langle \\cos\\!\\left(x\\right),\\sin^{6}\\!\\left(x\\right) \\rangle)=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ u=$ [ANS] $du=$ [ANS]\n$\\begin{array}{ccccccccc}\\hline & &=& & [ANS] \\int [ANS] & & [ANS] & & du \\\\ \\hline \\\\ \\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert \\cos\\!\\left(x\\right)\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "0", "pi", "cos(x)*[sin(x)]^6", "sin(x)", "cos(x)*dx", "0", "0", "u^6", "0", "0", "pi", "cos(x)*cos(x)", "sqrt(pi/2)" ], "answer_type_v1": [ "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "$0$ On the interval $[0,\\pi]$\n$\\begin{array}{ccccccc}\\hline \\langle \\cos\\!\\left(x\\right),\\sin^{2}\\!\\left(x\\right) \\rangle)=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ u=$ [ANS] $du=$ [ANS]\n$\\begin{array}{ccccccccc}\\hline & &=& & [ANS] \\int [ANS] & & [ANS] & & du \\\\ \\hline \\\\ \\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert \\cos\\!\\left(x\\right)\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "0", "pi", "cos(x)*[sin(x)]^2", "sin(x)", "cos(x)*dx", "0", "0", "u^2", "0", "0", "pi", "cos(x)*cos(x)", "sqrt(pi/2)" ], "answer_type_v2": [ "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "$0$ On the interval $[0,\\pi]$\n$\\begin{array}{ccccccc}\\hline \\langle \\cos\\!\\left(x\\right),\\sin^{3}\\!\\left(x\\right) \\rangle)=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ u=$ [ANS] $du=$ [ANS]\n$\\begin{array}{ccccccccc}\\hline & &=& & [ANS] \\int [ANS] & & [ANS] & & du \\\\ \\hline \\\\ \\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline \\Vert \\cos\\!\\left(x\\right)\\Vert=\\Bigg(& & [ANS] \\int [ANS] & & [ANS] & & dx\\Bigg)^ \\frac{1}{2} =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "0", "pi", "cos(x)*[sin(x)]^3", "sin(x)", "cos(x)*dx", "0", "0", "u^3", "0", "0", "pi", "cos(x)*cos(x)", "sqrt(pi/2)" ], "answer_type_v3": [ "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0394", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [ "inner product", "Hermitian inner product" ], "problem_v1": "Find the following (Hermitian) inner products:\n(a) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix} 8 \\\\ 7 \\\\ 4 \\\\ 6\\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 6 \\\\ 8 \\\\ 4 \\\\6 \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS]. (b) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix}-1 \\\\ 2 \\\\ 0 \\\\-1 \\\\ 0 \\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 5 \\\\ 3 \\\\ 5 \\\\5 \\\\ 2 \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS]. (c) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix} 2+4i \\\\ 4+4i \\\\ 8+3i \\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 5+7i \\\\ 4+8i \\\\ 1+3i \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS].", "answer_v1": [ "156", "-4", "103+31i" ], "answer_type_v1": [ "NV", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the following (Hermitian) inner products:\n(a) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix} 1 \\\\ 2 \\\\ 10 \\\\ 2\\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 10 \\\\ 4 \\\\ 4 \\\\4 \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS]. (b) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix} 1 \\\\ 2 \\\\ 3 \\\\-3 \\\\ 1 \\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 1 \\\\ 5 \\\\ 2 \\\\3 \\\\ 2 \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS]. (c) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix} 3+5i \\\\ 1+4i \\\\ 8+4i \\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 1+2i \\\\ 4+2i \\\\ 4+6i \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS].", "answer_v2": [ "66", "10", "81+19i" ], "answer_type_v2": [ "NV", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the following (Hermitian) inner products:\n(a) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix} 4 \\\\ 3 \\\\ 3 \\\\ 9\\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 7 \\\\ 6 \\\\ 4 \\\\10 \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS]. (b) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix} 4 \\\\-2 \\\\-5 \\\\ 5 \\\\ 1 \\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 3 \\\\ 3 \\\\ 6 \\\\8 \\\\ 2 \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS]. (c) $\\langle \\mathbf{u}, \\mathbf{v}\\rangle,$ where \\mathbf{u}=\\begin{bmatrix} 3+6i \\\\ 4+8i \\\\ 5+6i \\end{bmatrix} \\text{and} \\mathbf{v}=\\begin{bmatrix} 6+5i \\\\ 7+7i \\\\ 8+7i \\end{bmatrix}. Answer: $\\langle \\mathbf{u}, \\mathbf{v}\\rangle=$ [ANS].", "answer_v3": [ "148", "18", "214-62i" ], "answer_type_v3": [ "NV", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0395", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "2", "keywords": [ "" ], "problem_v1": "Consider the inner product $\\langle p(x),q(x)\\rangle:=\\int_0^1 p(x)q(x) dx$ on the vector space $V$ of real polynomials with degree less than 2. The linear functional $f:V\\to \\mathbb {R}$ is defined by $f(p(x))=p'(9)-3 p(-2)$. According to the Riesz Representation Theorem, there is an $r(x) \\in V$ such that $f(p(x)=\\langle p(x),r(x) \\rangle$ for all $p(x)\\in V$. Find this $r(x)$. $r(x)=$ [ANS]", "answer_v1": [ "-54+102*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Consider the inner product $\\langle p(x),q(x)\\rangle:=\\int_0^1 p(x)q(x) dx$ on the vector space $V$ of real polynomials with degree less than 2. The linear functional $f:V\\to \\mathbb {R}$ is defined by $f(p(x))=p'(-17)+7 p(-3)$. According to the Riesz Representation Theorem, there is an $r(x) \\in V$ such that $f(p(x)=\\langle p(x),r(x) \\rangle$ for all $p(x)\\in V$. Find this $r(x)$. $r(x)=$ [ANS]", "answer_v2": [ "148+(-282)*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Consider the inner product $\\langle p(x),q(x)\\rangle:=\\int_0^1 p(x)q(x) dx$ on the vector space $V$ of real polynomials with degree less than 2. The linear functional $f:V\\to \\mathbb {R}$ is defined by $f(p(x))=p'(-8)+2 p(-2)$. According to the Riesz Representation Theorem, there is an $r(x) \\in V$ such that $f(p(x)=\\langle p(x),r(x) \\rangle$ for all $p(x)\\in V$. Find this $r(x)$. $r(x)=$ [ANS]", "answer_v3": [ "26+(-48)*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0396", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "2", "keywords": [ "" ], "problem_v1": "Consider the inner product $\\langle X,Y \\rangle:=\\text{trace}(X^T Y)$ on the vector space $V=\\mathbb{R}^{2\\times 2}$. The linear operator $L:V \\to V$ is defined by $L(X)=\\left[\\begin{array}{cc} 4 &1\\cr 2 &4 \\end{array}\\right] X$. Find the adjoint $L^*$ of $L$. $L^*(\\left[\\matrix {a & b \\\\ c & d}\\right])=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "4*a+2*c", "4*b+2*d", "a+4*c", "b+4*d" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the inner product $\\langle X,Y \\rangle:=\\text{trace}(X^T Y)$ on the vector space $V=\\mathbb{R}^{2\\times 2}$. The linear operator $L:V \\to V$ is defined by $L(X)=\\left[\\begin{array}{cc}-7 &7\\cr-6 &-3 \\end{array}\\right] X$. Find the adjoint $L^*$ of $L$. $L^*(\\left[\\matrix {a & b \\\\ c & d}\\right])=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-(7*a+6*c)", "-(7*b+6*d)", "7*a-3*c", "7*b-3*d" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the inner product $\\langle X,Y \\rangle:=\\text{trace}(X^T Y)$ on the vector space $V=\\mathbb{R}^{2\\times 2}$. The linear operator $L:V \\to V$ is defined by $L(X)=\\left[\\begin{array}{cc}-3 &2\\cr-4 &1 \\end{array}\\right] X$. Find the adjoint $L^*$ of $L$. $L^*(\\left[\\matrix {a & b \\\\ c & d}\\right])=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-(3*a+4*c)", "-(3*b+4*d)", "2*a+c", "2*b+d" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0397", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "2", "keywords": [ "" ], "problem_v1": "Consider the inner product $\\langle p(x),q(x)\\rangle:=\\int_0^1 p(x)q(x) dx$ on the vector space $V$ of real polynomials with degree less than 2. The linear operator $L:V \\to V$ is defined by $L(p(x))=4 p'(x)+1 p(x)$. Find the adjoint $L^*$ of $L$. $L^*(a+bx)=$ [ANS]+[ANS] $x$", "answer_v1": [ "-(23*a+12*b)", "48*a+25*b" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Consider the inner product $\\langle p(x),q(x)\\rangle:=\\int_0^1 p(x)q(x) dx$ on the vector space $V$ of real polynomials with degree less than 2. The linear operator $L:V \\to V$ is defined by $L(p(x))=-7 p'(x)+7 p(x)$. Find the adjoint $L^*$ of $L$. $L^*(a+bx)=$ [ANS]+[ANS] $x$", "answer_v2": [ "49*a+21*b", "-(84*a+35*b)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Consider the inner product $\\langle p(x),q(x)\\rangle:=\\int_0^1 p(x)q(x) dx$ on the vector space $V$ of real polynomials with degree less than 2. The linear operator $L:V \\to V$ is defined by $L(p(x))=-3 p'(x)+2 p(x)$. Find the adjoint $L^*$ of $L$. $L^*(a+bx)=$ [ANS]+[ANS] $x$", "answer_v3": [ "20*a+9*b", "-(36*a+16*b)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0398", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [ "polynomials", "projection", "angles", "angle" ], "problem_v1": "Let the inner product on $P_4$ be with respect to $x_1=2$, $x_2=1$, $x_3=-1$, and $x_4=-2$. Find the angle between\np(x)=-x^{3} and\nq(x)=(-1)+x^{2}. Determine the vector projection of $p$ onto $q$. Use asin() or acos() to enter arcsine and arccosine and the projection should be a polynomial in the variable $x$. Angle: [ANS]\nProjection: [ANS]", "answer_v1": [ "1.5708", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let the inner product on $P_4$ be with respect to $x_1=-2$, $x_2=1$, $x_3=-1$, and $x_4=-3$. Find the angle between\np(x)=-\\left(1+x+x^{2}+x^{3}\\right) and\nq(x)=1-x+x^{2}+x^{3}. Determine the vector projection of $p$ onto $q$. Use asin() or acos() to enter arcsine and arccosine and the projection should be a polynomial in the variable $x$. Angle: [ANS]\nProjection: [ANS]", "answer_v2": [ "2.91517", "-[293*(1-x+x^2+x^3)/205]" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let the inner product on $P_4$ be with respect to $x_1=-2$, $x_2=1$, $x_3=-1$, and $x_4=-3$. Find the angle between\np(x)=x-1-x^{2}-x^{3} and\nq(x)=(-1)+x^{2}. Determine the vector projection of $p$ onto $q$. Use asin() or acos() to enter arcsine and arccosine and the projection should be a polynomial in the variable $x$. Angle: [ANS]\nProjection: [ANS]", "answer_v3": [ "0.347965", "115*[(-1)+x^2]/73" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0399", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Computing with inner products", "level": "3", "keywords": [], "problem_v1": "Let $\\lbrace \\mathbf{u_1}, \\mathbf{u_2}, \\mathbf{u_2} \\rbrace$ be an orthonormal basis for an inner product space $V$. Suppose\n\\mathbf{v}=a \\mathbf{u_1}+b \\mathbf{u_2}+c \\mathbf{u_3} is so that $||\\mathbf{v}||=\\sqrt{29}$, $\\langle \\mathbf{v},\\mathbf{u_2} \\rangle=5$, and $\\langle \\mathbf{v}, \\mathbf{u_3} \\rangle=2$. Find the possible values for $a$, $b$, and $c$. $a=$ [ANS], $b=$ [ANS], $c=$ [ANS]", "answer_v1": [ "0", "5", "2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $\\lbrace \\mathbf{u_1}, \\mathbf{u_2}, \\mathbf{u_2} \\rbrace$ be an orthonormal basis for an inner product space $V$. Suppose\n\\mathbf{v}=a \\mathbf{u_1}+b \\mathbf{u_2}+c \\mathbf{u_3} is so that $||\\mathbf{v}||=\\sqrt{58}$, $\\langle \\mathbf{v},\\mathbf{u_2} \\rangle=-7$, and $\\langle \\mathbf{v}, \\mathbf{u_3} \\rangle=-3$. Find the possible values for $a$, $b$, and $c$. $a=$ [ANS], $b=$ [ANS], $c=$ [ANS]", "answer_v2": [ "0", "-7", "-3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $\\lbrace \\mathbf{u_1}, \\mathbf{u_2}, \\mathbf{u_2} \\rbrace$ be an orthonormal basis for an inner product space $V$. Suppose\n\\mathbf{v}=a \\mathbf{u_1}+b \\mathbf{u_2}+c \\mathbf{u_3} is so that $||\\mathbf{v}||=\\sqrt{20}$, $\\langle \\mathbf{v},\\mathbf{u_2} \\rangle=-4$, and $\\langle \\mathbf{v}, \\mathbf{u_3} \\rangle=2$. Find the possible values for $a$, $b$, and $c$. $a=$ [ANS], $b=$ [ANS], $c=$ [ANS]", "answer_v3": [ "0", "-4", "2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0400", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "3", "keywords": [], "problem_v1": "The dot product of two vectors \\vec{x}=\\left\\lbrack \\begin{array}{c} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{array} \\right\\rbrack \\ \\mbox{and} \\ \\vec{y}=\\left\\lbrack \\begin{array}{c} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{array} \\right\\rbrack in ${\\mathbb R}^n$ is defined by $\\vec{x} \\cdot \\vec{y}=x_1 y_1+x_2 y_2+\\ldots+x_n y_n$. The vectors $\\vec{x}$ and $\\vec{y}$ are called perpendicular if $\\vec{x} \\cdot \\vec{y}=0$. Any vector in ${\\mathbb R}^3$ perpendicular to \\left[\\begin{array}{c} 5\\cr 2\\cr 3 \\end{array}\\right] can be written in the form\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $s+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $t.$", "answer_v1": [ "2", "-5", "0", "3", "0", "-5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The dot product of two vectors \\vec{x}=\\left\\lbrack \\begin{array}{c} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{array} \\right\\rbrack \\ \\mbox{and} \\ \\vec{y}=\\left\\lbrack \\begin{array}{c} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{array} \\right\\rbrack in ${\\mathbb R}^n$ is defined by $\\vec{x} \\cdot \\vec{y}=x_1 y_1+x_2 y_2+\\ldots+x_n y_n$. The vectors $\\vec{x}$ and $\\vec{y}$ are called perpendicular if $\\vec{x} \\cdot \\vec{y}=0$. Any vector in ${\\mathbb R}^3$ perpendicular to \\left[\\begin{array}{c}-8\\cr 8\\cr-7 \\end{array}\\right] can be written in the form\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $s+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $t.$", "answer_v2": [ "8", "8", "0", "-7", "0", "8" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The dot product of two vectors \\vec{x}=\\left\\lbrack \\begin{array}{c} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{array} \\right\\rbrack \\ \\mbox{and} \\ \\vec{y}=\\left\\lbrack \\begin{array}{c} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{array} \\right\\rbrack in ${\\mathbb R}^n$ is defined by $\\vec{x} \\cdot \\vec{y}=x_1 y_1+x_2 y_2+\\ldots+x_n y_n$. The vectors $\\vec{x}$ and $\\vec{y}$ are called perpendicular if $\\vec{x} \\cdot \\vec{y}=0$. Any vector in ${\\mathbb R}^3$ perpendicular to \\left[\\begin{array}{c}-4\\cr 2\\cr-5 \\end{array}\\right] can be written in the form\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $s+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $t.$", "answer_v3": [ "2", "4", "0", "-5", "0", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0401", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "3", "keywords": [], "problem_v1": "Any vector in ${\\mathbb R}^3$ perpendicular to \\left[\\begin{array}{c} 5\\cr 2\\cr 3 \\end{array}\\right] can be written in the form\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $s+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $t.$", "answer_v1": [ "2", "-5", "0", "3", "0", "-5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Any vector in ${\\mathbb R}^3$ perpendicular to \\left[\\begin{array}{c}-8\\cr 8\\cr-7 \\end{array}\\right] can be written in the form\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $s+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $t.$", "answer_v2": [ "8", "8", "0", "-7", "0", "8" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Any vector in ${\\mathbb R}^3$ perpendicular to \\left[\\begin{array}{c}-4\\cr 2\\cr-5 \\end{array}\\right] can be written in the form\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $s+$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $t.$", "answer_v3": [ "2", "4", "0", "-5", "0", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0402", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "3", "keywords": [], "problem_v1": "Let \\vec{u}=\\left[\\begin{array}{c}-1\\cr-2\\cr 1\\cr-1 \\end{array}\\right], \\vec{v}=\\left[\\begin{array}{c} 1\\cr 3\\cr 2\\cr-2 \\end{array}\\right], and let $W$ the subspace of ${\\mathbb R}^4$ spanned by $\\vec{u}$ and $\\vec{v}$. Find a basis of $W^{\\perp}$, the orthogonal complement of $W$ in $\\mathbb{R}^4$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "7", "-3", "1", "0", "-7", "3", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let \\vec{u}=\\left[\\begin{array}{c}-1\\cr-5\\cr-2\\cr 1 \\end{array}\\right], \\vec{v}=\\left[\\begin{array}{c}-1\\cr-4\\cr-1\\cr 4 \\end{array}\\right], and let $W$ the subspace of ${\\mathbb R}^4$ spanned by $\\vec{u}$ and $\\vec{v}$. Find a basis of $W^{\\perp}$, the orthogonal complement of $W$ in $\\mathbb{R}^4$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "3", "-1", "1", "0", "16", "-3", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let \\vec{u}=\\left[\\begin{array}{c} 2\\cr 7\\cr 3\\cr-3 \\end{array}\\right], \\vec{v}=\\left[\\begin{array}{c}-1\\cr-3\\cr-3\\cr-1 \\end{array}\\right], and let $W$ the subspace of ${\\mathbb R}^4$ spanned by $\\vec{u}$ and $\\vec{v}$. Find a basis of $W^{\\perp}$, the orthogonal complement of $W$ in $\\mathbb{R}^4$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-12", "3", "1", "0", "-16", "5", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0403", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "2", "keywords": [], "problem_v1": "Find a non-zero vector $\\vec{v}$ perpendicular to the vector $\\vec{u}=\\left[\\begin{array}{c} 5\\cr 2 \\end{array}\\right].$\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-2", "5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find a non-zero vector $\\vec{v}$ perpendicular to the vector $\\vec{u}=\\left[\\begin{array}{c}-8\\cr 8 \\end{array}\\right].$\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-8", "-8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find a non-zero vector $\\vec{v}$ perpendicular to the vector $\\vec{u}=\\left[\\begin{array}{c}-4\\cr 2 \\end{array}\\right].$\n$\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-2", "-4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0404", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "3", "keywords": [], "problem_v1": "Find two linearly independent vectors perpendicular to the vector \\vec{v}=\\left[\\begin{array}{c} 8\\cr 3\\cr 5 \\end{array}\\right].\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-5", "0", "8", "-3", "8", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find two linearly independent vectors perpendicular to the vector \\vec{v}=\\left[\\begin{array}{c} 1\\cr-7\\cr-3 \\end{array}\\right].\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "3", "0", "1", "7", "1", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find two linearly independent vectors perpendicular to the vector \\vec{v}=\\left[\\begin{array}{c} 2\\cr-5\\cr 1 \\end{array}\\right].\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-1", "0", "2", "5", "2", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0405", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "2", "keywords": [], "problem_v1": "Find the value of $k$ for which the vectors \\left[\\begin{array}{c} 3\\cr 1\\cr-2\\cr 1 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 1\\cr 2\\cr-2\\cr k\\cr \\end{array}\\right] are orthogonal.\n$k=$ [ANS].", "answer_v1": [ "-9/1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the value of $k$ for which the vectors \\left[\\begin{array}{c}-5\\cr-4\\cr 5\\cr-3 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 5\\cr-2\\cr-2\\cr k\\cr \\end{array}\\right] are orthogonal.\n$k=$ [ANS].", "answer_v2": [ "-(-27)/-3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the value of $k$ for which the vectors \\left[\\begin{array}{c}-2\\cr-2\\cr-3\\cr 3 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 1\\cr 1\\cr-2\\cr k\\cr \\end{array}\\right] are orthogonal.\n$k=$ [ANS].", "answer_v3": [ "-2/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0406", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "3", "keywords": [], "problem_v1": "Let $L$ be the line given by the span of $\\left[\\begin{array}{c} 5\\cr 2\\cr 2 \\end{array}\\right]$ in $\\mathbb{R}^3$. Find a basis for the orthogonal complement $L^{\\bot}$ of $L$.\nA basis for $L^{\\bot}$ is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v1": [ "2", "0", "-5", "2", "-5", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $L$ be the line given by the span of $\\left[\\begin{array}{c}-8\\cr 8\\cr-7 \\end{array}\\right]$ in $\\mathbb{R}^3$. Find a basis for the orthogonal complement $L^{\\bot}$ of $L$.\nA basis for $L^{\\bot}$ is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v2": [ "-7", "0", "8", "8", "8", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $L$ be the line given by the span of $\\left[\\begin{array}{c}-4\\cr 2\\cr-4 \\end{array}\\right]$ in $\\mathbb{R}^3$. Find a basis for the orthogonal complement $L^{\\bot}$ of $L$.\nA basis for $L^{\\bot}$ is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v3": [ "-4", "0", "4", "2", "4", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0407", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "1", "keywords": [ "Vector", "Parallel", "Perpendicular", "Cross Product", "Dot Product" ], "problem_v1": "Suppose $\\vec{v}=\\left<3,1,1\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<4.5,1.5,1.5\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<-6,-2,-2\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<0,8,-8\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<-2,4,2\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\end{array}$", "answer_v1": [ "parallel", "parallel", "perpendicular", "perpendicular" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ], "problem_v2": "Suppose $\\vec{v}=\\left<-5,5,-4\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<-3,17,25\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<10,-10,8\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<10,-2,-15\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<-7.5,7.5,-6\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\end{array}$", "answer_v2": [ "perpendicular", "parallel", "perpendicular", "parallel" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ], "problem_v3": "Suppose $\\vec{v}=\\left<-2,1,-2\\right>$. Then\n$\\begin{array}{ccc}\\hline \\left<-3,-2,-3\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<4,-2,4\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<-8,-6,5\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\\\ \\hline \\left<14,2,-13\\right>is & & [ANS]neitherto \\vec{v} \\\\ \\hline \\end{array}$", "answer_v3": [ "neither", "parallel", "perpendicular", "perpendicular" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ], [ "parallel", "perpendicular", "neither" ] ] }, { "id": "Linear_algebra_0408", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "1", "keywords": [ "Vector", "Perpendicular", "Dot Product" ], "problem_v1": "Suppose $\\vec{u}=\\left<4,2,1\\right>$. Mark each vector below with a \" T \" if it is perpendicular to $\\vec{u}$, and an \" F \" if it is not perpendicular to $\\vec{u}$:\n$\\begin{array}{ccc}\\hline [ANS] & 1. & \\left<2,-2,-2\\right> \\\\ \\hline [ANS] & 2. & \\left<1,2,-8\\right> \\\\ \\hline [ANS] & 3. & \\left<1,-2,0\\right> \\\\ \\hline [ANS] & 4. & \\left<1,1,-1\\right> \\\\ \\hline \\end{array}$", "answer_v1": [ "F", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose $\\vec{u}=\\left<-6,6,-4\\right>$. Mark each vector below with a \" T \" if it is perpendicular to $\\vec{u}$, and an \" F \" if it is not perpendicular to $\\vec{u}$:\n$\\begin{array}{ccc}\\hline [ANS] & 1. & \\left<3,3,0\\right> \\\\ \\hline [ANS] & 2. & \\left<-2,5,-2\\right> \\\\ \\hline [ANS] & 3. & \\left<10,-18,-42\\right> \\\\ \\hline [ANS] & 4. & \\left<-3,-2,1\\right> \\\\ \\hline \\end{array}$", "answer_v2": [ "T", "F", "T", "F" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose $\\vec{u}=\\left<-2,2,-2\\right>$. Mark each vector below with a \" T \" if it is perpendicular to $\\vec{u}$, and an \" F \" if it is not perpendicular to $\\vec{u}$:\n$\\begin{array}{ccc}\\hline [ANS] & 1. & \\left<1,-3,-2\\right> \\\\ \\hline [ANS] & 2. & \\left<4,-10,-14\\right> \\\\ \\hline [ANS] & 3. & \\left<3,5,4\\right> \\\\ \\hline [ANS] & 4. & \\left<1,1,0\\right> \\\\ \\hline \\end{array}$", "answer_v3": [ "F", "T", "F", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0409", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "2", "keywords": [ "inner product' 'orthogonal' 'projection' 'subspace" ], "problem_v1": "Suppose $v_1, v_2, v_3$ is an orthogonal set of vectors in ${\\mathbb R}^5$. Let $w$ be a vector in ${\\mathrm{Span}}(v_1, v_2, v_3)$ such that $v_1 \\cdot v_1=11, v_2 \\cdot v_2=30, v_3 \\cdot v_3=4$, $w \\cdot v_1=11, w \\cdot v_2=-60, w \\cdot v_3=4$, then $w=$ [ANS] $v_1+$ [ANS] $v_2+$ [ANS] $v_3.$", "answer_v1": [ "1", "-2", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $v_1, v_2, v_3$ is an orthogonal set of vectors in ${\\mathbb R}^5$. Let $w$ be a vector in ${\\mathrm{Span}}(v_1, v_2, v_3)$ such that $v_1 \\cdot v_1=45, v_2 \\cdot v_2=40.25, v_3 \\cdot v_3=4$, $w \\cdot v_1=-135, w \\cdot v_2=-80.5, w \\cdot v_3=4$, then $w=$ [ANS] $v_1+$ [ANS] $v_2+$ [ANS] $v_3.$", "answer_v2": [ "-3", "-2", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $v_1, v_2, v_3$ is an orthogonal set of vectors in ${\\mathbb R}^5$. Let $w$ be a vector in ${\\mathrm{Span}}(v_1, v_2, v_3)$ such that $v_1 \\cdot v_1=9, v_2 \\cdot v_2=9, v_3 \\cdot v_3=16$, $w \\cdot v_1=45, w \\cdot v_2=63, w \\cdot v_3=-48$, then $w=$ [ANS] $v_1+$ [ANS] $v_2+$ [ANS] $v_3.$", "answer_v3": [ "5", "7", "-3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0410", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Orthogonal and orthonormal sets", "level": "3", "keywords": [ "mapping", "orthonormal", "orthogonal", "projection" ], "problem_v1": "All vectors are in ${\\mathbb R}^n$.\nCheck the true statements below: [ANS] A\\. If $y$ is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix. B\\. If $L$ is a line through $0$ and if $\\hat{y}$ is the orthogonal projection of $y$ onto $L$, then $||\\hat{y}||$ gives the distance from $y$ to $L$. C\\. A matrix with orthonormal columns is an orthogonal matrix. D\\. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal. E\\. Not every linearly independent set in ${\\mathbb R}^n$ is an orthogonal set.", "answer_v1": [ "AE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "All vectors are in ${\\mathbb R}^n$.\nCheck the true statements below: [ANS] A\\. Not every linearly independent set in ${\\mathbb R}^n$ is an orthogonal set. B\\. If $L$ is a line through $0$ and if $\\hat{y}$ is the orthogonal projection of $y$ onto $L$, then $||\\hat{y}||$ gives the distance from $y$ to $L$. C\\. A matrix with orthonormal columns is an orthogonal matrix. D\\. If $y$ is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix. E\\. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.", "answer_v2": [ "AD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "All vectors are in ${\\mathbb R}^n$.\nCheck the true statements below: [ANS] A\\. If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal. B\\. Not every linearly independent set in ${\\mathbb R}^n$ is an orthogonal set. C\\. If $L$ is a line through $0$ and if $\\hat{y}$ is the orthogonal projection of $y$ onto $L$, then $||\\hat{y}||$ gives the distance from $y$ to $L$. D\\. A matrix with orthonormal columns is an orthogonal matrix. E\\. If $y$ is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.", "answer_v3": [ "BE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Linear_algebra_0411", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "2", "keywords": [ "dot product", "inner product", "orthogonal", "projection" ], "problem_v1": "Use the inner product \\langle f,g\\rangle=f(-1)g(-1)+f(0)g(0)+f(2)g(2) in $P_2$ to find the orthogonal projection of $f(x)=3x^2+5x-4$ onto the line $L$ spanned by $g(x)=2x^2-4x+1$.\n${\\rm proj}_L(f)=$ [ANS].", "answer_v1": [ "-28/51*(2*x^2+-4*x+1)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the inner product \\langle f,g\\rangle=f(-3)g(-3)+f(0)g(0)+f(3)g(3) in $P_2$ to find the orthogonal projection of $f(x)=2x^2+3x+8$ onto the line $L$ spanned by $g(x)=2x^2-6x-3$.\n${\\rm proj}_L(f)=$ [ANS].", "answer_v2": [ "432/1107*(2*x^2+-6*x+-3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the inner product \\langle f,g\\rangle=f(-3)g(-3)+f(0)g(0)+f(2)g(2) in $P_2$ to find the orthogonal projection of $f(x)=2x^2+4x-6$ onto the line $L$ spanned by $g(x)=3x^2-2x+8$.\n${\\rm proj}_L(f)=$ [ANS].", "answer_v3": [ "112/2001*(3*x^2+-2*x+8)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0412", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "3", "keywords": [], "problem_v1": "Let $L$ be the line in ${\\mathbb R} ^3$ that consists of all scalar multiples of the vector $\\left[\\begin{array}{c} 2\\cr-2\\cr-1 \\end{array}\\right]$. Find the reflection of the vector $\\vec{x}=\\left[\\begin{array}{c} 6\\cr 6\\cr 4 \\end{array}\\right]$ in the line $L$. ${\\rm Refl}_L(v)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-7.77778", "-4.22222", "-3.11111" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $L$ be the line in ${\\mathbb R} ^3$ that consists of all scalar multiples of the vector $\\left[\\begin{array}{c}-1\\cr 2\\cr-2 \\end{array}\\right]$. Find the reflection of the vector $\\vec{x}=\\left[\\begin{array}{c} 2\\cr 3\\cr 6 \\end{array}\\right]$ in the line $L$. ${\\rm Refl}_L(v)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-0.222222", "-6.55556", "-2.44444" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $L$ be the line in ${\\mathbb R} ^3$ that consists of all scalar multiples of the vector $\\left[\\begin{array}{c} 1\\cr-2\\cr-2 \\end{array}\\right]$. Find the reflection of the vector $\\vec{x}=\\left[\\begin{array}{c} 8\\cr 9\\cr 8 \\end{array}\\right]$ in the line $L$. ${\\rm Refl}_L(v)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-13.7778", "2.55556", "3.55556" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0413", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "3", "keywords": [], "problem_v1": "Find the orthogonal projection of \\vec{v}=\\left[\\begin{array}{c}-7\\cr 3\\cr 3 \\end{array}\\right] onto the subspace $W$ of ${\\mathbb R}^3$ spanned by \\left[\\begin{array}{c} 4\\cr 2\\cr-1 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 2\\cr 3\\cr 14 \\end{array}\\right].\n${\\rm proj}_W(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-4.40784", "-1.84985", "3.66895" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the orthogonal projection of \\vec{v}=\\left[\\begin{array}{c}-8\\cr-13\\cr-7 \\end{array}\\right] onto the subspace $W$ of ${\\mathbb R}^3$ spanned by \\left[\\begin{array}{c}-6\\cr-4\\cr 2 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 6\\cr-2\\cr 14 \\end{array}\\right].\n${\\rm proj}_W(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-12.2651", "-5.12591", "-4.04722" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the orthogonal projection of \\vec{v}=\\left[\\begin{array}{c}-6\\cr 13\\cr 17 \\end{array}\\right] onto the subspace $W$ of ${\\mathbb R}^3$ spanned by \\left[\\begin{array}{c}-3\\cr-4\\cr-1 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 2\\cr 1\\cr-10 \\end{array}\\right].\n${\\rm proj}_W(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "2.66557", "6.23663", "18.0568" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0414", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "3", "keywords": [], "problem_v1": "Let $W$ be the subspace of ${\\mathbb R}^4$ spanned by the vectors \\left[\\begin{array}{c}-1\\cr-1\\cr 1\\cr 1 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c}-2\\cr-4\\cr 4\\cr 2 \\end{array}\\right]. Find the matrix $A$ of the orthogonal projection onto $W$. $A=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0.5", "0", "0", "-0.5", "0", "0.5", "-0.5", "0", "0", "-0.5", "0.5", "0", "-0.5", "0", "0", "0.5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $W$ be the subspace of ${\\mathbb R}^4$ spanned by the vectors \\left[\\begin{array}{c} 1\\cr-1\\cr-1\\cr-1 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c}-8\\cr 9\\cr 1\\cr 2 \\end{array}\\right]. Find the matrix $A$ of the orthogonal projection onto $W$. $A=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "0.43", "-0.49", "-0.01", "-0.07", "-0.49", "0.57", "-0.07", "0.01", "-0.01", "-0.07", "0.57", "0.49", "-0.07", "0.01", "0.49", "0.43" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $W$ be the subspace of ${\\mathbb R}^4$ spanned by the vectors \\left[\\begin{array}{c} 1\\cr-1\\cr 1\\cr-1 \\end{array}\\right] \\ \\mbox{and} \\ \\left[\\begin{array}{c} 0\\cr 3\\cr-1\\cr 4 \\end{array}\\right]. Find the matrix $A$ of the orthogonal projection onto $W$. $A=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "0.65", "-0.05", "0.45", "0.15", "-0.05", "0.35", "-0.15", "0.45", "0.45", "-0.15", "0.35", "-0.05", "0.15", "0.45", "-0.05", "0.65" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0415", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "3", "keywords": [], "problem_v1": "Find bases of the kernel and image of the orthogonal projection onto the plane $8x+6 y+z=0$ in ${\\mathbb R}^3$.\nA basis for the kernel is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$\nA basis for the image is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v1": [ "8", "6", "1", "1", "0", "-8", "6", "-8", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find bases of the kernel and image of the orthogonal projection onto the plane $2x-3 y+z=0$ in ${\\mathbb R}^3$.\nA basis for the kernel is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$\nA basis for the image is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v2": [ "2", "-3", "1", "1", "0", "-2", "-3", "-2", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find bases of the kernel and image of the orthogonal projection onto the plane $4x+4 y+z=0$ in ${\\mathbb R}^3$.\nA basis for the kernel is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$\nA basis for the image is $\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v3": [ "4", "4", "1", "1", "0", "-4", "4", "-4", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0416", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "2", "keywords": [ "Vector", "Parallel", "Perpendicular", "Plane" ], "problem_v1": "Find a vector $\\vec{u}$ that is parallel to the $yz$-plane and perpendicular to the vector $\\vec{v}=\\left<3,1,1\\right>$.\n$\\vec{u}$=[ANS].", "answer_v1": [ "(0,1,-1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find a vector $\\vec{u}$ that is parallel to the $yz$-plane and perpendicular to the vector $\\vec{v}=\\left<-5,5,-4\\right>$.\n$\\vec{u}$=[ANS].", "answer_v2": [ "(0,-4,-5)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find a vector $\\vec{u}$ that is parallel to the $yz$-plane and perpendicular to the vector $\\vec{v}=\\left<-2,1,-2\\right>$.\n$\\vec{u}$=[ANS].", "answer_v3": [ "(0,-2,-1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0417", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "2", "keywords": [ "Vector", "Parallel", "Perpendicular" ], "problem_v1": "Find two vectors $\\vec{v}_1$ and $\\vec{v}_2$ whose sum is $\\left<3,1\\right>$, where $\\vec{v}_1$ is parallel to $\\left<1,2\\right>$ while $\\vec{v}_2$ is perpendicular to $\\left<1,2\\right>$.\n$\\vec{v}_1$=[ANS] and $\\vec{v}_2$=[ANS].", "answer_v1": [ "(1,2)", "(2,-1)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Find two vectors $\\vec{v}_1$ and $\\vec{v}_2$ whose sum is $\\left<-5,5\\right>$, where $\\vec{v}_1$ is parallel to $\\left<-4,-2\\right>$ while $\\vec{v}_2$ is perpendicular to $\\left<-4,-2\\right>$.\n$\\vec{v}_1$=[ANS] and $\\vec{v}_2$=[ANS].", "answer_v2": [ "(-2,-1)", "(-3,6)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Find two vectors $\\vec{v}_1$ and $\\vec{v}_2$ whose sum is $\\left<-2,1\\right>$, where $\\vec{v}_1$ is parallel to $\\left<-2,1\\right>$ while $\\vec{v}_2$ is perpendicular to $\\left<-2,1\\right>$.\n$\\vec{v}_1$=[ANS] and $\\vec{v}_2$=[ANS].", "answer_v3": [ "(-2,1)", "(0,0)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0418", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "2", "keywords": [ "Vector", "Projection", "Orthogonal" ], "problem_v1": "Suppose $\\vec{u}=\\left<3,1,1\\right>$ and $\\vec{v}=\\left<2,-2,-2\\right>$. Then:\nThe projection of $\\vec{u}$ along $\\vec{v}$ is [ANS].\nThe projection of $\\vec{u}$ orthogonal to $\\vec{v}$ is [ANS].", "answer_v1": [ "(0.333333,-0.333333,-0.333333)", "(2.66667,1.33333,1.33333)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $\\vec{u}=\\left<-5,5,-4\\right>$ and $\\vec{v}=\\left<-2,5,-2\\right>$. Then:\nThe projection of $\\vec{u}$ along $\\vec{v}$ is [ANS].\nThe projection of $\\vec{u}$ orthogonal to $\\vec{v}$ is [ANS].", "answer_v2": [ "(-2.60606,6.51515,-2.60606)", "(-2.39394,-1.51515,-1.39394)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $\\vec{u}=\\left<-2,1,-2\\right>$ and $\\vec{v}=\\left<1,-3,-2\\right>$. Then:\nThe projection of $\\vec{u}$ along $\\vec{v}$ is [ANS].\nThe projection of $\\vec{u}$ orthogonal to $\\vec{v}$ is [ANS].", "answer_v3": [ "(-0.0714286,0.214286,0.142857)", "(-1.92857,0.785714,-2.14286)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0419", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "3", "keywords": [ "plane", "distance", "projection" ], "problem_v1": "(a) Find the distance from the point $P_0=(5,6,7)$ to the plane $-3x+4 y+2 z=-1$. Use sqrt() to enter square roots. Distance: [ANS]\n(b) Find the equation of the plane that passes through the points $P=(2,-3,-1)$, $Q=(-1,-1,0)$, and $R=(-4,-4,-1)$. Write your answer in terms of the variables $x$, $y$, $z$. Answer: [ANS]", "answer_v1": [ "4.45669", "6*y-x-15*z = -5" ], "answer_type_v1": [ "NV", "EQ" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find the distance from the point $P_0=(-8,9,2)$ to the plane $3x-3 y-5 z=-2$. Use sqrt() to enter square roots. Distance: [ANS]\n(b) Find the equation of the plane that passes through the points $P=(-4,-4,-2)$, $Q=(-2,-1,2)$, and $R=(5,4,-5)$. Write your answer in terms of the variables $x$, $y$, $z$. Answer: [ANS]", "answer_v2": [ "8.99742", "41*x-42*y+11*z = -18" ], "answer_type_v2": [ "NV", "EQ" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find the distance from the point $P_0=(-4,6,3)$ to the plane $-4x+3 y+9 z=5$. Use sqrt() to enter square roots. Distance: [ANS]\n(b) Find the equation of the plane that passes through the points $P=(-4,0,-1)$, $Q=(-2,-5,-5)$, and $R=(3,-4,-2)$. Write your answer in terms of the variables $x$, $y$, $z$. Answer: [ANS]", "answer_v3": [ "5.4392", "11*x+26*y-27*z = -17" ], "answer_type_v3": [ "NV", "EQ" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0420", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "2", "keywords": [ "inner product' 'orthogonal' 'projection' 'subspace" ], "problem_v1": "Compute the orthogonal projection of $\\vec{v}=\\left[\\begin{array}{c} 5\\cr-3\\cr-4 \\end{array}\\right]$ onto the line $L$ through $\\left[\\begin{array}{c} 4\\cr 2\\cr 2 \\end{array}\\right]$ and the origin.\n${\\rm proj}_L(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "1", "0.5", "0.5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Compute the orthogonal projection of $\\vec{v}=\\left[\\begin{array}{c}-3\\cr 9\\cr-4 \\end{array}\\right]$ onto the line $L$ through $\\left[\\begin{array}{c}-6\\cr 6\\cr-5 \\end{array}\\right]$ and the origin.\n${\\rm proj}_L(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-5.69072", "5.69072", "-4.74227" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Compute the orthogonal projection of $\\vec{v}=\\left[\\begin{array}{c} 1\\cr-5\\cr-3 \\end{array}\\right]$ onto the line $L$ through $\\left[\\begin{array}{c}-3\\cr 2\\cr-3 \\end{array}\\right]$ and the origin.\n${\\rm proj}_L(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "0.545455", "-0.363636", "0.545455" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0421", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "3", "keywords": [ "inner product' 'orthogonal' 'projection' 'subspace" ], "problem_v1": "Let $\\vec{y}=\\left[\\begin{array}{c} 5\\cr-3\\cr-4 \\end{array}\\right]$ and $\\vec{u}=\\left[\\begin{array}{c} 4\\cr 2\\cr 2 \\end{array}\\right]$. Write $\\vec{y}$ as the sum of two orthogonal vectors, $\\vec{x}_{1}$ in $Span\\left\\lbrace u\\right\\rbrace$ and $\\vec{x}_{2}$ orthogonal to $\\vec{u}$.\n$\\vec{x}_{1}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\vec{x}_{2}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "1", "0.5", "0.5", "4", "-3.5", "-4.5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $\\vec{y}=\\left[\\begin{array}{c}-3\\cr 9\\cr-4 \\end{array}\\right]$ and $\\vec{u}=\\left[\\begin{array}{c}-6\\cr 6\\cr-5 \\end{array}\\right]$. Write $\\vec{y}$ as the sum of two orthogonal vectors, $\\vec{x}_{1}$ in $Span\\left\\lbrace u\\right\\rbrace$ and $\\vec{x}_{2}$ orthogonal to $\\vec{u}$.\n$\\vec{x}_{1}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\vec{x}_{2}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-5.69072", "5.69072", "-4.74227", "2.69072", "3.30928", "0.742268" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $\\vec{y}=\\left[\\begin{array}{c} 1\\cr-5\\cr-3 \\end{array}\\right]$ and $\\vec{u}=\\left[\\begin{array}{c}-3\\cr 2\\cr-3 \\end{array}\\right]$. Write $\\vec{y}$ as the sum of two orthogonal vectors, $\\vec{x}_{1}$ in $Span\\left\\lbrace u\\right\\rbrace$ and $\\vec{x}_{2}$ orthogonal to $\\vec{u}$.\n$\\vec{x}_{1}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\vec{x}_{2}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "0.545455", "-0.363636", "0.545455", "0.454545", "-4.63636", "-3.54545" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0422", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "2", "keywords": [ "inner product' 'orthogonal' 'projection' 'subspace' 'parametric" ], "problem_v1": "Find the projection of $\\vec{v}=\\left[\\begin{array}{c} 4\\cr 1\\cr 2 \\end{array}\\right]$ onto the line $\\ell$ of ${\\mathbb R}^3$ given by the parametric equation $\\ell=t\\vec{u}$, where $\\vec{u}=\\left[\\begin{array}{c} 3\\cr-3\\cr-3 \\end{array}\\right]$.\n$\\mathrm{proj}_{\\ell}(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.333333", "-0.333333", "-0.333333" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the projection of $\\vec{v}=\\left[\\begin{array}{c}-6\\cr 6\\cr-5 \\end{array}\\right]$ onto the line $\\ell$ of ${\\mathbb R}^3$ given by the parametric equation $\\ell=t\\vec{u}$, where $\\vec{u}=\\left[\\begin{array}{c}-2\\cr 7\\cr-3 \\end{array}\\right]$.\n$\\mathrm{proj}_{\\ell}(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-2.22581", "7.79032", "-3.33871" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the projection of $\\vec{v}=\\left[\\begin{array}{c}-3\\cr 2\\cr-3 \\end{array}\\right]$ onto the line $\\ell$ of ${\\mathbb R}^3$ given by the parametric equation $\\ell=t\\vec{u}$, where $\\vec{u}=\\left[\\begin{array}{c} 1\\cr-4\\cr-2 \\end{array}\\right]$.\n$\\mathrm{proj}_{\\ell}(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-0.238095", "0.952381", "0.47619" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0423", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "3", "keywords": [ "inner product' 'orthogonal' 'projection' 'subspace" ], "problem_v1": "Given $\\vec{v}=\\left[\\begin{array}{c} 2\\cr-3\\cr 0\\cr 3 \\end{array}\\right]$, find the closest point to $\\vec{v}$ in the subspace $W$ spanned by $\\left[\\begin{array}{c} 4\\cr 2\\cr-2\\cr 1 \\end{array}\\right]$ and $\\left[\\begin{array}{c} 2\\cr 3\\cr-3\\cr-20 \\end{array}\\right]$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "0.491943", "-0.0620853", "0.0620853", "3.28057" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Given $\\vec{v}=\\left[\\begin{array}{c}-4\\cr 1\\cr-9\\cr 3 \\end{array}\\right]$, find the closest point to $\\vec{v}$ in the subspace $W$ spanned by $\\left[\\begin{array}{c}-6\\cr-4\\cr 6\\cr-1 \\end{array}\\right]$ and $\\left[\\begin{array}{c} 6\\cr-2\\cr-3\\cr-46 \\end{array}\\right]$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "2.11471", "1.78948", "-2.30454", "3.32658" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Given $\\vec{v}=\\left[\\begin{array}{c} 9\\cr 8\\cr-6\\cr-4 \\end{array}\\right]$, find the closest point to $\\vec{v}$ in the subspace $W$ spanned by $\\left[\\begin{array}{c}-3\\cr-4\\cr-4\\cr 1 \\end{array}\\right]$ and $\\left[\\begin{array}{c} 2\\cr 1\\cr-2\\cr 2 \\end{array}\\right]$.\n\\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "7.4011", "6.02198", "-0.901099", "3.68681" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0424", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Projection and distance", "level": "2", "keywords": [ "inner product' 'orthogonal' 'projection' 'subspace' 'distance" ], "problem_v1": "Find the shortest distance from the point $P=(4, 1, 2)$ to a point on the line given by $l: (x,y,z)=(3 t,-3 t,-3 t)$. The distance is [ANS].", "answer_v1": [ "4.54606056566195" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the shortest distance from the point $P=(-6, 6,-5)$ to a point on the line given by $l: (x,y,z)=(-2 t, 7 t,-3 t)$. The distance is [ANS].", "answer_v2": [ "4.4955174807084" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the shortest distance from the point $P=(-3, 2,-3)$ to a point on the line given by $l: (x,y,z)=(1 t,-4 t,-2 t)$. The distance is [ANS].", "answer_v3": [ "4.56174569759471" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0425", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Gram-Schmidt process", "level": "3", "keywords": [ "linear algebra", "matrix", "basis", "dot product", "inner product", "orthonormal", "image" ], "problem_v1": "Let A=\\left[\\begin{array}{ccc} 1 &3 &-2\\cr 2 &2 &-2\\cr 1 &-1 &0 \\end{array}\\right]. Find an orthonormal basis of the column space of $A$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v1": [ "0.408248", "0.816497", "0.408248", "-0.707107", "0", "0.707107" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{ccc} 1 &-5 &0\\cr-3 &6 &-9\\cr 3 &-5 &10 \\end{array}\\right]. Find an orthonormal basis of the column space of $A$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v2": [ "0.229416", "-0.688247", "0.688247", "-0.948683", "0", "0.316228" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{ccc} 1 &-4 &1\\cr-1 &2 &1\\cr 1 &0 &-3 \\end{array}\\right]. Find an orthonormal basis of the column space of $A$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v3": [ "0.57735", "-0.57735", "0.57735", "-0.707107", "0", "0.707107" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0426", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Gram-Schmidt process", "level": "2", "keywords": [ "linear algebra", "vector space", "basis", "dot product", "inner product", "orthonormal" ], "problem_v1": "Let \\vec{x}=\\left[\\begin{array}{c} 2\\cr 1\\cr 1\\cr 0 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{y}=\\left[\\begin{array}{c}-1\\cr-1\\cr-3\\cr 2 \\end{array}\\right]. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of ${\\mathbb R}^4$ spanned by $\\vec{x}$ and $\\vec{y}$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v1": [ "0.816497", "0.408248", "0.408248", "0", "0.333333", "0", "-0.666667", "0.666667" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let \\vec{x}=\\left[\\begin{array}{c}-12\\cr-6\\cr 12\\cr 0 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{y}=\\left[\\begin{array}{c} 12\\cr 4\\cr-4\\cr-1 \\end{array}\\right]. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of ${\\mathbb R}^4$ spanned by $\\vec{x}$ and $\\vec{y}$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v2": [ "-0.666667", "-0.333333", "0.666667", "0", "0.696311", "0", "0.696311", "-0.174078" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let \\vec{x}=\\left[\\begin{array}{c}-4\\cr-4\\cr 2\\cr 0 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{y}=\\left[\\begin{array}{c}-1\\cr-4\\cr 8\\cr-9 \\end{array}\\right]. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of ${\\mathbb R}^4$ spanned by $\\vec{x}$ and $\\vec{y}$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v3": [ "-0.666667", "-0.666667", "0.333333", "0", "0.267261", "0", "0.534522", "-0.801784" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0427", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Gram-Schmidt process", "level": "3", "keywords": [ "linear algebra", "vector space", "basis", "dot product", "inner product", "orthonormal" ], "problem_v1": "Find an orthonormal basis of the plane $x_1+7x_2-x_3=0$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v1": [ "0.707107", "0", "0.707107", "-0.693103", "0.19803", "0.693103" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find an orthonormal basis of the plane $x_1+2x_2-x_3=0$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v2": [ "0.707107", "0", "0.707107", "-0.57735", "0.57735", "0.57735" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find an orthonormal basis of the plane $x_1+4x_2-x_3=0$.\n$\\Bigg\\lbrace$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\Bigg\\rbrace.$", "answer_v3": [ "0.707107", "0", "0.707107", "-0.666667", "0.333333", "0.666667" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0428", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Gram-Schmidt process", "level": "3", "keywords": [ "dot product", "inner product", "Gram-Schmidt" ], "problem_v1": "Let f(x)=2, \\ g(x)=6x-4 \\ \\mbox{and} \\ h(x)=4x^2. Consider the inner product \\langle p,q \\rangle=\\int_0^{4} p(x)q(x) \\, dx in the vector space $C^0\\lbrack 0,4\\rbrack$ of continuous functions on the domain $\\lbrack 0, 4 \\rbrack$. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of $C^0\\lbrack 0,4\\rbrack$ spanned by the functions $f(x)$, $g(x)$, and $h(x)$.\n$\\big\\lbrace$ [ANS], [ANS], [ANS] $\\big\\rbrace.$", "answer_v1": [ "1/2", "2*sqrt(3)*x/8-[sqrt(3)]/2", "6*sqrt(5)*x*x/32-6*sqrt(5)*x/8+[sqrt(5)]/2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let f(x)=8, \\ g(x)=-3x+8 \\ \\mbox{and} \\ h(x)=-4x^2. Consider the inner product \\langle p,q \\rangle=\\int_0^{1} p(x)q(x) \\, dx in the vector space $C^0\\lbrack 0,1\\rbrack$ of continuous functions on the domain $\\lbrack 0, 1 \\rbrack$. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of $C^0\\lbrack 0,1\\rbrack$ spanned by the functions $f(x)$, $g(x)$, and $h(x)$.\n$\\big\\lbrace$ [ANS], [ANS], [ANS] $\\big\\rbrace.$", "answer_v2": [ "1/1", "-2*sqrt(3)*x/1+[sqrt(3)]/1", "-6*sqrt(5)*x*x/1+6*sqrt(5)*x/1-[sqrt(5)]/1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let f(x)=2, \\ g(x)=4x-6 \\ \\mbox{and} \\ h(x)=4x^2. Consider the inner product \\langle p,q \\rangle=\\int_0^{1} p(x)q(x) \\, dx in the vector space $C^0\\lbrack 0,1\\rbrack$ of continuous functions on the domain $\\lbrack 0, 1 \\rbrack$. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of $C^0\\lbrack 0,1\\rbrack$ spanned by the functions $f(x)$, $g(x)$, and $h(x)$.\n$\\big\\lbrace$ [ANS], [ANS], [ANS] $\\big\\rbrace.$", "answer_v3": [ "1/1", "2*sqrt(3)*x/1-[sqrt(3)]/1", "6*sqrt(5)*x*x/1-6*sqrt(5)*x/1+[sqrt(5)]/1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0429", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Gram-Schmidt process", "level": "3", "keywords": [], "problem_v1": "Let M_1=\\left[\\begin{array}{cc} 1 &1\\cr 1 &1 \\end{array}\\right] \\ \\mbox{and} \\ M_2=\\left[\\begin{array}{cc}-4 &0\\cr 0 &-1 \\end{array}\\right]. Consider the inner product $\\langle A,B\\rangle={\\rm trace} (A^T B)$ in the vector space ${\\mathbb R}^{2\\times 2}$ of $2\\times 2$ matrices. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of ${\\mathbb R}^{2\\times 2}$ spanned by the matrices $M_1$ and $M_2$.\n$\\Bigg\\lbrace$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $\\Bigg\\rbrace$", "answer_v1": [ "0.5", "0.5", "0.5", "0.5", "-0.838742", "0.381246", "0.381246", "0.0762493" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let M_1=\\left[\\begin{array}{cc}-1 &1\\cr-1 &-1 \\end{array}\\right] \\ \\mbox{and} \\ M_2=\\left[\\begin{array}{cc}-6 &-2\\cr-1 &0 \\end{array}\\right]. Consider the inner product $\\langle A,B\\rangle={\\rm trace} (A^T B)$ in the vector space ${\\mathbb R}^{2\\times 2}$ of $2\\times 2$ matrices. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of ${\\mathbb R}^{2\\times 2}$ spanned by the matrices $M_1$ and $M_2$.\n$\\Bigg\\lbrace$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $\\Bigg\\rbrace$", "answer_v2": [ "-0.5", "0.5", "-0.5", "-0.5", "-0.805779", "-0.551323", "0.0424094", "0.212047" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let M_1=\\left[\\begin{array}{cc}-1 &1\\cr-1 &1 \\end{array}\\right] \\ \\mbox{and} \\ M_2=\\left[\\begin{array}{cc}-3 &2\\cr 2 &2 \\end{array}\\right]. Consider the inner product $\\langle A,B\\rangle={\\rm trace} (A^T B)$ in the vector space ${\\mathbb R}^{2\\times 2}$ of $2\\times 2$ matrices. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of ${\\mathbb R}^{2\\times 2}$ spanned by the matrices $M_1$ and $M_2$.\n$\\Bigg\\lbrace$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}, $\\Bigg\\rbrace$", "answer_v3": [ "-0.5", "0.5", "-0.5", "0.5", "-0.455661", "0.195283", "0.846228", "0.195283" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0430", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Gram-Schmidt process", "level": "3", "keywords": [ "Gram-Schmidt", "inner product" ], "problem_v1": "Let \\mathbf{v}_1=\\left[\\begin{array}{c}1\\\\4\\\\0\\\\1\\\\0\\\\2\\\\\\end{array}\\right], \\mathbf{v}_2=\\left[\\begin{array}{c}-2\\\\0\\\\2\\\\1\\\\0\\\\1\\\\\\end{array}\\right], \\text{and} \\mathbf{v}_3=\\left[\\begin{array}{c}0\\\\0\\\\0\\\\2\\\\2\\\\-3\\\\\\end{array}\\right]. Use the Gram-Schmidt procedure to produce an orthogonal set with the same span. (Hint: If you have unlimited submissions, it might be useful to submit to check your answer for each vector $\\mathbf{u}_j$ before continuing.) Note: The $\\mathbf{u}_j$ must be given in the same order as provided by the standard procedure. $\\mathbf{u}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\langle \\mathbf{u}_1, \\mathbf{v}_2 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle=$ [ANS]\n$\\mathbf{u}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\langle \\mathbf{u}_1, \\mathbf{v}_3 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_2, \\mathbf{v}_3 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle=$ [ANS]\n$\\mathbf{u}_3=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "1", "4", "0", "1", "0", "2", "-2.04545", "-0.181818", "2", "0.954545", "0", "0.909091", "0.0136986", "0.712329", "0.164384", "2.26027", "2", "-2.56164" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let \\mathbf{v}_1=\\left[\\begin{array}{c}5\\\\1\\\\0\\\\-4\\\\0\\\\-2\\\\\\end{array}\\right], \\mathbf{v}_2=\\left[\\begin{array}{c}-2\\\\0\\\\5\\\\-3\\\\0\\\\-2\\\\\\end{array}\\right], \\text{and} \\mathbf{v}_3=\\left[\\begin{array}{c}-5\\\\0\\\\0\\\\2\\\\3\\\\-1\\\\\\end{array}\\right]. Use the Gram-Schmidt procedure to produce an orthogonal set with the same span. (Hint: If you have unlimited submissions, it might be useful to submit to check your answer for each vector $\\mathbf{u}_j$ before continuing.) Note: The $\\mathbf{u}_j$ must be given in the same order as provided by the standard procedure. $\\mathbf{u}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\langle \\mathbf{u}_1, \\mathbf{v}_2 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle=$ [ANS]\n$\\mathbf{u}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\langle \\mathbf{u}_1, \\mathbf{v}_3 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_2, \\mathbf{v}_3 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle=$ [ANS]\n$\\mathbf{u}_3=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "5", "1", "0", "-4", "0", "-2", "-2.65217", "-0.130435", "5", "-2.47826", "0", "-1.73913", "-0.984177", "0.705696", "-1.21835", "-0.0917722", "3", "-1.92405" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let \\mathbf{v}_1=\\left[\\begin{array}{c}1\\\\2\\\\0\\\\-2\\\\0\\\\1\\\\\\end{array}\\right], \\mathbf{v}_2=\\left[\\begin{array}{c}-2\\\\0\\\\2\\\\3\\\\0\\\\5\\\\\\end{array}\\right], \\text{and} \\mathbf{v}_3=\\left[\\begin{array}{c}-3\\\\0\\\\0\\\\-2\\\\5\\\\-3\\\\\\end{array}\\right]. Use the Gram-Schmidt procedure to produce an orthogonal set with the same span. (Hint: If you have unlimited submissions, it might be useful to submit to check your answer for each vector $\\mathbf{u}_j$ before continuing.) Note: The $\\mathbf{u}_j$ must be given in the same order as provided by the standard procedure. $\\mathbf{u}_1=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\langle \\mathbf{u}_1, \\mathbf{v}_2 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle=$ [ANS]\n$\\mathbf{u}_2=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} $\\langle \\mathbf{u}_1, \\mathbf{v}_3 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_2, \\mathbf{v}_3 \\rangle=$ [ANS]\n$\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle=$ [ANS]\n$\\mathbf{u}_3=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "1", "2", "0", "-2", "0", "1", "-1.7", "0.6", "2", "2.4", "0", "5.3", "-3.44526", "0.627737", "0.759124", "-1.48905", "5", "-0.788321" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0431", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Find the Fourier coefficients of $f(x)=8x+2$, i.e. numbers $a_0$, $b_k$, $c_k$ (note that $b_k$ and $c_k$ may depend on $k$) such that f(x)=a_0 \\frac{1}{\\sqrt{2} }+b_1 \\sin(x)+c_1 \\cos(x)+b_2 \\sin(2x)+c_2 \\cos(2x)+\\ldots. $a_0=$ [ANS], $b_k=$ [ANS] if $k$ is odd, and [ANS] if $k$ is even, $c_k=$ [ANS] if $k$ is odd, and [ANS] if $k$ is even.", "answer_v1": [ "2.82843", "16/k", "-16/k", "0", "0" ], "answer_type_v1": [ "NV", "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Find the Fourier coefficients of $f(x)=2x-7$, i.e. numbers $a_0$, $b_k$, $c_k$ (note that $b_k$ and $c_k$ may depend on $k$) such that f(x)=a_0 \\frac{1}{\\sqrt{2} }+b_1 \\sin(x)+c_1 \\cos(x)+b_2 \\sin(2x)+c_2 \\cos(2x)+\\ldots. $a_0=$ [ANS], $b_k=$ [ANS] if $k$ is odd, and [ANS] if $k$ is even, $c_k=$ [ANS] if $k$ is odd, and [ANS] if $k$ is even.", "answer_v2": [ "-9.89949", "4/k", "-4/k", "0", "0" ], "answer_type_v2": [ "NV", "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Find the Fourier coefficients of $f(x)=4x-4$, i.e. numbers $a_0$, $b_k$, $c_k$ (note that $b_k$ and $c_k$ may depend on $k$) such that f(x)=a_0 \\frac{1}{\\sqrt{2} }+b_1 \\sin(x)+c_1 \\cos(x)+b_2 \\sin(2x)+c_2 \\cos(2x)+\\ldots. $a_0=$ [ANS], $b_k=$ [ANS] if $k$ is odd, and [ANS] if $k$ is even, $c_k=$ [ANS] if $k$ is odd, and [ANS] if $k$ is even.", "answer_v3": [ "-5.65685", "8/k", "-8/k", "0", "0" ], "answer_type_v3": [ "NV", "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Linear_algebra_0432", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Find the least-squares solution $\\vec{x}^*$ of the system \\left[\\begin{array}{cc} 2 &-1\\cr-2 &1\\cr 4 &5 \\end{array}\\right] \\vec{x}=\\left[\\begin{array}{c}-1\\cr 3\\cr-18 \\end{array}\\right]. $\\vec{x}^*=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-2", "-2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the least-squares solution $\\vec{x}^*$ of the system \\left[\\begin{array}{cc} 1 &-1\\cr-1 &1\\cr 3 &4 \\end{array}\\right] \\vec{x}=\\left[\\begin{array}{c} 3\\cr-11\\cr 7 \\end{array}\\right]. $\\vec{x}^*=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "5", "-2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the least-squares solution $\\vec{x}^*$ of the system \\left[\\begin{array}{cc} 1 &-1\\cr-1 &1\\cr 3 &4 \\end{array}\\right] \\vec{x}=\\left[\\begin{array}{c} 3\\cr 5\\cr-17 \\end{array}\\right]. $\\vec{x}^*=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-3", "-2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0433", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "During the summer months Terry makes and sells necklaces on the beach. Terry notices that if he lowers the price, he can sell more necklaces, and if he raises the price than he sells fewer necklaces. The table below shows how the number $n$ of necklaces sold in one day depends on the price $p$ (in dollars). \\begin{array}{|r|r|} \\hline \\mbox{Price} & \\mbox{Number of necklaces sold} \\cr \\hline 8 & 34 \\cr \\hline 12 & 21 \\cr \\hline 16 & 11 \\cr \\hline \\end{array}\n(a) Find a linear function of the form $n=c_0+c_1 p$ that best fits these data, using least squares. $n=n(p)=$ [ANS]\n(b) Find the revenue (number of items sold times the price of each item) as a function of price $p$. $R=R(p)=$ [ANS].\n(c) If the material for each necklace costs Terry $5$ dollars, find the profit (revenue minus cost of the material) as a function of price $p$. $P=P(p)=$ [ANS].\n(d) Finally, find the price that will maximize the profit. $p=$ [ANS].", "answer_v1": [ "56.5-2.875*p", "56.5*p+-2.875*p*p", "56.5*(p-5)+-2.875*p*(p-5)", "(-2.875*5-56.5)/2/-2.875" ], "answer_type_v1": [ "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "During the summer months Terry makes and sells necklaces on the beach. Terry notices that if he lowers the price, he can sell more necklaces, and if he raises the price than he sells fewer necklaces. The table below shows how the number $n$ of necklaces sold in one day depends on the price $p$ (in dollars). \\begin{array}{|r|r|} \\hline \\mbox{Price} & \\mbox{Number of necklaces sold} \\cr \\hline 7 & 32 \\cr \\hline 12 & 25 \\cr \\hline 15 & 11 \\cr \\hline \\end{array}\n(a) Find a linear function of the form $n=c_0+c_1 p$ that best fits these data, using least squares. $n=n(p)=$ [ANS]\n(b) Find the revenue (number of items sold times the price of each item) as a function of price $p$. $R=R(p)=$ [ANS].\n(c) If the material for each necklace costs Terry $4$ dollars, find the profit (revenue minus cost of the material) as a function of price $p$. $P=P(p)=$ [ANS].\n(d) Finally, find the price that will maximize the profit. $p=$ [ANS].", "answer_v2": [ "51-2.5*p", "51*p+-2.5*p*p", "51*(p-4)+-2.5*p*(p-4)", "(-2.5*4-51)/2/-2.5" ], "answer_type_v2": [ "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "During the summer months Terry makes and sells necklaces on the beach. Terry notices that if he lowers the price, he can sell more necklaces, and if he raises the price than he sells fewer necklaces. The table below shows how the number $n$ of necklaces sold in one day depends on the price $p$ (in dollars). \\begin{array}{|r|r|} \\hline \\mbox{Price} & \\mbox{Number of necklaces sold} \\cr \\hline 7 & 33 \\cr \\hline 12 & 21 \\cr \\hline 15 & 12 \\cr \\hline \\end{array}\n(a) Find a linear function of the form $n=c_0+c_1 p$ that best fits these data, using least squares. $n=n(p)=$ [ANS]\n(b) Find the revenue (number of items sold times the price of each item) as a function of price $p$. $R=R(p)=$ [ANS].\n(c) If the material for each necklace costs Terry $6$ dollars, find the profit (revenue minus cost of the material) as a function of price $p$. $P=P(p)=$ [ANS].\n(d) Finally, find the price that will maximize the profit. $p=$ [ANS].", "answer_v3": [ "51.4898-2.60204*p", "51.4898*p+-2.60204*p*p", "51.4898*(p-6)+-2.60204*p*(p-6)", "(-2.60204*6-51.4898)/2/-2.60204" ], "answer_type_v3": [ "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0434", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Fit a trigonometric function of the form $f(t)=c_0+c_1 \\sin(t)+c_2 \\cos(t)$ to the data points $(0, 7.5)$, $( \\frac{\\pi}{2} ,2.5)$, $(\\pi,-0.5)$, $( \\frac{3\\pi}{2} ,-1.5)$, using least squares.\n$f(t)=$ [ANS]", "answer_v1": [ "2+2*sin(t)+4*cos(t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Fit a trigonometric function of the form $f(t)=c_0+c_1 \\sin(t)+c_2 \\cos(t)$ to the data points $(0, 2.5)$, $( \\frac{\\pi}{2} ,3.5)$, $(\\pi, 8.5)$, $( \\frac{3\\pi}{2} , 17.5)$, using least squares.\n$f(t)=$ [ANS]", "answer_v2": [ "8-7*sin(t)-3*cos(t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Fit a trigonometric function of the form $f(t)=c_0+c_1 \\sin(t)+c_2 \\cos(t)$ to the data points $(0, 2)$, $( \\frac{\\pi}{2} ,-1)$, $(\\pi, 0)$, $( \\frac{3\\pi}{2} , 7)$, using least squares.\n$f(t)=$ [ANS]", "answer_v3": [ "2-4*sin(t)+cos(t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0435", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Let $S(t)$ be the number of daylight hours on the $t$ th day of the year in Miami. We are given the following data for $S(t)$: \\begin{array}{|l|r|r|} \\hline \\mbox{Day} & t & S(t) \\cr \\hline \\mbox{January} 24 & 24 & 11 \\cr \\hline \\mbox{March} 19 & 78 & 12 \\cr \\hline \\mbox{May} 20 & 140 & 13 \\cr \\hline \\mbox{July} 23 & 204 & 14 \\cr \\hline \\end{array} We wish to fit a trigonometric function of the form f(t)=a+b \\sin \\left( \\frac{2\\pi}{365} t \\right)+c \\cos \\left( \\frac{2\\pi}{365} t \\right) to these data. Find the best approximation of this form, using least squares.\n$f(t)=$ [ANS]", "answer_v1": [ "12.5089-0.442025*sin(2*pi*t/365)-1.32453*cos(2*pi*t/365)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $S(t)$ be the number of daylight hours on the $t$ th day of the year in Manley Hot Springs. We are given the following data for $S(t)$: \\begin{array}{|l|r|r|} \\hline \\mbox{Day} & t & S(t) \\cr \\hline \\mbox{January} 3 & 3 & 5 \\cr \\hline \\mbox{March} 29 & 88 & 12 \\cr \\hline \\mbox{May} 5 & 125 & 19 \\cr \\hline \\mbox{July} 11 & 192 & 20 \\cr \\hline \\end{array} We wish to fit a trigonometric function of the form f(t)=a+b \\sin \\left( \\frac{2\\pi}{365} t \\right)+c \\cos \\left( \\frac{2\\pi}{365} t \\right) to these data. Find the best approximation of this form, using least squares.\n$f(t)=$ [ANS]", "answer_v2": [ "12.6533+0.888386*sin(2*pi*t/365)-8.01788*cos(2*pi*t/365)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $S(t)$ be the number of daylight hours on the $t$ th day of the year in New York. We are given the following data for $S(t)$: \\begin{array}{|l|r|r|} \\hline \\mbox{Day} & t & S(t) \\cr \\hline \\mbox{January} 10 & 10 & 10 \\cr \\hline \\mbox{March} 19 & 78 & 12 \\cr \\hline \\mbox{May} 9 & 129 & 14 \\cr \\hline \\mbox{July} 18 & 199 & 15 \\cr \\hline \\end{array} We wish to fit a trigonometric function of the form f(t)=a+b \\sin \\left( \\frac{2\\pi}{365} t \\right)+c \\cos \\left( \\frac{2\\pi}{365} t \\right) to these data. Find the best approximation of this form, using least squares.\n$f(t)=$ [ANS]", "answer_v3": [ "12.5317-0.0155183*sin(2*pi*t/365)-2.54165*cos(2*pi*t/365)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0436", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "The table below lists the height $h$ (in cm), the age $a$ (in years), the gender $g$ (0=\"Female\", 1=\"Male\"), and the weight $w$ (in kg) of some college students. \\begin{array}{|r|r|r|r|} \\hline \\mbox{Height} & \\mbox{Age} & \\mbox{Gender} & \\mbox{Weight} \\cr \\hline 171 & 19 & 1 & 80 \\cr \\hline 176 & 20 & 1 & 85 \\cr \\hline 161 & 21 & 0 & 59 \\cr \\hline 158 & 23 & 0 & 55 \\cr \\hline 166 & 20 & 0 & 64 \\cr \\hline \\end{array} We wish to fit a linear function of the form w=c_0+c_1 h+c_2 a+c_3 g which predicts the weight from the rest of the data. Find the best approximation of this function, using least squares.\n$w=w(h,a,g)=$ [ANS]", "answer_v1": [ "1.02455*h-100.35-0.279018*a+10.5313*g" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The table below lists the height $h$ (in cm), the age $a$ (in years), the gender $g$ (0=\"Female\", 1=\"Male\"), and the weight $w$ (in kg) of some college students. \\begin{array}{|r|r|r|r|} \\hline \\mbox{Height} & \\mbox{Age} & \\mbox{Gender} & \\mbox{Weight} \\cr \\hline 155 & 22 & 0 & 46 \\cr \\hline 161 & 20 & 0 & 50 \\cr \\hline 167 & 19 & 0 & 57 \\cr \\hline 172 & 20 & 1 & 76 \\cr \\hline 175 & 19 & 1 & 80 \\cr \\hline \\end{array} We wish to fit a linear function of the form w=c_0+c_1 h+c_2 a+c_3 g which predicts the weight from the rest of the data. Find the best approximation of this function, using least squares.\n$w=w(h,a,g)=$ [ANS]", "answer_v2": [ "1.4*h-211+1.8*a+11*g" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The table below lists the height $h$ (in cm), the age $a$ (in years), the gender $g$ (0=\"Female\", 1=\"Male\"), and the weight $w$ (in kg) of some college students. \\begin{array}{|r|r|r|r|} \\hline \\mbox{Height} & \\mbox{Age} & \\mbox{Gender} & \\mbox{Weight} \\cr \\hline 179 & 21 & 1 & 83 \\cr \\hline 169 & 19 & 0 & 61 \\cr \\hline 161 & 22 & 0 & 55 \\cr \\hline 180 & 20 & 1 & 84 \\cr \\hline 171 & 18 & 1 & 75 \\cr \\hline \\end{array} We wish to fit a linear function of the form w=c_0+c_1 h+c_2 a+c_3 g which predicts the weight from the rest of the data. Find the best approximation of this function, using least squares.\n$w=w(h,a,g)=$ [ANS]", "answer_v3": [ "0.893665*h-96.5602+0.346606*a+12.5294*g" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0437", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Fit a linear function of the form $f(t)=c_0+c_1 t$ to the data points $(-7, 33)$, $(0,2)$, $(7,-23)$, using least squares.\n$f(t)=$ [ANS]", "answer_v1": [ "4-4*t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Fit a linear function of the form $f(t)=c_0+c_1 t$ to the data points $(-1,-2)$, $(0,-11)$, $(1,-8)$, using least squares.\n$f(t)=$ [ANS]", "answer_v2": [ "-(7+3*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Fit a linear function of the form $f(t)=c_0+c_1 t$ to the data points $(-3,-6)$, $(0,-6)$, $(3, 0)$, using least squares.\n$f(t)=$ [ANS]", "answer_v3": [ "t-4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0438", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Among all unit vectors $\\vec{u}=\\left\\lbrack\\begin{array}{c} x \\\\ y \\\\ z\\end{array} \\right\\rbrack$ in ${\\mathbb R}^3$, find the one for which the sum $x+8 y+6 z$ is minimal.\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "-0.0995037", "-0.79603", "-0.597022" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Among all unit vectors $\\vec{u}=\\left\\lbrack\\begin{array}{c} x \\\\ y \\\\ z\\end{array} \\right\\rbrack$ in ${\\mathbb R}^3$, find the one for which the sum $x+2 y+9 z$ is minimal.\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-0.107833", "-0.215666", "-0.970495" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Among all unit vectors $\\vec{u}=\\left\\lbrack\\begin{array}{c} x \\\\ y \\\\ z\\end{array} \\right\\rbrack$ in ${\\mathbb R}^3$, find the one for which the sum $x+4 y+6 z$ is minimal.\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-0.137361", "-0.549442", "-0.824163" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0439", "subject": "Linear_algebra", "topic": "Inner products", "subtopic": "Applications", "level": "3", "keywords": [ "algebra" ], "problem_v1": "By using the method of least squares, find the best line through the points: $(1,-1)$, $(0, 0)$, $(-1, 1)$.\nStep 1. The general equation of a line is $c_0+c_1x=y$. Plugging the data points into this formula gives a matrix equation $\\mathrm{A}\\mathbf{c}=\\mathbf{y}$.\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $\\left[\\begin{matrix}c_0\\\\c_1\\end{matrix}\\right]$ $=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nStep 2. The matrix equation $\\mathrm{A}\\mathbf{c}=\\mathbf{y}$ has no solution, so instead we use the normal equation $\\mathrm{A^T} \\mathrm{A}\\, \\hat{\\mathbf{c}}=\\mathrm{A^T} \\mathbf{y}$ $\\mathrm{A^T} \\mathrm{A}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$\\mathrm{A^T} \\mathbf{y}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\nStep 3. Solving the normal equation gives the answer $\\hat{\\mathbf{c}}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} which corresponds to the formula $y=$ [ANS]\nAnalysis. Compute the predicted $y$ values: $\\hat{\\mathbf{y}}=\\mathrm{A}\\hat{\\mathbf{c}}$. $\\hat{\\mathbf{y}}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nCompute the error vector: $\\mathbf{e}=\\mathbf{y}-\\hat{\\mathbf{y}}$. $\\mathbf{e}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} Compute the total error: $\\mathrm{SSE}=e_1^2+e_2^2+e_3^2$. $\\mathrm{SSE}=$ [ANS]", "answer_v1": [ "1", "1", "1", "0", "1", "-1", "-1", "0", "1", "3", "0", "0", "2", "0", "-2", "0", "-1", "-1", "0", "1", "0", "0", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "By using the method of least squares, find the best line through the points: $(-2,-1)$, $(2,-2)$, $(-1,-1)$.\nStep 1. The general equation of a line is $c_0+c_1x=y$. Plugging the data points into this formula gives a matrix equation $\\mathrm{A}\\mathbf{c}=\\mathbf{y}$.\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $\\left[\\begin{matrix}c_0\\\\c_1\\end{matrix}\\right]$ $=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nStep 2. The matrix equation $\\mathrm{A}\\mathbf{c}=\\mathbf{y}$ has no solution, so instead we use the normal equation $\\mathrm{A^T} \\mathrm{A}\\, \\hat{\\mathbf{c}}=\\mathrm{A^T} \\mathbf{y}$ $\\mathrm{A^T} \\mathrm{A}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$\\mathrm{A^T} \\mathbf{y}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\nStep 3. Solving the normal equation gives the answer $\\hat{\\mathbf{c}}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} which corresponds to the formula $y=$ [ANS]\nAnalysis. Compute the predicted $y$ values: $\\hat{\\mathbf{y}}=\\mathrm{A}\\hat{\\mathbf{c}}$. $\\hat{\\mathbf{y}}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nCompute the error vector: $\\mathbf{e}=\\mathbf{y}-\\hat{\\mathbf{y}}$. $\\mathbf{e}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} Compute the total error: $\\mathrm{SSE}=e_1^2+e_2^2+e_3^2$. $\\mathrm{SSE}=$ [ANS]", "answer_v2": [ "1", "-2", "1", "2", "1", "-1", "-1", "-2", "-1", "3", "-1", "-1", "9", "-4", "-1", "-1.42308", "-0.269231", "-0.884615", "-1.96154", "-1.15385", "-0.115385", "-0.0384615", "0.153846" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "By using the method of least squares, find the best line through the points: $(-1,-1)$, $(1, 2)$, $(-2, 3)$.\nStep 1. The general equation of a line is $c_0+c_1x=y$. Plugging the data points into this formula gives a matrix equation $\\mathrm{A}\\mathbf{c}=\\mathbf{y}$.\n\\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $\\left[\\begin{matrix}c_0\\\\c_1\\end{matrix}\\right]$ $=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nStep 2. The matrix equation $\\mathrm{A}\\mathbf{c}=\\mathbf{y}$ has no solution, so instead we use the normal equation $\\mathrm{A^T} \\mathrm{A}\\, \\hat{\\mathbf{c}}=\\mathrm{A^T} \\mathbf{y}$ $\\mathrm{A^T} \\mathrm{A}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$\\mathrm{A^T} \\mathbf{y}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}\nStep 3. Solving the normal equation gives the answer $\\hat{\\mathbf{c}}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array} which corresponds to the formula $y=$ [ANS]\nAnalysis. Compute the predicted $y$ values: $\\hat{\\mathbf{y}}=\\mathrm{A}\\hat{\\mathbf{c}}$. $\\hat{\\mathbf{y}}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\nCompute the error vector: $\\mathbf{e}=\\mathbf{y}-\\hat{\\mathbf{y}}$. $\\mathbf{e}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array} Compute the total error: $\\mathrm{SSE}=e_1^2+e_2^2+e_3^2$. $\\mathrm{SSE}=$ [ANS]", "answer_v3": [ "1", "-1", "1", "1", "1", "-2", "-1", "2", "3", "3", "-2", "-2", "6", "4", "-3", "1.28571", "-0.0714286", "1.35714", "1.21429", "1.42857", "-2.35714", "0.785714", "1.57143" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0440", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Properties", "level": "4", "keywords": [], "problem_v1": "Let \\vec{v}_1=\\left[\\begin{array}{c}-1\\cr-2 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{v}_2=\\left[\\begin{array}{c}-1\\cr-1 \\end{array}\\right]. Let $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be the linear transformation satisfying T(\\vec{v}_1)=\\left[\\begin{array}{c}-9\\cr-10 \\end{array}\\right] \\ \\mbox{and} \\ T(\\vec{v}_2)=\\left[\\begin{array}{c}-7\\cr-6 \\end{array}\\right]. Find the image of an arbitrary vector $\\left[\\begin{array}{c} x\\cr y\\cr \\end{array}\\right].$\n$T \\left(\\left[\\begin{array}{c} x\\cr y\\cr \\end{array}\\right]\\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "5*x+2*y", "2*x+4*y" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let \\vec{v}_1=\\left[\\begin{array}{c} 3\\cr 2 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{v}_2=\\left[\\begin{array}{c}-1\\cr-1 \\end{array}\\right]. Let $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be the linear transformation satisfying T(\\vec{v}_1)=\\left[\\begin{array}{c}-8\\cr-27 \\end{array}\\right] \\ \\mbox{and} \\ T(\\vec{v}_2)=\\left[\\begin{array}{c} 0\\cr 10 \\end{array}\\right]. Find the image of an arbitrary vector $\\left[\\begin{array}{c} x\\cr y\\cr \\end{array}\\right].$\n$T \\left(\\left[\\begin{array}{c} x\\cr y\\cr \\end{array}\\right]\\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "-8*x+8*y", "-7*x+(-3)*y" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let \\vec{v}_1=\\left[\\begin{array}{c}-3\\cr 1 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{v}_2=\\left[\\begin{array}{c} 2\\cr-1 \\end{array}\\right]. Let $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be the linear transformation satisfying T(\\vec{v}_1)=\\left[\\begin{array}{c} 14\\cr 13 \\end{array}\\right] \\ \\mbox{and} \\ T(\\vec{v}_2)=\\left[\\begin{array}{c}-10\\cr-9 \\end{array}\\right]. Find the image of an arbitrary vector $\\left[\\begin{array}{c} x\\cr y\\cr \\end{array}\\right].$\n$T \\left(\\left[\\begin{array}{c} x\\cr y\\cr \\end{array}\\right]\\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-4*x+2*y", "-4*x+1*y" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0441", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Properties", "level": "3", "keywords": [], "problem_v1": "If $T: {\\mathbb R}^3 \\rightarrow {\\mathbb R}^3$ is a linear transformation such that T\\left(\\left[\\begin{array}{c} 1\\cr 0\\cr 0 \\end{array}\\right] \\right)=\\left[\\begin{array}{c} 2\\cr-2\\cr-1 \\end{array}\\right], \\ \\ \\ T\\left(\\left[\\begin{array}{c} 0\\cr 1\\cr 0 \\end{array}\\right] \\right)=\\left[\\begin{array}{c} 1\\cr-2\\cr 0 \\end{array}\\right], \\ \\ \\ T\\left(\\left[\\begin{array}{c} 0\\cr 0\\cr 1 \\end{array}\\right] \\right)=\\left[\\begin{array}{c} 1\\cr 1\\cr 1 \\end{array}\\right], then $\\ T \\left(\\left[\\begin{array}{c} 2\\cr 1\\cr-3 \\end{array}\\right]\\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "2", "-9", "-5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If $T: {\\mathbb R}^3 \\rightarrow {\\mathbb R}^3$ is a linear transformation such that T\\left(\\left[\\begin{array}{c} 1\\cr 0\\cr 0 \\end{array}\\right] \\right)=\\left[\\begin{array}{c}-4\\cr 4\\cr 1 \\end{array}\\right], \\ \\ \\ T\\left(\\left[\\begin{array}{c} 0\\cr 1\\cr 0 \\end{array}\\right] \\right)=\\left[\\begin{array}{c} 4\\cr-2\\cr-4 \\end{array}\\right], \\ \\ \\ T\\left(\\left[\\begin{array}{c} 0\\cr 0\\cr 1 \\end{array}\\right] \\right)=\\left[\\begin{array}{c}-3\\cr-3\\cr 1 \\end{array}\\right], then $\\ T \\left(\\left[\\begin{array}{c}-2\\cr-2\\cr-1 \\end{array}\\right]\\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "3", "-1", "5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If $T: {\\mathbb R}^3 \\rightarrow {\\mathbb R}^3$ is a linear transformation such that T\\left(\\left[\\begin{array}{c} 1\\cr 0\\cr 0 \\end{array}\\right] \\right)=\\left[\\begin{array}{c}-2\\cr-3\\cr 3 \\end{array}\\right], \\ \\ \\ T\\left(\\left[\\begin{array}{c} 0\\cr 1\\cr 0 \\end{array}\\right] \\right)=\\left[\\begin{array}{c} 1\\cr-1\\cr-3 \\end{array}\\right], \\ \\ \\ T\\left(\\left[\\begin{array}{c} 0\\cr 0\\cr 1 \\end{array}\\right] \\right)=\\left[\\begin{array}{c}-2\\cr 3\\cr-2 \\end{array}\\right], then $\\ T \\left(\\left[\\begin{array}{c} 1\\cr 5\\cr-3 \\end{array}\\right]\\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "9", "-17", "-6" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0442", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Properties", "level": "2", "keywords": [ "linear transformation" ], "problem_v1": "Which of the following transformations are linear? [ANS] A\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 8x_1+x_2 \\cr y_2 &=&-x_1 \\end{array} \\right.$ B\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 4x_1-5x_2+10x_3 \\cr y_2 &=& 2x_2-8x_3 \\cr y_3 &=&-7x_1-6x_2 \\end{array} \\right.$ C\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 8 \\cr y_2 &=& 6 \\cr y_3 &=& 7 \\cr \\end{array} \\right.$ D\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& x_1+2 \\cr y_2 &=& x_2 \\end{array} \\right.$ E\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 7x_2 \\cr y_2 &=&-9x_3 \\cr y_3 &=&-3x_1 \\end{array} \\right.$ F\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 0 \\cr y_2 &=& x_1x_2 \\cr \\end{array} \\right.$", "answer_v1": [ "ABE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Which of the following transformations are linear? [ANS] A\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 0 \\cr y_2 &=& x_1x_2 \\cr \\end{array} \\right.$ B\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& x_2^2 \\cr y_2 &=& x_3 \\cr y_3 &=& x_1 \\cr \\end{array} \\right.$ C\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 9x_1 \\cr y_2 &=& 5 \\cr \\end{array} \\right.$ D\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 5x_1-3x_2+7x_3 \\cr y_2 &=& 8x_2-2x_3 \\cr y_3 &=&-4x_1-10x_2 \\end{array} \\right.$ E\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 4x_2 \\cr y_2 &=&-6x_3 \\cr y_3 &=&-9x_1 \\end{array} \\right.$ F\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 2x_1+x_2 \\cr y_2 &=&-x_1 \\end{array} \\right.$", "answer_v2": [ "DEF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of the following transformations are linear? [ANS] A\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 3x_2 \\cr y_2 &=&-8x_3 \\cr y_3 &=&-5x_1 \\end{array} \\right.$ B\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 5x_1 \\cr y_2 &=& 6 \\cr \\end{array} \\right.$ C\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 4 \\cr y_2 &=& 7 \\cr y_3 &=& 3 \\cr \\end{array} \\right.$ D\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& x_2^2 \\cr y_2 &=& x_3 \\cr y_3 &=& x_1 \\cr \\end{array} \\right.$ E\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 0 \\cr y_2 &=& 7x_2 \\end{array} \\right.$ F\\. $\\left\\{\\begin{array}{r@{}r@{}l} y_1 &=& 4x_1+x_2 \\cr y_2 &=&-x_1 \\end{array} \\right.$", "answer_v3": [ "AEF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0443", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Properties", "level": "2", "keywords": [], "problem_v1": "Let $T: P_3 \\rightarrow P_3$ be the linear transformation satisfying T(1)=4x^2+2, \\ \\ \\ T(x)=3x-4, \\ \\ \\ T(x^2)=3x^2+x-2. Find the image of an arbitrary quadratic polynomial $ax^2+bx+c$.\n$T(ax^2+bx+c)=$ [ANS].", "answer_v1": [ "a*(3*x^2+x-2)+b*(3*x-4)+c*(4*x*x+2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $T: P_3 \\rightarrow P_3$ be the linear transformation satisfying T(1)=2x^2+8, \\ \\ \\ T(x)=-2x+8, \\ \\ \\ T(x^2)=-3x^2-x+1. Find the image of an arbitrary quadratic polynomial $ax^2+bx+c$.\n$T(ax^2+bx+c)=$ [ANS].", "answer_v2": [ "a*[1-(3*x^2+x)]+b*(8-2*x)+c*(2*x*x+8)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $T: P_3 \\rightarrow P_3$ be the linear transformation satisfying T(1)=2x^2+2, \\ \\ \\ T(x)=2x-6, \\ \\ \\ T(x^2)=3x^2+x+7. Find the image of an arbitrary quadratic polynomial $ax^2+bx+c$.\n$T(ax^2+bx+c)=$ [ANS].", "answer_v3": [ "a*(3*x^2+x+7)+b*(2*x-6)+c*(2*x*x+2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0444", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Properties", "level": "3", "keywords": [], "problem_v1": "If $T: P_1 \\rightarrow P_1$ is a linear transformation such that $T(1+4x)=1-4x \\ $ and $\\ T(5+19x)=-2+3x, \\ $ then\n$T(-1-3x)=$ [ANS].", "answer_v1": [ "6+-19*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $T: P_1 \\rightarrow P_1$ is a linear transformation such that $T(1+5x)=-3+3x \\ $ and $\\ T(2+9x)=-2-2x, \\ $ then\n$T(1-2x)=$ [ANS].", "answer_v2": [ "25+-53*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If $T: P_1 \\rightarrow P_1$ is a linear transformation such that $T(1+4x)=-2-3x \\ $ and $\\ T(3+11x)=-1+4x, \\ $ then\n$T(3-2x)=$ [ANS].", "answer_v3": [ "64+173*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0445", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Properties", "level": "2", "keywords": [], "problem_v1": "Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be sure you can justify your answers. [ANS] A\\. $ T(f(t))=f'(t)+3f(t)$ from $C^{\\infty}$ to $C^{\\infty}$ B\\. $T(x+iy)=8x-iy$ from ${\\mathbb C}$ to ${\\mathbb C}$ C\\. $T(x_0, x_1, x_2, \\ldots)=(1, x_0,x_1, x_2, \\ldots)$ from the space of infinite sequences into itself D\\. $ T(f(t))=f'(t)+5f(t)+7$ from $C^{\\infty}$ to $C^{\\infty}$ E\\. $ T(f(t))=f(-t)$ from $P_{2}$ to $P_{2}$ F\\. $ T(f(t))=f(4)$ from $P_{9}$ to ${\\mathbb R}$", "answer_v1": [ "ABEF" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be sure you can justify your answers. [ANS] A\\. $ T(f(t))=f'(t)+8f(t)$ from $C^{\\infty}$ to $C^{\\infty}$ B\\. $ T(f(t))=f(7)$ from $P_{6}$ to ${\\mathbb R}$ C\\. $ T(f(t))=f'(t)+3f(t)+5$ from $C^{\\infty}$ to $C^{\\infty}$ D\\. $T(x+iy)=2x-iy$ from ${\\mathbb C}$ to ${\\mathbb C}$ E\\. $ T(f(t))=f(-t)$ from $P_{4}$ to $P_{4}$ F\\. $ T(f(t))=f(t)f'(t)$ from $P_{9}$ to $\\ P_{17}$", "answer_v2": [ "ABDE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be sure you can justify your answers. [ANS] A\\. $T(x+iy)=4x-iy$ from ${\\mathbb C}$ to ${\\mathbb C}$ B\\. $T(x_0, x_1, x_2, \\ldots)=(1, x_0,x_1, x_2, \\ldots)$ from the space of infinite sequences into itself C\\. $ T(f(t))=(f(t))^3+2(f(t))^2+8f(t)$ from $C^{\\infty}$ to $C^{\\infty}$ D\\. $ T(f(t))=\\int_{-7}^{3} f(t)dt$ from $P_{6}$ to ${\\mathbb R}$ E\\. $ T(f(t))=t^{4}f'(t)$ from $P_2$ to $P_{5}$ F\\. $ T(f(t))=f'(t)+2f(t)$ from $C^{\\infty}$ to $C^{\\infty}$", "answer_v3": [ "ADEF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0446", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Properties", "level": "3", "keywords": [], "problem_v1": "Let $V$ be a vector space, and $T:V \\rightarrow V$ a linear transformation such that $T(5 \\vec{v}_1+3 \\vec{v}_2)=4 \\vec{v}_1-3 \\vec{v}_2$ and $T(3 \\vec{v}_1+2 \\vec{v}_2)=4 \\vec{v}_1-3 \\vec{v}_2$. Then\n$T(\\vec{v}_1)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$,\n$T(\\vec{v}_2)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$,\n$T(2 \\vec{v}_1-2 \\vec{v}_2)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$.", "answer_v1": [ "-4", "3", "8", "-6", "-24", "18" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $V$ be a vector space, and $T:V \\rightarrow V$ a linear transformation such that $T(2 \\vec{v}_1+3 \\vec{v}_2)=-2 \\vec{v}_1-5 \\vec{v}_2$ and $T(3 \\vec{v}_1+5 \\vec{v}_2)=-2 \\vec{v}_1-4 \\vec{v}_2$. Then\n$T(\\vec{v}_1)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$,\n$T(\\vec{v}_2)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$,\n$T(2 \\vec{v}_1+4 \\vec{v}_2)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$.", "answer_v2": [ "-4", "-13", "2", "7", "0", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $V$ be a vector space, and $T:V \\rightarrow V$ a linear transformation such that $T(2 \\vec{v}_1+3 \\vec{v}_2)=3 \\vec{v}_1-2 \\vec{v}_2$ and $T(3 \\vec{v}_1+5 \\vec{v}_2)=5 \\vec{v}_1-5 \\vec{v}_2$. Then\n$T(\\vec{v}_1)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$,\n$T(\\vec{v}_2)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$,\n$T(-2 \\vec{v}_1+2 \\vec{v}_2)=$ [ANS] $\\vec{v}_1+$ [ANS] $\\vec{v}_2$.", "answer_v3": [ "0", "5", "1", "-4", "2", "-18" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0448", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Properties", "level": "3", "keywords": [ "vector space", "linear transformation' 'matrix' 'image" ], "problem_v1": "Let $\\vec{e}_1=(1,0)$, $\\vec{e}_2=(0,1)$, $\\vec{x}_1=(5, 2)$ and $\\vec{x}_2=(2, 4)$.\nLet $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation that sends $\\vec{e}_1$ to $\\vec{x}_1$ and $\\vec{e}_2$ to $\\vec{x}_2$. If $T$ maps $(-4,-3)$ to the vector $\\vec{y}$, then\n$\\vec{y}=$ [ANS]. (Enter your answer as an ordered pair, such as (1,2), including the parentheses.)", "answer_v1": [ "(-26,-20)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Let $\\vec{e}_1=(1,0)$, $\\vec{e}_2=(0,1)$, $\\vec{x}_1=(-8, 8)$ and $\\vec{x}_2=(-7,-3)$.\nLet $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation that sends $\\vec{e}_1$ to $\\vec{x}_1$ and $\\vec{e}_2$ to $\\vec{x}_2$. If $T$ maps $(8,-3)$ to the vector $\\vec{y}$, then\n$\\vec{y}=$ [ANS]. (Enter your answer as an ordered pair, such as (1,2), including the parentheses.)", "answer_v2": [ "(-43,73)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Let $\\vec{e}_1=(1,0)$, $\\vec{e}_2=(0,1)$, $\\vec{x}_1=(-4, 2)$ and $\\vec{x}_2=(-4, 1)$.\nLet $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation that sends $\\vec{e}_1$ to $\\vec{x}_1$ and $\\vec{e}_2$ to $\\vec{x}_2$. If $T$ maps $(-6,-3)$ to the vector $\\vec{y}$, then\n$\\vec{y}=$ [ANS]. (Enter your answer as an ordered pair, such as (1,2), including the parentheses.)", "answer_v3": [ "(36,-15)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0449", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Evaluating linear transformations", "level": "3", "keywords": [], "problem_v1": "Let $V$ be a vector space, $v, u \\in V$, and let $T_1: V \\rightarrow V$ and $T_2: V \\rightarrow V$ be linear transformations such that T_1(v)=6 v+5 u, \\ \\ \\ T_1(u)=-4 v-3 u, T_2(v)=3 v-5 u, \\ \\ \\ T_2(u)=-6 v-5 u. Find the images of $v$ and $u$ under the composite of $T_1$ and $T_2$.\n$(T_2 T_1)(v)=$ [ANS], $(T_2 T_1)(u)=$ [ANS].", "answer_v1": [ "-12*v+-55*u", "6*v+35*u" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $V$ be a vector space, $v, u \\in V$, and let $T_1: V \\rightarrow V$ and $T_2: V \\rightarrow V$ be linear transformations such that T_1(v)=2 v-2 u, \\ \\ \\ T_1(u)=-2 v-7 u, T_2(v)=-2 v-5 u, \\ \\ \\ T_2(u)=-6 v-6 u. Find the images of $v$ and $u$ under the composite of $T_1$ and $T_2$.\n$(T_2 T_1)(v)=$ [ANS], $(T_2 T_1)(u)=$ [ANS].", "answer_v2": [ "8*v+2*u", "46*v+52*u" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $V$ be a vector space, $v, u \\in V$, and let $T_1: V \\rightarrow V$ and $T_2: V \\rightarrow V$ be linear transformations such that T_1(v)=3 v+3 u, \\ \\ \\ T_1(u)=-4 v-3 u, T_2(v)=4 v-3 u, \\ \\ \\ T_2(u)=-7 v+2 u. Find the images of $v$ and $u$ under the composite of $T_1$ and $T_2$.\n$(T_2 T_1)(v)=$ [ANS], $(T_2 T_1)(u)=$ [ANS].", "answer_v3": [ "-9*v+-3*u", "5*v+6*u" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0450", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Evaluating linear transformations", "level": "3", "keywords": [ "vector space", "linear transformation", "image' 'matrix" ], "problem_v1": "Let A=\\left[\\begin{array}{cc} 5 &2\\cr 2 &4\\cr-4 &-3 \\end{array}\\right]. Define the linear transformation $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^3$ by $T(\\vec{x})=A\\vec{x}$. Find the images of $\\vec{u}=\\left[\\begin{array}{c} 1\\cr 1 \\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{c} a\\cr b\\cr \\end{array}\\right]$ under $T$.\n$T(\\vec{u})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$T(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v1": [ "7", "6", "-7", "5*a+2*b", "2*a+4*b", "-(4*a+3*b)" ], "answer_type_v1": [ "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{cc}-8 &8\\cr-7 &-3\\cr 8 &-3 \\end{array}\\right]. Define the linear transformation $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^3$ by $T(\\vec{x})=A\\vec{x}$. Find the images of $\\vec{u}=\\left[\\begin{array}{c}-3\\cr-2 \\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{c} a\\cr b\\cr \\end{array}\\right]$ under $T$.\n$T(\\vec{u})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$T(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v2": [ "8", "27", "-18", "8*b-8*a", "-(7*a+3*b)", "8*a-3*b" ], "answer_type_v2": [ "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{cc}-4 &2\\cr-4 &1\\cr-6 &-3 \\end{array}\\right]. Define the linear transformation $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^3$ by $T(\\vec{x})=A\\vec{x}$. Find the images of $\\vec{u}=\\left[\\begin{array}{c} 3\\cr 5 \\end{array}\\right]$ and $\\vec{v}=\\left[\\begin{array}{c} a\\cr b\\cr \\end{array}\\right]$ under $T$.\n$T(\\vec{u})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}\n$T(\\vec{v})=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}", "answer_v3": [ "-2", "-7", "-33", "2*b-4*a", "b-4*a", "-(6*a+3*b)" ], "answer_type_v3": [ "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0451", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "2", "keywords": [], "problem_v1": "Find the matrix $M$ of the linear transformation $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ given by T\\left(\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]\\right)=\\left[\\begin{array}{c} 8x_{1}+6x_{2}\\cr-4x_{1}+x_{2}\\cr \\end{array}\\right]. $M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "8", "6", "-4", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the matrix $M$ of the linear transformation $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ given by T\\left(\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]\\right)=\\left[\\begin{array}{c} 2x_{1}+\\left(-3\\right)x_{2}\\cr-9x_{1}-x_{2}\\cr \\end{array}\\right]. $M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "2", "-3", "-9", "-1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the matrix $M$ of the linear transformation $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ given by T\\left(\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right]\\right)=\\left[\\begin{array}{c} 4x_{1}+4x_{2}\\cr-3x_{1}+x_{2}\\cr \\end{array}\\right]. $M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "4", "4", "-3", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0452", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [], "problem_v1": "Let A=\\left[\\begin{array}{cc} 7 &5\\cr 4 &8 \\end{array}\\right] \\ \\mbox{and} B=\\left[\\begin{array}{cc} 1 &2\\cr 6 &3 \\end{array}\\right]. Find the matrix $C$ of the linear transformation $\\ T(x)=B(A(x)).$\n$C=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "15", "21", "54", "54" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{cc} 0 &9\\cr 2 &4 \\end{array}\\right] \\ \\mbox{and} B=\\left[\\begin{array}{cc} 8 &3\\cr 1 &5 \\end{array}\\right]. Find the matrix $C$ of the linear transformation $\\ T(x)=B(A(x)).$\n$C=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "6", "84", "10", "29" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{cc} 3 &6\\cr 2 &5 \\end{array}\\right] \\ \\mbox{and} B=\\left[\\begin{array}{cc} 1 &4\\cr 9 &8 \\end{array}\\right]. Find the matrix $C$ of the linear transformation $\\ T(x)=B(A(x)).$\n$C=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "11", "26", "43", "94" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0453", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [], "problem_v1": "Find the matrix $A$ of the linear transformation $T$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ that rotates any vector through an angle of $120 ^\\circ$ in the clockwise direction.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "-0.5", "0.866025", "-0.866025", "-0.5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the matrix $A$ of the linear transformation $T$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ that rotates any vector through an angle of $150 ^\\circ$ in the counterclockwise direction.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-0.866025", "-0.5", "0.5", "-0.866025" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the matrix $A$ of the linear transformation $T$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ that rotates any vector through an angle of $120 ^\\circ$ in the counterclockwise direction.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-0.5", "-0.866025", "0.866025", "-0.5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0454", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [], "problem_v1": "Find the matrices of the following linear transformations from ${\\mathbb R}^3$ to ${\\mathbb R}^3$.\nThe orthogonal projection onto the $y$-axis:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nThe reflection in the $xz$-plane:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "0", "0", "0", "0", "1", "0", "0", "0", "0", "1", "0", "0", "0", "-1", "0", "0", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the matrices of the following linear transformations from ${\\mathbb R}^3$ to ${\\mathbb R}^3$.\nThe orthogonal projection onto the $xy$-plane:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nThe reflection in the $z$-axis:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "0", "0", "1", "0", "0", "0", "0", "-1", "0", "0", "0", "-1", "0", "0", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the matrices of the following linear transformations from ${\\mathbb R}^3$ to ${\\mathbb R}^3$.\nThe orthogonal projection onto the $yz$-plane:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\nThe reflection in the $x$-axis:\n\\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "0", "0", "0", "0", "1", "0", "0", "0", "1", "1", "0", "0", "0", "-1", "0", "0", "0", "-1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0455", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "2", "keywords": [], "problem_v1": "Find the matrix $A$ of the linear transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^3$ given by T \\left(\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right] \\right)=\\left[\\begin{array}{c} 5\\cr 2\\cr 2 \\end{array}\\right] x_1+\\left[\\begin{array}{c} 4\\cr-4\\cr-3 \\end{array}\\right] x_2.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "5", "4", "2", "-4", "2", "-3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find the matrix $A$ of the linear transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^3$ given by T \\left(\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right] \\right)=\\left[\\begin{array}{c}-8\\cr 8\\cr-7 \\end{array}\\right] x_1+\\left[\\begin{array}{c}-3\\cr 8\\cr-3 \\end{array}\\right] x_2.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-8", "-3", "8", "8", "-7", "-3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find the matrix $A$ of the linear transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^3$ given by T \\left(\\left[\\begin{array}{c} x_{1}\\cr x_{2}\\cr \\end{array}\\right] \\right)=\\left[\\begin{array}{c}-4\\cr 2\\cr-4 \\end{array}\\right] x_1+\\left[\\begin{array}{c} 1\\cr-6\\cr-3 \\end{array}\\right] x_2.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-4", "1", "2", "-6", "-4", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0456", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [], "problem_v1": "Find the matrix $A$ of the orthogonal projection onto the line $L$ in ${\\mathbb R} ^2$ that consists of all scalar multiples of the vector $\\left[\\begin{array}{c} 5\\cr 3 \\end{array}\\right]$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "0.735294", "0.441176", "0.441176", "0.264706" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the matrix $A$ of the orthogonal projection onto the line $L$ in ${\\mathbb R} ^2$ that consists of all scalar multiples of the vector $\\left[\\begin{array}{c} 1\\cr 5 \\end{array}\\right]$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "0.0384615", "0.192308", "0.192308", "0.961538" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the matrix $A$ of the orthogonal projection onto the line $L$ in ${\\mathbb R} ^2$ that consists of all scalar multiples of the vector $\\left[\\begin{array}{c} 2\\cr 3 \\end{array}\\right]$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "0.307692", "0.461538", "0.461538", "0.692308" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0457", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Let $\\vec{b}_1=\\left[\\begin{array}{c}-1\\cr-1 \\end{array}\\right]$ and $\\vec{b}_2=\\left[\\begin{array}{c}-1\\cr-2 \\end{array}\\right].$ The set $B=\\left\\lbrace \\vec{b}_1, \\vec{b}_2 \\right\\rbrace$ is a basis for ${\\mathbb R}^2.$ Let $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation such that $T(\\vec{b}_1)=7 \\vec{b}_1+6 \\vec{b}_2$ and $T(\\vec{b}_2)=6 \\vec{b}_1+7 \\vec{b}_2.$\n(a) The matrix of $T$ relative to the basis $B$ is $[T]_B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n(b) The matrix of $T$ relative to the standard basis $E$ for ${\\mathbb R}^2$ is $[T]_E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "7", "6", "6", "7", "13", "0", "18", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $\\vec{b}_1=\\left[\\begin{array}{c} 1\\cr-1 \\end{array}\\right]$ and $\\vec{b}_2=\\left[\\begin{array}{c}-2\\cr 1 \\end{array}\\right].$ The set $B=\\left\\lbrace \\vec{b}_1, \\vec{b}_2 \\right\\rbrace$ is a basis for ${\\mathbb R}^2.$ Let $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation such that $T(\\vec{b}_1)=2 \\vec{b}_1+3 \\vec{b}_2$ and $T(\\vec{b}_2)=8 \\vec{b}_1+4 \\vec{b}_2.$\n(a) The matrix of $T$ relative to the basis $B$ is $[T]_B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n(b) The matrix of $T$ relative to the standard basis $E$ for ${\\mathbb R}^2$ is $[T]_E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "2", "8", "3", "4", "4", "8", "3", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $\\vec{b}_1=\\left[\\begin{array}{c}-1\\cr-1 \\end{array}\\right]$ and $\\vec{b}_2=\\left[\\begin{array}{c}-2\\cr-1 \\end{array}\\right].$ The set $B=\\left\\lbrace \\vec{b}_1, \\vec{b}_2 \\right\\rbrace$ is a basis for ${\\mathbb R}^2.$ Let $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation such that $T(\\vec{b}_1)=4 \\vec{b}_1+3 \\vec{b}_2$ and $T(\\vec{b}_2)=6 \\vec{b}_1+5 \\vec{b}_2.$\n(a) The matrix of $T$ relative to the basis $B$ is $[T]_B=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.\n(b) The matrix of $T$ relative to the standard basis $E$ for ${\\mathbb R}^2$ is $[T]_E=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "4", "6", "3", "5", "6", "4", "4", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0458", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [], "problem_v1": "Match each linear transformation with its matrix. $\\begin{array}{cccc}\\hline & [ANS]EFF 1. \\left[\\begin{array}{cc} 1 &0\\cr 0 &-1 \\end{array}\\right] [ANS]EFF 2. \\left[\\begin{array}{cc}-1 &0\\cr 0 &1 \\end{array}\\right] [ANS]EFF 3. \\left[\\begin{array}{cc} 0 &1\\cr 1 &0 \\end{array}\\right] [ANS]EFF 4. \\left[\\begin{array}{cc} 1 &0\\cr 0 &1 \\end{array}\\right] [ANS]EFF 5. \\left[\\begin{array}{cc} 0 &1\\cr-1 &0 \\end{array}\\right] [ANS]EFF 6. \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right] & & A.Rotation through an angle of 90^\\circin the clockwise direction B.Identity transformation C.Projection onto the y-axis D.Reflection in the y-axis E.Reflection in the line y=x F.Reflection in the x-axis \\\\ \\hline \\end{array}$", "answer_v1": [ "F", "D", "E", "B", "A", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Match each linear transformation with its matrix. $\\begin{array}{cccc}\\hline & [ANS]EFF 1. \\left[\\begin{array}{cc} 2 &0\\cr 0 &2 \\end{array}\\right] [ANS]EFF 2. \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right] [ANS]EFF 3. \\left[\\begin{array}{cc} 0 &1\\cr 1 &0 \\end{array}\\right] [ANS]EFF 4. \\left[\\begin{array}{cc} 0 &1\\cr-1 &0 \\end{array}\\right] [ANS]EFF 5. \\left[\\begin{array}{cc} 0 &-1\\cr 1 &0 \\end{array}\\right] [ANS]EFF 6. \\left[\\begin{array}{cc} 1 &0\\cr 0 &-1 \\end{array}\\right] & & A.Rotation through an angle of 90^\\circin the counterclockwise direction B.Reflection in the x-axis C.Reflection in the line y=x D.Projection onto the y-axis E.Rotation through an angle of 90^\\circin the clockwise direction F.Dilation by a factor of 2 \\\\ \\hline \\end{array}$", "answer_v2": [ "F", "D", "C", "E", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Match each linear transformation with its matrix. $\\begin{array}{cccc}\\hline & [ANS]EFF 1. \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right] [ANS]EFF 2. \\left[\\begin{array}{cc} 1 &0\\cr 0 &1 \\end{array}\\right] [ANS]EFF 3. \\left[\\begin{array}{cc} 0.5 &0\\cr 0 &0.5 \\end{array}\\right] [ANS]EFF 4. \\left[\\begin{array}{cc}-1 &0\\cr 0 &1 \\end{array}\\right] [ANS]EFF 5. \\left[\\begin{array}{cc}-1 &0\\cr 0 &-1 \\end{array}\\right] [ANS]EFF 6. \\left[\\begin{array}{cc} 0 &1\\cr-1 &0 \\end{array}\\right] & & A.Rotation through an angle of 90^\\circin the clockwise direction B.Contraction by a factor of 2 C.Identity transformation D.Projection onto the y-axis E.Reflection in the origin F.Reflection in the y-axis \\\\ \\hline \\end{array}$", "answer_v3": [ "D", "C", "B", "F", "E", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0459", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [], "problem_v1": "Find a $3\\times 3$ matrix $A$ such that $A\\vec{x}=9 \\vec{x}$ for all $\\vec{x}$ in ${\\mathbb R}^3$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "9", "0", "0", "0", "9", "0", "0", "0", "9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find a $3\\times 3$ matrix $A$ such that $A\\vec{x}=4 \\vec{x}$ for all $\\vec{x}$ in ${\\mathbb R}^3$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "4", "0", "0", "0", "4", "0", "0", "0", "4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find a $3\\times 3$ matrix $A$ such that $A\\vec{x}=6 \\vec{x}$ for all $\\vec{x}$ in ${\\mathbb R}^3$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "6", "0", "0", "0", "6", "0", "0", "0", "6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0460", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "The dot product of two vectors in ${\\mathbb R}^3$ is defined by \\left\\lbrack \\begin{array}{c} a_1 \\\\ a_2 \\\\ a_3 \\end{array} \\right\\rbrack \\cdot \\left\\lbrack \\begin{array}{c} b_1 \\\\ b_2 \\\\ b_3 \\end{array} \\right\\rbrack=a_1 b_1+a_2 b_2+a_3 b_3. Let $\\vec{v}=\\left[\\begin{array}{c} 5\\cr 6\\cr-3 \\end{array}\\right]$. Find the matrix $A$ of the linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}$ given by $T(\\vec{x})=\\vec{v} \\cdot \\vec{x}.$\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "5", "6", "-3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The dot product of two vectors in ${\\mathbb R}^3$ is defined by \\left\\lbrack \\begin{array}{c} a_1 \\\\ a_2 \\\\ a_3 \\end{array} \\right\\rbrack \\cdot \\left\\lbrack \\begin{array}{c} b_1 \\\\ b_2 \\\\ b_3 \\end{array} \\right\\rbrack=a_1 b_1+a_2 b_2+a_3 b_3. Let $\\vec{v}=\\left[\\begin{array}{c}-8\\cr 9\\cr-8 \\end{array}\\right]$. Find the matrix $A$ of the linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}$ given by $T(\\vec{x})=\\vec{v} \\cdot \\vec{x}.$\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-8", "9", "-8" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The dot product of two vectors in ${\\mathbb R}^3$ is defined by \\left\\lbrack \\begin{array}{c} a_1 \\\\ a_2 \\\\ a_3 \\end{array} \\right\\rbrack \\cdot \\left\\lbrack \\begin{array}{c} b_1 \\\\ b_2 \\\\ b_3 \\end{array} \\right\\rbrack=a_1 b_1+a_2 b_2+a_3 b_3. Let $\\vec{v}=\\left[\\begin{array}{c}-4\\cr 6\\cr-6 \\end{array}\\right]$. Find the matrix $A$ of the linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}$ given by $T(\\vec{x})=\\vec{v} \\cdot \\vec{x}.$\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-4", "6", "-6" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0461", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [ "linear algebra", "vector space", "linear transformation" ], "problem_v1": "Find a matrix $A$ such that A \\left\\lbrack \\begin{array}{r} 1 \\\\ 0 \\\\ 0 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c} 3\\cr 2\\cr 1 \\end{array}\\right], \\ \\ \\ \\ A \\left\\lbrack \\begin{array}{r} 0 \\\\ 1 \\\\ 0 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c} 1\\cr-2\\cr 1 \\end{array}\\right], \\ \\ \\ \\ A \\left\\lbrack \\begin{array}{r} 0 \\\\ 0 \\\\ 1 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c} 1\\cr-2\\cr-1 \\end{array}\\right].\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "3", "1", "1", "2", "-2", "-2", "1", "1", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find a matrix $A$ such that A \\left\\lbrack \\begin{array}{r} 1 \\\\ 0 \\\\ 0 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c}-5\\cr-2\\cr-3 \\end{array}\\right], \\ \\ \\ \\ A \\left\\lbrack \\begin{array}{r} 0 \\\\ 1 \\\\ 0 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c} 5\\cr 5\\cr-2 \\end{array}\\right], \\ \\ \\ \\ A \\left\\lbrack \\begin{array}{r} 0 \\\\ 0 \\\\ 1 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c}-4\\cr-2\\cr 1 \\end{array}\\right].\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-5", "5", "-4", "-2", "5", "-2", "-3", "-2", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find a matrix $A$ such that A \\left\\lbrack \\begin{array}{r} 1 \\\\ 0 \\\\ 0 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c}-2\\cr 1\\cr 3 \\end{array}\\right], \\ \\ \\ \\ A \\left\\lbrack \\begin{array}{r} 0 \\\\ 1 \\\\ 0 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c} 1\\cr-3\\cr 5 \\end{array}\\right], \\ \\ \\ \\ A \\left\\lbrack \\begin{array}{r} 0 \\\\ 0 \\\\ 1 \\end{array} \\right\\rbrack=\\left[\\begin{array}{c}-2\\cr-2\\cr 4 \\end{array}\\right].\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-2", "1", "-2", "1", "-3", "-2", "3", "5", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0462", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Find the determinant of the linear transformation T(M)=\\left[\\begin{array}{cc} 4 &-4\\cr 0 &8 \\end{array}\\right] M from the space $V$ of upper triangular $2\\times 2$ matrices to $V$.\n$\\det=$ [ANS]", "answer_v1": [ "4*4*8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the linear transformation T(M)=\\left[\\begin{array}{cc} 1 &-1\\cr 0 &5 \\end{array}\\right] M from the space $V$ of upper triangular $2\\times 2$ matrices to $V$.\n$\\det=$ [ANS]", "answer_v2": [ "1*1*5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the linear transformation T(M)=\\left[\\begin{array}{cc} 2 &-4\\cr 0 &6 \\end{array}\\right] M from the space $V$ of upper triangular $2\\times 2$ matrices to $V$.\n$\\det=$ [ANS]", "answer_v3": [ "2*2*6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0463", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Find the determinant of the linear transformation $T(f(t))=f(6 t)-4 f(t)$ from $P_2$ to $P_2$.\n$\\det=$ [ANS]", "answer_v1": [ "(1+-4)*(6+-4)*(6^2+-4)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the linear transformation $T(f(t))=f(2 t)-2 f(t)$ from $P_2$ to $P_2$.\n$\\det=$ [ANS]", "answer_v2": [ "(1+-2)*(2+-2)*(2^2+-2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the linear transformation $T(f(t))=f(3 t)-4 f(t)$ from $P_2$ to $P_2$.\n$\\det=$ [ANS]", "answer_v3": [ "(1+-4)*(3+-4)*(3^2+-4)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0464", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Find the determinant of the linear transformation T(\\vec{v})=\\left[\\begin{array}{c} 1\\cr 4\\cr 4 \\end{array}\\right] \\times \\vec{v} from the plane $E$ given by $x+4 y+4 z=0$ to $E$.\n$\\det=$ [ANS]", "answer_v1": [ "1+4*4+4*4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the linear transformation T(\\vec{v})=\\left[\\begin{array}{c} 1\\cr-5\\cr-3 \\end{array}\\right] \\times \\vec{v} from the plane $E$ given by $x-5 y-3 z=0$ to $E$.\n$\\det=$ [ANS]", "answer_v2": [ "1+-5*-5+-3*-3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the linear transformation T(\\vec{v})=\\left[\\begin{array}{c} 1\\cr-4\\cr-4 \\end{array}\\right] \\times \\vec{v} from the plane $E$ given by $x-4 y-4 z=0$ to $E$.\n$\\det=$ [ANS]", "answer_v3": [ "1+-4*-4+-4*-4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0465", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Consider a linear transformation $T(x)=Ax$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$. Suppose for two vectors $\\vec{v}_1$ and $\\vec{v}_2$ in ${\\mathbb R}^2$ we have $T(\\vec{v}_1)=8 \\vec{v}_2$ and $T(\\vec{v}_2)=-5 \\vec{v}_1$. Find the determinant of the matrix $A$.\n$\\det(A)=$ [ANS]", "answer_v1": [ "40" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider a linear transformation $T(x)=Ax$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$. Suppose for two vectors $\\vec{v}_1$ and $\\vec{v}_2$ in ${\\mathbb R}^2$ we have $T(\\vec{v}_1)=2 \\vec{v}_2$ and $T(\\vec{v}_2)=-2 \\vec{v}_1$. Find the determinant of the matrix $A$.\n$\\det(A)=$ [ANS]", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider a linear transformation $T(x)=Ax$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$. Suppose for two vectors $\\vec{v}_1$ and $\\vec{v}_2$ in ${\\mathbb R}^2$ we have $T(\\vec{v}_1)=4 \\vec{v}_2$ and $T(\\vec{v}_2)=-5 \\vec{v}_1$. Find the determinant of the matrix $A$.\n$\\det(A)=$ [ANS]", "answer_v3": [ "20" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0466", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Find the determinant of the linear transformation $T(f)=8 f+6 f'+4 f''$ from the space $V$ spanned by $\\cos(x)$ and $\\sin(x)$ to $V$.\n$\\det=$ [ANS]", "answer_v1": [ "(8-4)^2+6^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the linear transformation $T(f)=2 f-3 f'+9 f''$ from the space $V$ spanned by $\\cos(x)$ and $\\sin(x)$ to $V$.\n$\\det=$ [ANS]", "answer_v2": [ "(2-9)^2+-3^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the linear transformation $T(f)=4 f+4 f'+3 f''$ from the space $V$ spanned by $\\cos(x)$ and $\\sin(x)$ to $V$.\n$\\det=$ [ANS]", "answer_v3": [ "(4-3)^2+4^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0467", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Find the determinant of the linear transformation $T(z)=(5+6 i) z$ from ${\\mathbb C}$ to ${\\mathbb C}$.\n$\\det=$ [ANS]", "answer_v1": [ "5^2+6^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the linear transformation $T(z)=(-8-9 i) z$ from ${\\mathbb C}$ to ${\\mathbb C}$.\n$\\det=$ [ANS]", "answer_v2": [ "-8^2+-9^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the linear transformation $T(z)=(-4-6 i) z$ from ${\\mathbb C}$ to ${\\mathbb C}$.\n$\\det=$ [ANS]", "answer_v3": [ "-4^2+-6^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0468", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Find the matrix $A$ of the linear transformation $T(f(t))=f(8 t+6)$ from $P_2$ to $P_2$ with respect to the standard basis for $P_2$, $\\left\\lbrace 1, t, t^2 \\right\\rbrace$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "6", "36", "0", "8", "96", "0", "0", "64" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the matrix $A$ of the linear transformation $T(f(t))=f(2 t+9)$ from $P_2$ to $P_2$ with respect to the standard basis for $P_2$, $\\left\\lbrace 1, t, t^2 \\right\\rbrace$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "9", "81", "0", "2", "36", "0", "0", "4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the matrix $A$ of the linear transformation $T(f(t))=f(4 t+6)$ from $P_2$ to $P_2$ with respect to the standard basis for $P_2$, $\\left\\lbrace 1, t, t^2 \\right\\rbrace$.\n$A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "6", "36", "0", "4", "48", "0", "0", "16" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0469", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Find the matrix $A$ of the linear transformation $T(z)=(8+6 i)z$ from ${\\mathbb C}$ to ${\\mathbb C}$ with respect to the standard basis ${\\mathbb C}$, $\\left\\lbrace 1, i \\right\\rbrace$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "8", "-6", "6", "8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the matrix $A$ of the linear transformation $T(z)=(2+9 i)z$ from ${\\mathbb C}$ to ${\\mathbb C}$ with respect to the standard basis ${\\mathbb C}$, $\\left\\lbrace 1, i \\right\\rbrace$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "2", "-9", "9", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the matrix $A$ of the linear transformation $T(z)=(4+6 i)z$ from ${\\mathbb C}$ to ${\\mathbb C}$ with respect to the standard basis ${\\mathbb C}$, $\\left\\lbrace 1, i \\right\\rbrace$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "4", "-6", "6", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0470", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "The matrices A_1=\\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\ A_2=\\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right], A_3=\\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right], \\ A_4=\\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right], form a basis for the linear space $V={\\mathbb R}^{2\\times 2}.$ Write the matrix of the linear transformation $T:{\\mathbb R}^{2\\times 2} \\rightarrow {\\mathbb R}^{2\\times 2}$ such that $T(A)=12 A+10 A^T$ relative to this basis.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "22", "0", "0", "0", "0", "12", "10", "0", "0", "10", "12", "0", "0", "0", "0", "22" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "The matrices A_1=\\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\ A_2=\\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right], A_3=\\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right], \\ A_4=\\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right], form a basis for the linear space $V={\\mathbb R}^{2\\times 2}.$ Write the matrix of the linear transformation $T:{\\mathbb R}^{2\\times 2} \\rightarrow {\\mathbb R}^{2\\times 2}$ such that $T(A)=3 A+15 A^T$ relative to this basis.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "18", "0", "0", "0", "0", "3", "15", "0", "0", "15", "3", "0", "0", "0", "0", "18" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "The matrices A_1=\\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\ A_2=\\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right], A_3=\\left[\\begin{array}{cc} 0 &0\\cr 1 &0 \\end{array}\\right], \\ A_4=\\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right], form a basis for the linear space $V={\\mathbb R}^{2\\times 2}.$ Write the matrix of the linear transformation $T:{\\mathbb R}^{2\\times 2} \\rightarrow {\\mathbb R}^{2\\times 2}$ such that $T(A)=6 A+10 A^T$ relative to this basis.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "16", "0", "0", "0", "0", "6", "10", "0", "0", "10", "6", "0", "0", "0", "0", "16" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0471", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Let $V$ be the space spanned by the two functions $\\cos(t)$ and $\\sin(t)$. Find the matrix $A$ of the linear transformation $T(f(t))=f''(t)+8 f'(t)+6 f(t)$ from $V$ into itself with respect to the basis $\\left\\lbrace \\cos(t), \\sin(t) \\right\\rbrace$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "5", "8", "-8", "5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $V$ be the space spanned by the two functions $\\cos(t)$ and $\\sin(t)$. Find the matrix $A$ of the linear transformation $T(f(t))=f''(t)+2 f'(t)+9 f(t)$ from $V$ into itself with respect to the basis $\\left\\lbrace \\cos(t), \\sin(t) \\right\\rbrace$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "8", "2", "-2", "8" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $V$ be the space spanned by the two functions $\\cos(t)$ and $\\sin(t)$. Find the matrix $A$ of the linear transformation $T(f(t))=f''(t)+4 f'(t)+6 f(t)$ from $V$ into itself with respect to the basis $\\left\\lbrace \\cos(t), \\sin(t) \\right\\rbrace$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "5", "4", "-4", "5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0472", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Find the matrix $A$ of the linear transformation T(M)=\\left[\\begin{array}{cc} 7 &5\\cr 0 &6 \\end{array}\\right] M {\\left[\\begin{array}{cc} 7 &5\\cr 0 &6 \\end{array}\\right]}^{-1} from $U^{2\\times 2}$ to $U^{2\\times 2}$ (upper triangular matrices) with respect to the standard basis for $U^{2\\times 2}$ given by \\left\\lbrace \\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right] \\right\\rbrace. $A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "0", "0", "-0.833333", "1.16667", "0.833333", "0", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the matrix $A$ of the linear transformation T(M)=\\left[\\begin{array}{cc} 1 &9\\cr 0 &3 \\end{array}\\right] M {\\left[\\begin{array}{cc} 1 &9\\cr 0 &3 \\end{array}\\right]}^{-1} from $U^{2\\times 2}$ to $U^{2\\times 2}$ (upper triangular matrices) with respect to the standard basis for $U^{2\\times 2}$ given by \\left\\lbrace \\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right] \\right\\rbrace. $A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "0", "-3", "0.333333", "3", "0", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the matrix $A$ of the linear transformation T(M)=\\left[\\begin{array}{cc} 3 &6\\cr 0 &2 \\end{array}\\right] M {\\left[\\begin{array}{cc} 3 &6\\cr 0 &2 \\end{array}\\right]}^{-1} from $U^{2\\times 2}$ to $U^{2\\times 2}$ (upper triangular matrices) with respect to the standard basis for $U^{2\\times 2}$ given by \\left\\lbrace \\left[\\begin{array}{cc} 1 &0\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &1\\cr 0 &0 \\end{array}\\right], \\left[\\begin{array}{cc} 0 &0\\cr 0 &1 \\end{array}\\right] \\right\\rbrace. $A=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "0", "0", "-3", "1.5", "3", "0", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0473", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Let $V$ be the plane with equation $x_1+4x_2+3x_3=0$ in ${\\mathbb R}^3$. The linear transformation T\\left(\\begin{bmatrix}x_1\\\\x_2\\\\x_3\\end{bmatrix}\\right)=\\left[\\begin{array}{ccc} 7 &2 &10\\cr-1 &1 &-1\\cr-1 &-2 &-2 \\end{array}\\right] \\begin{bmatrix}x_1\\\\x_2\\\\x_3\\end{bmatrix} maps $V$ into $V$ so, by restricting it to $V$, we may regard it as a linear transformation $\\tilde T \\colon V \\to V$. Find the matrix $A$ of the restricted map $\\tilde T \\colon V \\to V$ with respect to the basis $\\left\\lbrace \\left[\\begin{array}{c}-4\\cr 1\\cr 0 \\end{array}\\right], \\left[\\begin{array}{c}-3\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace.$\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "5", "2", "2", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $V$ be the plane with equation $x_1-2x_2-4x_3=0$ in ${\\mathbb R}^3$. The linear transformation T\\left(\\begin{bmatrix}x_1\\\\x_2\\\\x_3\\end{bmatrix}\\right)=\\left[\\begin{array}{ccc} 0 &6 &-12\\cr-2 &-1 &-2\\cr 1 &2 &-2 \\end{array}\\right] \\begin{bmatrix}x_1\\\\x_2\\\\x_3\\end{bmatrix} maps $V$ into $V$ so, by restricting it to $V$, we may regard it as a linear transformation $\\tilde T \\colon V \\to V$. Find the matrix $A$ of the restricted map $\\tilde T \\colon V \\to V$ with respect to the basis $\\left\\lbrace \\left[\\begin{array}{c} 2\\cr 1\\cr 0 \\end{array}\\right], \\left[\\begin{array}{c} 4\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace.$\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-5", "-10", "4", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $V$ be the plane with equation $x_1-2x_2+3x_3=0$ in ${\\mathbb R}^3$. The linear transformation T\\left(\\begin{bmatrix}x_1\\\\x_2\\\\x_3\\end{bmatrix}\\right)=\\left[\\begin{array}{ccc} 7 &7 &7\\cr 2 &2 &2\\cr-1 &-1 &-1 \\end{array}\\right] \\begin{bmatrix}x_1\\\\x_2\\\\x_3\\end{bmatrix} maps $V$ into $V$ so, by restricting it to $V$, we may regard it as a linear transformation $\\tilde T \\colon V \\to V$. Find the matrix $A$ of the restricted map $\\tilde T \\colon V \\to V$ with respect to the basis $\\left\\lbrace \\left[\\begin{array}{c} 2\\cr 1\\cr 0 \\end{array}\\right], \\left[\\begin{array}{c}-3\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace.$\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "6", "-4", "-3", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0474", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [], "problem_v1": "Let $V$ be the plane with equation $x_1+5x_2+4x_3=0$ in ${\\mathbb R}^3$. Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $\\vec{v}=\\left[\\begin{array}{c}-2\\cr 2\\cr-2 \\end{array}\\right]$ with respect to the basis $\\left\\lbrace \\left[\\begin{array}{c}-5\\cr 1\\cr 0 \\end{array}\\right], \\left[\\begin{array}{c}-4\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace.$\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "2", "1", "-2", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $V$ be the plane with equation $x_1-2x_2-5x_3=0$ in ${\\mathbb R}^3$. Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $\\vec{v}=\\left[\\begin{array}{c}-2\\cr 4\\cr-2 \\end{array}\\right]$ with respect to the basis $\\left\\lbrace \\left[\\begin{array}{c} 2\\cr 1\\cr 0 \\end{array}\\right], \\left[\\begin{array}{c} 5\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace.$\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "0", "-2", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $V$ be the plane with equation $x_1-3x_2+4x_3=0$ in ${\\mathbb R}^3$. Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $\\vec{v}=\\left[\\begin{array}{c} 14\\cr 2\\cr-2 \\end{array}\\right]$ with respect to the basis $\\left\\lbrace \\left[\\begin{array}{c} 3\\cr 1\\cr 0 \\end{array}\\right], \\left[\\begin{array}{c}-4\\cr 0\\cr 1 \\end{array}\\right] \\right\\rbrace.$\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "0.431373", "-0.568627", "-0.431373", "0.568627" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0475", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [], "problem_v1": "Let $\\mathcal{P}_{n}$ be the vector space of all polynomials of degree $n$ or less in the variable $x$. Let $D: \\mathcal{P}_{3} \\to \\mathcal{P}_{2}$ be the linear transformation defined by $D(p(x))=p'(x)$. That is, $D$ is the derivative operator. Let \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace 1+x+x^{2}+x^{3},-1-2x-x^{2}-x^{3},-2-2x-x^{2}-x^{3}, 2+3x+2x^{2}+x^{3} \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace 1, x, x^{2} \\rbrace, \\end{array} be ordered bases for $\\mathcal{P}_{3}$ and $\\mathcal{P}_{2}$, respectively. Find the matrix $\\lbrack D \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}$ for $D$ relative to the basis $\\mathcal{B}$ in the domain and $\\mathcal{C}$ in the codomain.\n$\\lbrack D \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "-2", "-2", "3", "2", "-2", "-2", "4", "3", "-3", "-3", "3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $\\mathcal{P}_{n}$ be the vector space of all polynomials of degree $n$ or less in the variable $x$. Let $D: \\mathcal{P}_{3} \\to \\mathcal{P}_{2}$ be the linear transformation defined by $D(p(x))=p'(x)$. That is, $D$ is the derivative operator. Let \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace-2+x-x^{2},-2-x^{2}, 3-x+2x^{2}, 1+x^{2}-x^{3} \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace 1, x, x^{2} \\rbrace, \\end{array} be ordered bases for $\\mathcal{P}_{3}$ and $\\mathcal{P}_{2}$, respectively. Find the matrix $\\lbrack D \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}$ for $D$ relative to the basis $\\mathcal{B}$ in the domain and $\\mathcal{C}$ in the codomain.\n$\\lbrack D \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "-1", "0", "-2", "-2", "4", "2", "0", "0", "0", "-3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $\\mathcal{P}_{n}$ be the vector space of all polynomials of degree $n$ or less in the variable $x$. Let $D: \\mathcal{P}_{3} \\to \\mathcal{P}_{2}$ be the linear transformation defined by $D(p(x))=p'(x)$. That is, $D$ is the derivative operator. Let \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace-1+x-x^{2}, 1-2x+x^{2},-1-x, 1+2x+x^{3} \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace 1, x, x^{2} \\rbrace, \\end{array} be ordered bases for $\\mathcal{P}_{3}$ and $\\mathcal{P}_{2}$, respectively. Find the matrix $\\lbrack D \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}$ for $D$ relative to the basis $\\mathcal{B}$ in the domain and $\\mathcal{C}$ in the codomain.\n$\\lbrack D \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "-2", "-1", "2", "-2", "2", "0", "0", "0", "0", "0", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0476", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [], "problem_v1": "Let $f: \\mathbb{R}^{2} \\to \\mathbb{R}^{3}$ be the linear transformation defined by f(\\vec{x})=\\left[\\begin{array}{cc} 3 &1\\cr 1 &2\\cr-2 &-2 \\end{array}\\right] \\vec{x}. Let \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace \\left<1,-1\\right>, \\left<1,-2\\right> \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace \\left<1,-1,1\\right>, \\left<-1,0,-1\\right>, \\left<-2,0,-1\\right> \\rbrace, \\end{array} be bases for $\\mathbb{R}^{2}$ and $\\mathbb{R}^{3}$, respectively. Find the matrix $\\lbrack f \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}$ for $f$ relative to the basis $\\mathcal{B}$ in the domain and $\\mathcal{C}$ in the codomain.\n$\\lbrack f \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "3", "3", "0", "-2", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $f: \\mathbb{R}^{2} \\to \\mathbb{R}^{3}$ be the linear transformation defined by f(\\vec{x})=\\left[\\begin{array}{cc}-5 &5\\cr-4 &-2\\cr 5 &-2 \\end{array}\\right] \\vec{x}. Let \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace \\left<-1,2\\right>, \\left<-1,3\\right> \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace \\left<0,1,-1\\right>, \\left<0,-2,1\\right>, \\left<-1,2,-1\\right> \\rbrace, \\end{array} be bases for $\\mathbb{R}^{2}$ and $\\mathbb{R}^{3}$, respectively. Find the matrix $\\lbrack f \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}$ for $f$ relative to the basis $\\mathcal{B}$ in the domain and $\\mathcal{C}$ in the codomain.\n$\\lbrack f \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "18", "24", "-6", "-7", "-15", "-20" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $f: \\mathbb{R}^{2} \\to \\mathbb{R}^{3}$ be the linear transformation defined by f(\\vec{x})=\\left[\\begin{array}{cc}-2 &1\\cr-2 &1\\cr-3 &-2 \\end{array}\\right] \\vec{x}. Let \\begin{array}{lcl} \\mathcal{B} &=& \\lbrace \\left<1,2\\right>, \\left<3,5\\right> \\rbrace, \\\\ \\mathcal{C} &=& \\lbrace \\left<-1,-1,-1\\right>, \\left<-1,0,-1\\right>, \\left<-3,-1,-2\\right> \\rbrace, \\end{array} be bases for $\\mathbb{R}^{2}$ and $\\mathbb{R}^{3}$, respectively. Find the matrix $\\lbrack f \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}$ for $f$ relative to the basis $\\mathcal{B}$ in the domain and $\\mathcal{C}$ in the codomain.\n$\\lbrack f \\rbrack_{\\mathcal{B}}^{\\mathcal{C}}=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "7", "19", "14", "36", "-7", "-18" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0478", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [ "linear algebra", "similar" ], "problem_v1": "Recall that similarity of matrices is an equivalence relation, that is, the relation is reflexive, symmetric and transitive. Verify that $A=\\left[\\begin{array}{cc}-2 &2\\cr 2 &1 \\end{array}\\right]$ is similar to itself by finding a $T$ such that $A=T^{-1} A T$. $T=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} We know that $A$ and $B=\\left[\\begin{array}{cc} 0 &-3\\cr-2 &-1 \\end{array}\\right]$ are similar since $A=P^{-1} B P$ where $P=\\left[\\begin{array}{cc}-1 &-1\\cr 0 &1 \\end{array}\\right]$. Verify that $B\\sim A$ by finding an $S$ such that $B=S^{-1} A S$. $S=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} We also know that $B$ and $C=\\left[\\begin{array}{cc}-1 &-2\\cr-3 &0 \\end{array}\\right]$ are similar since $B=Q^{-1} C Q$ where $Q=\\left[\\begin{array}{cc} 0 &-1\\cr-1 &0 \\end{array}\\right]$. Verify that $A\\sim C$ by finding an $R$ such that $A=R^{-1} C R$. $R=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "0", "0", "1", "-1", "-1", "0", "1", "0", "-1", "1", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Recall that similarity of matrices is an equivalence relation, that is, the relation is reflexive, symmetric and transitive. Verify that $A=\\left[\\begin{array}{cc} 7 &-24\\cr 3 &-10 \\end{array}\\right]$ is similar to itself by finding a $T$ such that $A=T^{-1} A T$. $T=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} We know that $A$ and $B=\\left[\\begin{array}{cc}-1 &-2\\cr 0 &-2 \\end{array}\\right]$ are similar since $A=P^{-1} B P$ where $P=\\left[\\begin{array}{cc}-1 &2\\cr 1 &-3 \\end{array}\\right]$. Verify that $B\\sim A$ by finding an $S$ such that $B=S^{-1} A S$. $S=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} We also know that $B$ and $C=\\left[\\begin{array}{cc}-10 &-3\\cr 24 &7 \\end{array}\\right]$ are similar since $B=Q^{-1} C Q$ where $Q=\\left[\\begin{array}{cc} 1 &1\\cr-3 &-2 \\end{array}\\right]$. Verify that $A\\sim C$ by finding an $R$ such that $A=R^{-1} C R$. $R=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "1", "0", "0", "1", "-3", "-2", "-1", "-1", "0", "-1", "1", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Recall that similarity of matrices is an equivalence relation, that is, the relation is reflexive, symmetric and transitive. Verify that $A=\\left[\\begin{array}{cc}-1 &-3\\cr 3 &6 \\end{array}\\right]$ is similar to itself by finding a $T$ such that $A=T^{-1} A T$. $T=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} We know that $A$ and $B=\\left[\\begin{array}{cc} 3 &1\\cr 3 &2 \\end{array}\\right]$ are similar since $A=P^{-1} B P$ where $P=\\left[\\begin{array}{cc} 1 &1\\cr-1 &0 \\end{array}\\right]$. Verify that $B\\sim A$ by finding an $S$ such that $B=S^{-1} A S$. $S=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} We also know that $B$ and $C=\\left[\\begin{array}{cc} 2 &3\\cr 1 &3 \\end{array}\\right]$ are similar since $B=Q^{-1} C Q$ where $Q=\\left[\\begin{array}{cc} 3 &1\\cr 2 &1 \\end{array}\\right]$. Verify that $A\\sim C$ by finding an $R$ such that $A=R^{-1} C R$. $R=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "0", "0", "1", "0", "-1", "1", "1", "2", "3", "1", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0479", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "2", "keywords": [ "vector space", "linear transformation", "matrix" ], "problem_v1": "Let $A$ be a $8 \\times 6$ matrix. What must $a$ and $b$ be if we define the linear transformation by $T: {\\mathbb R}^a \\rightarrow {\\mathbb R}^b$ as $T(x)=Ax\\,?$ $a=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "6", "8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $A$ be a $2 \\times 9$ matrix. What must $a$ and $b$ be if we define the linear transformation by $T: {\\mathbb R}^a \\rightarrow {\\mathbb R}^b$ as $T(x)=Ax\\,?$ $a=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "9", "2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $A$ be a $4 \\times 6$ matrix. What must $a$ and $b$ be if we define the linear transformation by $T: {\\mathbb R}^a \\rightarrow {\\mathbb R}^b$ as $T(x)=Ax\\,?$ $a=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "6", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0480", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [ "vector space", "linear transformation' 'matrix' 'image" ], "problem_v1": "Let \\vec{x}=\\left\\lbrack \\begin{array}{c} a \\\\ b \\end{array} \\right\\rbrack, \\ \\vec{v}=\\left\\lbrack \\begin{array}{c} 5 \\\\ 2 \\end{array} \\right\\rbrack, \\ \\mbox{and} \\ \\vec{w}=\\left\\lbrack \\begin{array}{c} 2 \\\\ 4 \\end{array} \\right\\rbrack. Let $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation defined by mapping every $\\vec{x}=\\left\\lbrack \\begin{array}{c} a \\\\ b \\end{array} \\right\\rbrack$ to $a \\vec{v}+b \\vec{w}$. Find a matrix $A$ such that $T(\\vec{x})=A\\vec{x}$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "5", "2", "2", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let \\vec{x}=\\left\\lbrack \\begin{array}{c} a \\\\ b \\end{array} \\right\\rbrack, \\ \\vec{v}=\\left\\lbrack \\begin{array}{c}-8 \\\\ 8 \\end{array} \\right\\rbrack, \\ \\mbox{and} \\ \\vec{w}=\\left\\lbrack \\begin{array}{c}-7 \\\\-3 \\end{array} \\right\\rbrack. Let $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation defined by mapping every $\\vec{x}=\\left\\lbrack \\begin{array}{c} a \\\\ b \\end{array} \\right\\rbrack$ to $a \\vec{v}+b \\vec{w}$. Find a matrix $A$ such that $T(\\vec{x})=A\\vec{x}$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-8", "-7", "8", "-3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let \\vec{x}=\\left\\lbrack \\begin{array}{c} a \\\\ b \\end{array} \\right\\rbrack, \\ \\vec{v}=\\left\\lbrack \\begin{array}{c}-4 \\\\ 2 \\end{array} \\right\\rbrack, \\ \\mbox{and} \\ \\vec{w}=\\left\\lbrack \\begin{array}{c}-4 \\\\ 1 \\end{array} \\right\\rbrack. Let $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be a linear transformation defined by mapping every $\\vec{x}=\\left\\lbrack \\begin{array}{c} a \\\\ b \\end{array} \\right\\rbrack$ to $a \\vec{v}+b \\vec{w}$. Find a matrix $A$ such that $T(\\vec{x})=A\\vec{x}$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "-4", "-4", "2", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0481", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [ "vector' 'linear transformation' 'matrix' 'composition" ], "problem_v1": "Let $S$ be a linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}^2$ with associated matrix A=\\left[\\begin{array}{ccc} 2 &1 &1\\cr 2 &-1 &-1 \\end{array}\\right]. Let $T$ be a linear transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ with associated matrix B=\\left[\\begin{array}{cc} 0 &1\\cr-1 &0 \\end{array}\\right]. Determine the matrix $C$ of the composition $T\\circ S$.\n$C=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "2", "-1", "-1", "-2", "-1", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $S$ be a linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}^2$ with associated matrix A=\\left[\\begin{array}{ccc}-3 &3 &-2\\cr-1 &3 &-1 \\end{array}\\right]. Let $T$ be a linear transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ with associated matrix B=\\left[\\begin{array}{cc}-2 &-1\\cr 0 &-3 \\end{array}\\right]. Determine the matrix $C$ of the composition $T\\circ S$.\n$C=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "7", "-9", "5", "3", "-9", "3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $S$ be a linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}^2$ with associated matrix A=\\left[\\begin{array}{ccc}-1 &1 &-2\\cr 0 &-2 &-1 \\end{array}\\right]. Let $T$ be a linear transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ with associated matrix B=\\left[\\begin{array}{cc} 2 &3\\cr 3 &-2 \\end{array}\\right]. Determine the matrix $C$ of the composition $T\\circ S$.\n$C=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-2", "-4", "-7", "-3", "7", "-4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0482", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [ "vector' 'linear transformation' 'matrix" ], "problem_v1": "Let $T$ be the linear transformation defined by\n$T(x, y)=(5y-7x, 4x-8y, 2y-x, 6x-3y)$.\nFind its associated matrix $A$. $A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v1": [ "-7", "5", "4", "-8", "-1", "2", "6", "-3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $T$ be the linear transformation defined by\n$T(x, y)=(9y, 2x+4y,-8x-3y,-x-5y)$.\nFind its associated matrix $A$. $A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v2": [ "0", "9", "2", "4", "-8", "-3", "-1", "-5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $T$ be the linear transformation defined by\n$T(x, y)=(-3x-6y, 5y-2x, x+4y, 8y-9x)$.\nFind its associated matrix $A$. $A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v3": [ "-3", "-6", "-2", "5", "1", "4", "-9", "8" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0483", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "3", "keywords": [ "vector' 'linear transformation' 'matrix' 'reflection' 'rotation" ], "problem_v1": "To every linear transformation $T$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$, there is an associated $2 \\times 2$ matrix. Match the following linear transformations with their associated matrix. [ANS] 1. Reflection about the line y=x [ANS] 2. Clockwise rotation by $\\pi/2$ radians [ANS] 3. Counter-clockwise rotation by $\\pi/2$ radians [ANS] 4. Reflection about the $x$-axis [ANS] 5. The projection onto the x-axis given by T(x,y)=(x,0) [ANS] 6. Reflection about the y-axis\nA. $\\begin{pmatrix}1&0\\\\ 0&0 \\end{pmatrix}$ B. $\\begin{pmatrix}1&0\\\\ 0&-1 \\end{pmatrix}$ C. $\\begin{pmatrix}-1&0\\\\ 0&1 \\end{pmatrix}$ D. $\\begin{pmatrix}0&1\\\\-1&0 \\end{pmatrix}$ E. $\\begin{pmatrix}0&-1\\\\ 1&0 \\end{pmatrix}$ F. $\\begin{pmatrix}0&1\\\\ 1&0 \\end{pmatrix}$ G. None of the above", "answer_v1": [ "F", "D", "E", "B", "A", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "To every linear transformation $T$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$, there is an associated $2 \\times 2$ matrix. Match the following linear transformations with their associated matrix. [ANS] 1. Reflection about the $x$-axis [ANS] 2. Clockwise rotation by $\\pi/2$ radians [ANS] 3. Counter-clockwise rotation by $\\pi/2$ radians [ANS] 4. Reflection about the y-axis [ANS] 5. The projection onto the x-axis given by T(x,y)=(x,0) [ANS] 6. Reflection about the line y=x\nA. $\\begin{pmatrix}1&0\\\\ 0&0 \\end{pmatrix}$ B. $\\begin{pmatrix}0&1\\\\ 1&0 \\end{pmatrix}$ C. $\\begin{pmatrix}0&-1\\\\ 1&0 \\end{pmatrix}$ D. $\\begin{pmatrix}0&1\\\\-1&0 \\end{pmatrix}$ E. $\\begin{pmatrix}-1&0\\\\ 0&1 \\end{pmatrix}$ F. $\\begin{pmatrix}1&0\\\\ 0&-1 \\end{pmatrix}$ G. None of the above", "answer_v2": [ "F", "D", "C", "E", "A", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "To every linear transformation $T$ from ${\\mathbb R}^2$ to ${\\mathbb R}^2$, there is an associated $2 \\times 2$ matrix. Match the following linear transformations with their associated matrix. [ANS] 1. Reflection about the line y=x [ANS] 2. Reflection about the $x$-axis [ANS] 3. Reflection about the y-axis [ANS] 4. Counter-clockwise rotation by $\\pi/2$ radians [ANS] 5. Clockwise rotation by $\\pi/2$ radians [ANS] 6. The projection onto the x-axis given by T(x,y)=(x,0)\nA. $\\begin{pmatrix}1&0\\\\ 0&0 \\end{pmatrix}$ B. $\\begin{pmatrix}-1&0\\\\ 0&1 \\end{pmatrix}$ C. $\\begin{pmatrix}1&0\\\\ 0&-1 \\end{pmatrix}$ D. $\\begin{pmatrix}0&1\\\\ 1&0 \\end{pmatrix}$ E. $\\begin{pmatrix}0&1\\\\-1&0 \\end{pmatrix}$ F. $\\begin{pmatrix}0&-1\\\\ 1&0 \\end{pmatrix}$ G. None of the above", "answer_v3": [ "D", "C", "B", "F", "E", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0484", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Associated matrices", "level": "4", "keywords": [ "linear", "transformation", "matrix" ], "problem_v1": "Let $T: {\\mathbb R}^2\\rightarrow {\\mathbb R}^2$ be the linear transformation that first rotates points clockwise through $135^\\circ$ ($3\\pi/4$ radians) and then reflects points through the line $y=x$. Find the standard matrix $A$ for $T$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v1": [ "-0.707107", "-0.707107", "-0.707107", "0.707107" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $T: {\\mathbb R}^2\\rightarrow {\\mathbb R}^2$ be the linear transformation that first rotates points clockwise through $30^\\circ$ ($\\pi/6$ radians) and then reflects points through the line $y=x$. Find the standard matrix $A$ for $T$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v2": [ "-0.5", "0.866025", "0.866025", "0.5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $T: {\\mathbb R}^2\\rightarrow {\\mathbb R}^2$ be the linear transformation that first rotates points clockwise through $45^\\circ$ ($\\pi/4$ radians) and then reflects points through the line $y=x$. Find the standard matrix $A$ for $T$.\n$A=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}.", "answer_v3": [ "-0.707107", "0.707107", "0.707107", "0.707107" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0486", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "One-to-one and onto", "level": "3", "keywords": [], "problem_v1": "A linear transformation $T:{\\mathbb R}^3 \\rightarrow {\\mathbb R}^2$ whose matrix is \\left[\\begin{array}{ccc} 2 &-1 &4\\cr 1 &-0.5 &-1+k\\cr \\end{array}\\right] is onto if and only if $k \\ne$ [ANS].", "answer_v1": [ "3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A linear transformation $T:{\\mathbb R}^3 \\rightarrow {\\mathbb R}^2$ whose matrix is \\left[\\begin{array}{ccc}-3 &12 &6\\cr 3 &-12 &-15+k\\cr \\end{array}\\right] is onto if and only if $k \\ne$ [ANS].", "answer_v2": [ "9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A linear transformation $T:{\\mathbb R}^3 \\rightarrow {\\mathbb R}^2$ whose matrix is \\left[\\begin{array}{ccc}-1 &2 &1.5\\cr 1 &-2 &-3.5+k\\cr \\end{array}\\right] is onto if and only if $k \\ne$ [ANS].", "answer_v3": [ "2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0487", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "One-to-one and onto", "level": "3", "keywords": [], "problem_v1": "Let $f: \\mathbb{R}^{2} \\to \\mathbb{R}^{3}$ be the linear transformation determined by f \\left(\\begin{array}{c} 1 \\\\ 0 \\end{array} \\right)=\\left(\\begin{array}{r} 3 \\\\ 1 \\\\-2 \\end{array} \\right), \\ \\ \\ \\ \\ f \\left(\\begin{array}{c} 0 \\\\ 1 \\end{array} \\right)=\\left(\\begin{array}{r} 1 \\\\ 2 \\\\-2 \\end{array} \\right).\nFind $ f \\left(\\begin{array}{c} 8 \\\\-7 \\end{array} \\right)$.\n$ f \\left(\\begin{array}{c} 8 \\\\-7 \\end{array} \\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nFind the matrix of the linear transformation $f$.\n$ f \\left(\\begin{array}{c} x \\\\ y \\end{array} \\right)=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\end{array} \\right\\rbrack.$\nThe linear transformation $f$ is [ANS] [ANS] [ANS] [ANS]", "answer_v1": [ "17", "-6", "-2", "3", "1", "1", "2", "-2", "-2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $f: \\mathbb{R}^{2} \\to \\mathbb{R}^{3}$ be the linear transformation determined by f \\left(\\begin{array}{c} 1 \\\\ 0 \\end{array} \\right)=\\left(\\begin{array}{r}-5 \\\\-4 \\\\ 5 \\end{array} \\right), \\ \\ \\ \\ \\ f \\left(\\begin{array}{c} 0 \\\\ 1 \\end{array} \\right)=\\left(\\begin{array}{r} 5 \\\\-2 \\\\-2 \\end{array} \\right).\nFind $ f \\left(\\begin{array}{c}-7 \\\\ 6 \\end{array} \\right)$.\n$ f \\left(\\begin{array}{c}-7 \\\\ 6 \\end{array} \\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nFind the matrix of the linear transformation $f$.\n$ f \\left(\\begin{array}{c} x \\\\ y \\end{array} \\right)=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\end{array} \\right\\rbrack.$\nThe linear transformation $f$ is [ANS] [ANS] [ANS] [ANS]", "answer_v2": [ "65", "16", "-47", "-5", "5", "-4", "-2", "5", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $f: \\mathbb{R}^{2} \\to \\mathbb{R}^{3}$ be the linear transformation determined by f \\left(\\begin{array}{c} 1 \\\\ 0 \\end{array} \\right)=\\left(\\begin{array}{r} 3 \\\\ 4 \\\\-2 \\end{array} \\right), \\ \\ \\ \\ \\ f \\left(\\begin{array}{c} 0 \\\\ 1 \\end{array} \\right)=\\left(\\begin{array}{r} 5 \\\\-3 \\\\-3 \\end{array} \\right).\nFind $ f \\left(\\begin{array}{c}-8 \\\\ 9 \\end{array} \\right)$.\n$ f \\left(\\begin{array}{c}-8 \\\\ 9 \\end{array} \\right)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nFind the matrix of the linear transformation $f$.\n$ f \\left(\\begin{array}{c} x \\\\ y \\end{array} \\right)=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array} $ \\left\\lbrack \\begin{array}{c} x \\\\ y \\end{array} \\right\\rbrack.$\nThe linear transformation $f$ is [ANS] [ANS] [ANS] [ANS]", "answer_v3": [ "21", "-59", "-11", "3", "5", "4", "-3", "-2", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0489", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "One-to-one and onto", "level": "3", "keywords": [], "problem_v1": "Let $V$ be a vector space and let $\\vec{v}_1, \\vec{v}_2, \\vec{v}_3 \\in V$. Suppose $f: V \\to \\mathbb{R}^{3}$ is a linear transformation and f(\\vec{v}_1)=\\left(\\begin{array}{r} 3 \\\\ 2 \\\\ 1 \\end{array} \\right), \\ \\ \\ f(\\vec{v}_2)=\\left(\\begin{array}{r} 1 \\\\-2 \\\\ 1 \\end{array} \\right), \\ \\ \\ f(\\vec{v}_3)=\\left(\\begin{array}{r} 1 \\\\-2 \\\\-1 \\end{array} \\right).\nFind $f(3 \\vec{v}_1+6 \\vec{v}_2-2 \\vec{v}_3)$.\n$f(3 \\vec{v}_1+6 \\vec{v}_2-2 \\vec{v}_3)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nIf $V$ has dimension $3$ and $\\lbrace \\vec{v}_1, \\vec{v}_2, \\vec{v}_3 \\rbrace$ is a linearly independent set, then the linear transformation $f$ is [ANS] [ANS] [ANS] [ANS]", "answer_v1": [ "13", "-2", "11" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $V$ be a vector space and let $\\vec{v}_1, \\vec{v}_2, \\vec{v}_3 \\in V$. Suppose $f: V \\to \\mathbb{R}^{3}$ is a linear transformation and f(\\vec{v}_1)=\\left(\\begin{array}{r}-5 \\\\-2 \\\\-3 \\end{array} \\right), \\ \\ \\ f(\\vec{v}_2)=\\left(\\begin{array}{r} 5 \\\\ 5 \\\\-2 \\end{array} \\right), \\ \\ \\ f(\\vec{v}_3)=\\left(\\begin{array}{r}-4 \\\\-2 \\\\ 1 \\end{array} \\right).\nFind $f(2 \\vec{v}_1+6 \\vec{v}_2-3 \\vec{v}_3)$.\n$f(2 \\vec{v}_1+6 \\vec{v}_2-3 \\vec{v}_3)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nIf $V$ has dimension $3$ and $\\lbrace \\vec{v}_1, \\vec{v}_2, \\vec{v}_3 \\rbrace$ is a linearly independent set, then the linear transformation $f$ is [ANS] [ANS] [ANS] [ANS]", "answer_v2": [ "32", "32", "-21" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $V$ be a vector space and let $\\vec{v}_1, \\vec{v}_2, \\vec{v}_3 \\in V$. Suppose $f: V \\to \\mathbb{R}^{3}$ is a linear transformation and f(\\vec{v}_1)=\\left(\\begin{array}{r}-2 \\\\ 1 \\\\ 3 \\end{array} \\right), \\ \\ \\ f(\\vec{v}_2)=\\left(\\begin{array}{r} 1 \\\\-3 \\\\ 5 \\end{array} \\right), \\ \\ \\ f(\\vec{v}_3)=\\left(\\begin{array}{r}-2 \\\\-2 \\\\ 4 \\end{array} \\right).\nFind $f(2 \\vec{v}_1+5 \\vec{v}_2-2 \\vec{v}_3)$.\n$f(2 \\vec{v}_1+5 \\vec{v}_2-2 \\vec{v}_3)=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.\nIf $V$ has dimension $3$ and $\\lbrace \\vec{v}_1, \\vec{v}_2, \\vec{v}_3 \\rbrace$ is a linearly independent set, then the linear transformation $f$ is [ANS] [ANS] [ANS] [ANS]", "answer_v3": [ "5", "-9", "23" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0490", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "One-to-one and onto", "level": "3", "keywords": [ "linear", "transformation", "matrix", "echelon", "pivots" ], "problem_v1": "Let $T$ be an linear transformation from ${\\mathbb R}^r$ to ${\\mathbb R}^s$. Let $A$ be the matrix associated to $T$.\nFill in the correct answer for each of the following situations. [ANS] 1. Two rows in the row-echelon form of $A$ do not have pivots. [ANS] 2. The row-echelon form of $A$ has a row of zeros. [ANS] 3. The row-echelon form of $A$ has a pivot in every column. [ANS] 4. Every row in the row-echelon form of $A$ has a pivot.\nA. T is not onto B. T is onto C. There is not enough information to tell.", "answer_v1": [ "A", "A", "C", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Let $T$ be an linear transformation from ${\\mathbb R}^r$ to ${\\mathbb R}^s$. Let $A$ be the matrix associated to $T$.\nFill in the correct answer for each of the following situations. [ANS] 1. The row-echelon form of $A$ has a row of zeros. [ANS] 2. The row-echelon form of $A$ has a pivot in every column. [ANS] 3. Two rows in the row-echelon form of $A$ do not have pivots. [ANS] 4. Every row in the row-echelon form of $A$ has a pivot.\nA. T is onto B. T is not onto C. There is not enough information to tell.", "answer_v2": [ "B", "C", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Let $T$ be an linear transformation from ${\\mathbb R}^r$ to ${\\mathbb R}^s$. Let $A$ be the matrix associated to $T$.\nFill in the correct answer for each of the following situations. [ANS] 1. The row-echelon form of $A$ has a row of zeros. [ANS] 2. Every row in the row-echelon form of $A$ has a pivot. [ANS] 3. Two rows in the row-echelon form of $A$ do not have pivots. [ANS] 4. The row-echelon form of $A$ has a pivot in every column.\nA. T is onto B. T is not onto C. There is not enough information to tell.", "answer_v3": [ "B", "A", "B", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Linear_algebra_0491", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "One-to-one and onto", "level": "3", "keywords": [ "vector' 'linear transformation' 'one to one' 'onto" ], "problem_v1": "Let $T$ be a linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}^3$. Determine whether or not $T$ is onto in each of the following situations: [ANS] 1. Suppose $T(2, 1, 1)=u$, $T(2,-2,-2)=v$, $T(4, 0,-1)=u+v$. [ANS] 2. Suppose $T$ is a one-to-one function [ANS] 3. Suppose $T(a)=u$, $T(b)=v$, $T(c)=u+v$, where $a, b, c, u,v$ are vectors in ${\\mathbb R}^3$.\nA. T is not onto. B. T is onto. C. There is not enough information to tell", "answer_v1": [ "A", "B", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Let $T$ be a linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}^3$. Determine whether or not $T$ is onto in each of the following situations: [ANS] 1. Suppose $T$ is a one-to-one function [ANS] 2. Suppose $T(a)=u$, $T(b)=v$, $T(c)=u+v$, where $a, b, c, u,v$ are vectors in ${\\mathbb R}^3$. [ANS] 3. Suppose $T(-4, 4,-3)=u$, $T(-1, 4,-2)=v$, $T(-5, 9,-5)=u+v$.\nA. T is onto. B. T is not onto. C. There is not enough information to tell", "answer_v2": [ "A", "C", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Let $T$ be a linear transformation from ${\\mathbb R}^3$ to ${\\mathbb R}^3$. Determine whether or not $T$ is onto in each of the following situations: [ANS] 1. Suppose $T(a)=u$, $T(b)=v$, $T(c)=u+v$, where $a, b, c, u,v$ are vectors in ${\\mathbb R}^3$. [ANS] 2. Suppose $T$ is a one-to-one function [ANS] 3. Suppose $T(-2, 1,-2)=u$, $T(0,-3,-1)=v$, $T(-2,-1,-3)=u+v$.\nA. T is not onto. B. T is onto. C. There is not enough information to tell", "answer_v3": [ "C", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Linear_algebra_0492", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "One-to-one and onto", "level": "3", "keywords": [ "vector' 'linear transformation' 'one to one" ], "problem_v1": "Let $T$ be a linear transformation from ${\\mathbb R}^r$ to ${\\mathbb R}^s$. Determine whether or not $T$ is one-to-one in each of the following situations: [ANS] 1. $r>s$ [ANS] 2. $rs$ [ANS] 3. $r=s$\nA. T is not a one-to-one transformation B. T is a one-to-one transformation C. There is not enough information to tell", "answer_v2": [ "C", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Let $T$ be a linear transformation from ${\\mathbb R}^r$ to ${\\mathbb R}^s$. Determine whether or not $T$ is one-to-one in each of the following situations: [ANS] 1. $r>s$ [ANS] 2. $r$ B\\. $<6,0,3>$ C\\. $<0,-2,1,1>$ D\\. $<-4,0,1,1>$ E\\. $<0,0,1,0>$ F\\. $<2,-2,1>$\nSelect all of the vectors that are in the image of $f$. You should be able to justify your answers (there may be more than one correct answer). [ANS] A\\. $<6,0,3>$ B\\. $<0,0,1,0>$ C\\. $<0,-2,1,1>$ D\\. $<3,4,3>$ E\\. $<2,-2,1>$ F\\. $<-4,0,1,1>$\n\\", "answer_v1": [ "R^k", "R^n", "null space", "column space", "EFG", "ACDE", "R^4", "R^3", "CD", "AE" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCM", "MCM", "EX", "EX", "MCM", "MCM" ], "options_v1": [ [ "R^k", "R^n" ], [ "R^k", "R^n" ], [ "column space", "null space", "row space", "subspace" ], [ "column space", "null space", "row space", "subspace" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [], [], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Part 1: Basic properties of linear transformations Part 1: Basic properties of linear transformations\n\\$(\"#Prob-1_SC-1_SECT-1\").canopen() If a linear transformation $f: V \\to W$ is given by $f(\\mathbf{x})=A \\mathbf{x}$ for some $A \\in M_{n,k}(\\mathbb{R})$, then:\nthe domain of $f$ is $V=$ [ANS],\nthe codomain of $f$ is $W=$ [ANS],\nthe kernel of $f$ is equal to the [ANS] of $A$, and\nthe image of $f$ is equal to the [ANS] of $A$. Part 2: Show you know definitions Part 2: Show you know definitions\n\\$(\"#Prob-1_SC-1_SECT-2\").canopen() Suppose $A \\in M_{n,k}(\\mathbb{R})$. Write $A=(\\mathbf{a_1} \\mid \\cdots \\mid \\mathbf{a_k})$.\nA vector $\\mathbf{x} \\in \\mathbb{R}^k$ is in the null space of $A$ if (select all that apply): [ANS] A\\. $x_1 \\mathbf{a_1}+\\cdots+x_k \\mathbf{a_k}=\\mathbf{0}_n$ B\\. $\\mathbf{y}=A \\mathbf{x}$ for some $\\mathbf{x} \\in \\mathbb{R}^k$. C\\. $\\mathbf{y} \\in \\mathrm{span} \\lbrace \\mathbf{a_1}, \\ldots, \\mathbf{a_k} \\rbrace$. D\\. $\\mathbf{y}=x_1 \\mathbf{a_1}+\\cdots+x_k \\mathbf{a_k}$ for some $\\langle x_1, \\ldots, x_k \\rangle \\in \\mathbb{R}^k$. E\\. $A \\mathbf{x}=\\mathbf{0}_n$. F\\. $\\mathrm{rref}(A \\mid \\mathbf{y})$ has no rows of the form $(0 \\ldots 0 \\mid 1)$. G\\. The dot product of $\\mathbf{x}$ with every row vector from $A$ is $0$.\nA vector $\\mathbf{y} \\in \\mathbb{R}^n$ is in the column space of $A$ if (select all that apply): [ANS] A\\. $x_1 \\mathbf{a_1}+\\cdots+x_k \\mathbf{a_k}=\\mathbf{0}_n$ B\\. $\\mathbf{y} \\in \\mathrm{span} \\lbrace \\mathbf{a_1}, \\ldots, \\mathbf{a_k} \\rbrace$. C\\. $\\mathbf{y}=A \\mathbf{x}$ for some $\\mathbf{x} \\in \\mathbb{R}^k$. D\\. $\\mathbf{y}=x_1 \\mathbf{a_1}+\\cdots+x_k \\mathbf{a_k}$ for some $\\langle x_1, \\ldots, x_k \\rangle \\in \\mathbb{R}^k$. E\\. The dot product of $\\mathbf{x}$ with every row vector from $A$ is $0$. F\\. $\\mathrm{rref}(A \\mid \\mathbf{y})$ has no rows of the form $(0 \\ldots 0 \\mid 1)$. G\\. $A \\mathbf{x}=\\mathbf{0}_n$.\nPart 3: Apply your knowledge of definitions Part 3: Apply your knowledge of definitions\n\\$(\"#Prob-1_SC-1_SECT-3\").canopen() Suppose $f$ is the function defined by $f(\\mathbf{x})=A \\mathbf{x}$, where\n${A={\\left[\\begin{array}{cccc}-3 &-6 &1 &-13\\cr 0 &0 &-1 &1\\cr-1 &-2 &1 &-5 \\end{array}\\right]}.}$\nThe domain of $f$ is [ANS]. For instance, enter R^5 for $\\mathbb{R}^5$.\nThe codomain of $f$ is [ANS]. For instance, enter R^5 for $\\mathbb{R}^5$.\nSelect all of the vectors that are in the kernel of $f$. You should be able to justify your answers (there may be more than one correct answer). [ANS] A\\. $<-9,0,-3>$ B\\. $<0,-2,1,1>$ C\\. $<0,0,1,0>$ D\\. $<-4,0,1,1>$ E\\. $<-10,4,-5>$ F\\. $<-8,5,-6>$\nSelect all of the vectors that are in the image of $f$. You should be able to justify your answers (there may be more than one correct answer). [ANS] A\\. $<-4,0,1,1>$ B\\. $<0,-2,1,1>$ C\\. $<0,0,1,0>$ D\\. $<-9,0,-3>$ E\\. $<-10,4,-5>$ F\\. $<-8,5,-6>$\n\\", "answer_v2": [ "R^k", "R^n", "null space", "column space", "AEG", "BCDF", "R^4", "R^3", "BD", "DF" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCM", "MCM", "EX", "EX", "MCM", "MCM" ], "options_v2": [ [ "R^k", "R^n" ], [ "R^k", "R^n" ], [ "column space", "null space", "row space", "subspace" ], [ "column space", "null space", "row space", "subspace" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [], [], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Part 1: Basic properties of linear transformations Part 1: Basic properties of linear transformations\n\\$(\"#Prob-1_SC-1_SECT-1\").canopen() If a linear transformation $f: V \\to W$ is given by $f(\\mathbf{x})=A \\mathbf{x}$ for some $A \\in M_{n,k}(\\mathbb{R})$, then:\nthe domain of $f$ is $V=$ [ANS],\nthe codomain of $f$ is $W=$ [ANS],\nthe kernel of $f$ is equal to the [ANS] of $A$, and\nthe image of $f$ is equal to the [ANS] of $A$. Part 2: Show you know definitions Part 2: Show you know definitions\n\\$(\"#Prob-1_SC-1_SECT-2\").canopen() Suppose $A \\in M_{n,k}(\\mathbb{R})$. Write $A=(\\mathbf{a_1} \\mid \\cdots \\mid \\mathbf{a_k})$.\nA vector $\\mathbf{x} \\in \\mathbb{R}^k$ is in the null space of $A$ if (select all that apply): [ANS] A\\. $\\mathbf{y}=x_1 \\mathbf{a_1}+\\cdots+x_k \\mathbf{a_k}$ for some $\\langle x_1, \\ldots, x_k \\rangle \\in \\mathbb{R}^k$. B\\. The dot product of $\\mathbf{x}$ with every row vector from $A$ is $0$. C\\. $\\mathrm{rref}(A \\mid \\mathbf{y})$ has no rows of the form $(0 \\ldots 0 \\mid 1)$. D\\. $\\mathbf{y} \\in \\mathrm{span} \\lbrace \\mathbf{a_1}, \\ldots, \\mathbf{a_k} \\rbrace$. E\\. $\\mathbf{y}=A \\mathbf{x}$ for some $\\mathbf{x} \\in \\mathbb{R}^k$. F\\. $A \\mathbf{x}=\\mathbf{0}_n$. G\\. $x_1 \\mathbf{a_1}+\\cdots+x_k \\mathbf{a_k}=\\mathbf{0}_n$\nA vector $\\mathbf{y} \\in \\mathbb{R}^n$ is in the column space of $A$ if (select all that apply): [ANS] A\\. $\\mathrm{rref}(A \\mid \\mathbf{y})$ has no rows of the form $(0 \\ldots 0 \\mid 1)$. B\\. $\\mathbf{y}=A \\mathbf{x}$ for some $\\mathbf{x} \\in \\mathbb{R}^k$. C\\. $\\mathbf{y} \\in \\mathrm{span} \\lbrace \\mathbf{a_1}, \\ldots, \\mathbf{a_k} \\rbrace$. D\\. $A \\mathbf{x}=\\mathbf{0}_n$. E\\. $\\mathbf{y}=x_1 \\mathbf{a_1}+\\cdots+x_k \\mathbf{a_k}$ for some $\\langle x_1, \\ldots, x_k \\rangle \\in \\mathbb{R}^k$. F\\. $x_1 \\mathbf{a_1}+\\cdots+x_k \\mathbf{a_k}=\\mathbf{0}_n$ G\\. The dot product of $\\mathbf{x}$ with every row vector from $A$ is $0$.\nPart 3: Apply your knowledge of definitions Part 3: Apply your knowledge of definitions\n\\$(\"#Prob-1_SC-1_SECT-3\").canopen() Suppose $f$ is the function defined by $f(\\mathbf{x})=A \\mathbf{x}$, where\n${A={\\left[\\begin{array}{cccc}-1 &-2 &-1 &-3\\cr 1 &2 &0 &4\\cr-2 &-4 &-2 &-6 \\end{array}\\right]}.}$\nThe domain of $f$ is [ANS]. For instance, enter R^5 for $\\mathbb{R}^5$.\nThe codomain of $f$ is [ANS]. For instance, enter R^5 for $\\mathbb{R}^5$.\nSelect all of the vectors that are in the kernel of $f$. You should be able to justify your answers (there may be more than one correct answer). [ANS] A\\. $<0,-2,1,1>$ B\\. $<-1,4,-2>$ C\\. $<0,0,1,0>$ D\\. $<-4,0,1,1>$ E\\. $<-3,3,-6>$ F\\. $<-3,4,-3>$\nSelect all of the vectors that are in the image of $f$. You should be able to justify your answers (there may be more than one correct answer). [ANS] A\\. $<-3,4,-3>$ B\\. $<0,0,1,0>$ C\\. $<-3,3,-6>$ D\\. $<0,-2,1,1>$ E\\. $<-1,4,-2>$ F\\. $<-4,0,1,1>$\n\\", "answer_v3": [ "R^k", "R^n", "null space", "column space", "BFG", "ABCE", "R^4", "R^3", "AD", "CE" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCM", "MCM", "EX", "EX", "MCM", "MCM" ], "options_v3": [ [ "R^k", "R^n" ], [ "R^k", "R^n" ], [ "column space", "null space", "row space", "subspace" ], [ "column space", "null space", "row space", "subspace" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [], [], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0498", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Kernel and image", "level": "4", "keywords": [], "problem_v1": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ defined by $L_A(x)=Ax$, where $A=\\left[\\begin{array}{cccc} 0 &-1 &-1 &0\\cr 1 &2 &1 &0\\cr-1 &0 &-1 &1\\cr-3 &-7 &-2 &-1 \\end{array}\\right]$. a. Find the smallest nonnegative $k$ such that the ranks of $L_A^k$ and $L_A^{k+1}$ are the same. $k=$ [ANS]\nIt can shown that $\\mathbb{R}^4=\\ker(L_A^k)\\oplus\\hbox{ran}(L_A^k)$. This means $\\ker(L_A^k)\\cap\\hbox{ran}(L_A^k)=\\lbrace \\underline{0}\\}$ and $\\mathbb{R}^4=\\ker(L_A^k)+\\hbox{ran}(L_A^k)$. b. Find a matrix $K$ whose row space is $\\ker(L_A^k)$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}, c. Find a matrix $R$ whose row space is $\\hbox{ran}(L_A^k)$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}, d. Find a vector $u\\in \\ker(L_A^k)$ and a vector $v\\in \\hbox{ran}(L_A^k)$ such that $u+v=\\left(2,-2,0,5\\right)$ $u=$ [ANS], $v=$ [ANS]", "answer_v1": [ "1", "-1", "1", "2", "1", "-1", "0", "2", "0", "1", "-2", "-2", "1", "0", "0", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ defined by $L_A(x)=Ax$, where $A=\\left[\\begin{array}{cccc} 1 &1 &0 &0\\cr-6 &-4 &-7 &-2\\cr 2 &1 &3 &1\\cr 0 &0 &1 &0 \\end{array}\\right]$. a. Find the smallest nonnegative $k$ such that the ranks of $L_A^k$ and $L_A^{k+1}$ are the same. $k=$ [ANS]\nIt can shown that $\\mathbb{R}^4=\\ker(L_A^k)\\oplus\\hbox{ran}(L_A^k)$. This means $\\ker(L_A^k)\\cap\\hbox{ran}(L_A^k)=\\lbrace \\underline{0}\\}$ and $\\mathbb{R}^4=\\ker(L_A^k)+\\hbox{ran}(L_A^k)$. b. Find a matrix $K$ whose row space is $\\ker(L_A^k)$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}, c. Find a matrix $R$ whose row space is $\\hbox{ran}(L_A^k)$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}, d. Find a vector $u\\in \\ker(L_A^k)$ and a vector $v\\in \\hbox{ran}(L_A^k)$ such that $u+v=\\left(1,1,-1,0\\right)$ $u=$ [ANS], $v=$ [ANS]", "answer_v2": [ "1", "-1", "0", "-1", "0", "1", "-1", "2", "1", "1", "-1", "-1", "2", "-1", "-1", "-1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the multiplication operator $L_A:\\mathbb{R}^4\\to\\mathbb{R}^4$ defined by $L_A(x)=Ax$, where $A=\\left[\\begin{array}{cccc}-3 &-5 &-6 &-2\\cr 2 &3 &4 &1\\cr 0 &0 &-1 &0\\cr 0 &1 &3 &1 \\end{array}\\right]$. a. Find the smallest nonnegative $k$ such that the ranks of $L_A^k$ and $L_A^{k+1}$ are the same. $k=$ [ANS]\nIt can shown that $\\mathbb{R}^4=\\ker(L_A^k)\\oplus\\hbox{ran}(L_A^k)$. This means $\\ker(L_A^k)\\cap\\hbox{ran}(L_A^k)=\\lbrace \\underline{0}\\}$ and $\\mathbb{R}^4=\\ker(L_A^k)+\\hbox{ran}(L_A^k)$. b. Find a matrix $K$ whose row space is $\\ker(L_A^k)$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}, c. Find a matrix $R$ whose row space is $\\hbox{ran}(L_A^k)$.\n\\begin {array}{cccc} [ANS] & [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] & [ANS] \\end{array}, d. Find a vector $u\\in \\ker(L_A^k)$ and a vector $v\\in \\hbox{ran}(L_A^k)$ such that $u+v=\\left(-1,1,-2,4\\right)$ $u=$ [ANS], $v=$ [ANS]", "answer_v3": [ "1", "-1", "0", "1", "-2", "1", "0", "0", "-1", "1", "-1", "2", "0", "-1", "1", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0500", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Kernel and image", "level": "3", "keywords": [ "vector space", "linear transformation' 'matrix' 'image" ], "problem_v1": "Let A=\\left[\\begin{array}{cc}-2 &3\\cr-3 &-4\\cr-20 &-4 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{b}=\\left[\\begin{array}{c}-1\\cr 7\\cr 24 \\end{array}\\right]. Define the linear transformation $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^3$ by $T(\\vec{x})=A\\vec{x}$. Find a vector $\\vec{x}$ whose image under $T$ is $\\vec{b}$.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}. Is the vector $\\vec{x}$ unique? [ANS]", "answer_v1": [ "-1", "-1" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "unique", "not unique" ] ], "problem_v2": "Let A=\\left[\\begin{array}{cc}-6 &-1\\cr-3 &-4\\cr-3 &3 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{b}=\\left[\\begin{array}{c}-15\\cr 3\\cr-18 \\end{array}\\right]. Define the linear transformation $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^3$ by $T(\\vec{x})=A\\vec{x}$. Find a vector $\\vec{x}$ whose image under $T$ is $\\vec{b}$.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}. Is the vector $\\vec{x}$ unique? [ANS]", "answer_v2": [ "3", "-3" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "unique", "not unique" ] ], "problem_v3": "Let A=\\left[\\begin{array}{cc}-2 &5\\cr-5 &-2\\cr-14 &6 \\end{array}\\right] \\ \\mbox{and} \\ \\vec{b}=\\left[\\begin{array}{c} 15\\cr 23\\cr 76 \\end{array}\\right]. Define the linear transformation $T: {\\mathbb R}^2 \\rightarrow {\\mathbb R}^3$ by $T(\\vec{x})=A\\vec{x}$. Find a vector $\\vec{x}$ whose image under $T$ is $\\vec{b}$.\n$\\vec{x}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\end{array}. Is the vector $\\vec{x}$ unique? [ANS]", "answer_v3": [ "-5", "1" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "unique", "not unique" ] ] }, { "id": "Linear_algebra_0501", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Kernel and image", "level": "4", "keywords": [ "matrix", "basis", "kernel", "linear", "transformation" ], "problem_v1": "Let A=\\left[\\begin{array}{cccc} 1 &3 &1 &1\\cr 0 &1 &2 &-2\\cr-1 &-2 &1 &-3 \\end{array}\\right]. Find a pair of vectors $\\vec{u}, \\vec{v}$ in ${\\mathbb R}^4$ that span the set of all $\\vec{x}\\in {\\mathbb R}^4$ that are mapped into the zero vector by the transformation $\\vec{x} \\mapsto A\\vec{x}$.\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v1": [ "-7", "2", "0", "1", "5", "-2", "1", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{cccc} 1 &-5 &5 &-4\\cr 0 &1 &-2 &5\\cr-1 &3 &-1 &-6 \\end{array}\\right]. Find a pair of vectors $\\vec{u}, \\vec{v}$ in ${\\mathbb R}^4$ that span the set of all $\\vec{x}\\in {\\mathbb R}^4$ that are mapped into the zero vector by the transformation $\\vec{x} \\mapsto A\\vec{x}$.\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v2": [ "-21", "-5", "0", "1", "5", "2", "1", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{cccc} 1 &-2 &1 &-2\\cr 0 &1 &1 &-3\\cr-1 &4 &1 &-4 \\end{array}\\right]. Find a pair of vectors $\\vec{u}, \\vec{v}$ in ${\\mathbb R}^4$ that span the set of all $\\vec{x}\\in {\\mathbb R}^4$ that are mapped into the zero vector by the transformation $\\vec{x} \\mapsto A\\vec{x}$.\n$\\vec{u}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}, $\\vec{v}=$ \\begin {array}{c} [ANS] \\\\ [ANS] \\\\ [ANS] \\\\ [ANS] \\end{array}.", "answer_v3": [ "8", "3", "0", "1", "-3", "-1", "1", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0503", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Inverses", "level": "3", "keywords": [], "problem_v1": "Select all of the linear transformations from ${\\mathbb R}^3$ to ${\\mathbb R}^3$ that are invertible. There may be more than one correct answer. [ANS] A\\. Reflection in the $z$-axis B\\. Dilation by a factor of $5$ C\\. Identity transformation (i.e. $T(\\vec{v})=\\vec{v}$ for all $\\vec{v}$) D\\. Trivial transformation (i.e. $T(\\vec{v})=\\vec{0}$ for all $\\vec{v}$) E\\. Projection onto the $xz$-plane F\\. Rotation about the $x$-axis", "answer_v1": [ "ABCF" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Select all of the linear transformations from ${\\mathbb R}^3$ to ${\\mathbb R}^3$ that are invertible. There may be more than one correct answer. [ANS] A\\. Trivial transformation (i.e. $T(\\vec{v})=\\vec{0}$ for all $\\vec{v}$) B\\. Rotation about the $z$-axis C\\. Dilation by a factor of $3$ D\\. Projection onto the $x$-axis E\\. Reflection in the $y$-axis F\\. Identity transformation (i.e. $T(\\vec{v})=\\vec{v}$ for all $\\vec{v}$)", "answer_v2": [ "BCEF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Select all of the linear transformations from ${\\mathbb R}^3$ to ${\\mathbb R}^3$ that are invertible. There may be more than one correct answer. [ANS] A\\. Reflection in the $xy$-plane B\\. Projection onto the $yz$-plane C\\. Identity transformation (i.e. $T(\\vec{v})=\\vec{v}$ for all $\\vec{v}$) D\\. Dilation by a factor of $7$ E\\. Projection onto the $z$-axis F\\. Rotation about the $x$-axis", "answer_v3": [ "ACDF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Linear_algebra_0504", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Inverses", "level": "3", "keywords": [], "problem_v1": "Find the inverse of the (nonlinear) transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ given by \\begin{array}{r@{}c@{}l} u &=& 8 y \\cr v &=& 6x^{7}-6 y \\end{array} $x=$ [ANS], $y=$ [ANS].", "answer_v1": [ "[(v+6*u/8)/6]^(1/7)", "u/8" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find the inverse of the (nonlinear) transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ given by \\begin{array}{r@{}c@{}l} u &=& 2 y \\cr v &=& 9x^{3}-4 y \\end{array} $x=$ [ANS], $y=$ [ANS].", "answer_v2": [ "[(v+4*u/2)/9]^(1/3)", "u/2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find the inverse of the (nonlinear) transformation from ${\\mathbb R}^2$ to ${\\mathbb R}^2$ given by \\begin{array}{r@{}c@{}l} u &=& 4 y \\cr v &=& 6x^{5}-6 y \\end{array} $x=$ [ANS], $y=$ [ANS].", "answer_v3": [ "[(v+6*u/4)/6]^(1/5)", "u/4" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0505", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Inverses", "level": "3", "keywords": [], "problem_v1": "Let $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be given by T(\\vec{x})=\\left[\\begin{array}{cc} 3 &1\\cr 1 &2 \\end{array}\\right] \\vec{x}. Find the matrix $M$ of the inverse linear transformation, $T^{-1}$.\n$M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "0.4", "-0.2", "-0.2", "0.6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be given by T(\\vec{x})=\\left[\\begin{array}{cc}-5 &5\\cr-4 &-2 \\end{array}\\right] \\vec{x}. Find the matrix $M$ of the inverse linear transformation, $T^{-1}$.\n$M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-0.0666667", "-0.166667", "0.133333", "-0.166667" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $T:{\\mathbb R}^2 \\rightarrow {\\mathbb R}^2$ be given by T(\\vec{x})=\\left[\\begin{array}{cc} 0 &1\\cr-2 &3 \\end{array}\\right] \\vec{x}. Find the matrix $M$ of the inverse linear transformation, $T^{-1}$.\n$M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "1.5", "-0.5", "1", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0506", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Inverses", "level": "4", "keywords": [], "problem_v1": "Let $T:P_3 \\rightarrow P_3$ be defined by T(ax^2+bx+c)=(5 a+b)x^2+(4 a-4 b+c) x-a. Find the inverse of $T$.\n$T^{-1}(ax^2+bx+c)=$ [ANS].", "answer_v1": [ "-c*x^2+(a+5*c)*x+-(-4)*a+b+(4--4*5)*c" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $T:P_3 \\rightarrow P_3$ be defined by T(ax^2+bx+c)=(2 a+b)x^2+(-5 a+3 b+c) x-a. Find the inverse of $T$.\n$T^{-1}(ax^2+bx+c)=$ [ANS].", "answer_v2": [ "-c*x^2+(a+2*c)*x+-3*a+b+(-5-3*2)*c" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $T:P_3 \\rightarrow P_3$ be defined by T(ax^2+bx+c)=(3 a+b)x^2+(-4 a-4 b+c) x-a. Find the inverse of $T$.\n$T^{-1}(ax^2+bx+c)=$ [ANS].", "answer_v3": [ "-c*x^2+(a+3*c)*x+-(-4)*a+b+(-4--4*3)*c" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0507", "subject": "Linear_algebra", "topic": "Linear transformations", "subtopic": "Inverses", "level": "3", "keywords": [ "matrix' 'inverse" ], "problem_v1": "a. The linear transformation $T_1: R^2 \\rightarrow R^2$ is given by:\n$T_1(x, y)=(2x+9 y, 8x+37 y)$.\nFind $T_1^{-1}(x, y)$.\n$T_1^{-1}(x, y)=($ [ANS] x+[ANS] y, [ANS] x+[ANS] y $)$\nb. The linear transformation $T_2: R^3 \\rightarrow R^3$ is given by:\n$T_2(x, y, z)=(x+2 z, 2x+y, 1 y+z)$.\nFind $T_2^{-1}(x, y, z)$.\n$T_2^{-1}(x, y, z)=($ [ANS] x+[ANS] y+[ANS] z, [ANS] x+[ANS] y+[ANS] z, [ANS] x+[ANS] y+[ANS] z $)$\nc. Using $T_1$ from part a, it is given that:\n$T_1(x, y)=(2,-3)$\nFind x and y.\n$x=$ [ANS] $y=$ [ANS]\nd. Using $T_2$ from part b, it is given that:\n$T_2(x, y, z)=(5,-5, 1)$\nFind x, y, and z.\n$x=$ [ANS] $y=$ [ANS] $z=$ [ANS]", "answer_v1": [ "18.5", "-4.5", "-4", "1", "0.2", "0.4", "-0.4", "-0.4", "0.2", "0.8", "0.4", "-0.2", "0.2", "50.5", "-11", "-1.4", "-2.2", "3.2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "a. The linear transformation $T_1: R^2 \\rightarrow R^2$ is given by:\n$T_1(x, y)=(2x+3 y, 10x+16 y)$.\nFind $T_1^{-1}(x, y)$.\n$T_1^{-1}(x, y)=($ [ANS] x+[ANS] y, [ANS] x+[ANS] y $)$\nb. The linear transformation $T_2: R^3 \\rightarrow R^3$ is given by:\n$T_2(x, y, z)=(x+1 z, 1x+y, 2 y+z)$.\nFind $T_2^{-1}(x, y, z)$.\n$T_2^{-1}(x, y, z)=($ [ANS] x+[ANS] y+[ANS] z, [ANS] x+[ANS] y+[ANS] z, [ANS] x+[ANS] y+[ANS] z $)$\nc. Using $T_1$ from part a, it is given that:\n$T_1(x, y)=(2,-5)$\nFind x and y.\n$x=$ [ANS] $y=$ [ANS]\nd. Using $T_2$ from part b, it is given that:\n$T_2(x, y, z)=(4,-4,-2)$\nFind x, y, and z.\n$x=$ [ANS] $y=$ [ANS] $z=$ [ANS]", "answer_v2": [ "8", "-1.5", "-5", "1", "0.333333333333333", "0.666666666666667", "-0.333333333333333", "-0.333333333333333", "0.333333333333333", "0.333333333333333", "0.666666666666667", "-0.666666666666667", "0.333333333333333", "23.5", "-15", "-0.666666666666667", "-3.33333333333333", "4.66666666666667" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "a. The linear transformation $T_1: R^2 \\rightarrow R^2$ is given by:\n$T_1(x, y)=(2x+5 y, 8x+21 y)$.\nFind $T_1^{-1}(x, y)$.\n$T_1^{-1}(x, y)=($ [ANS] x+[ANS] y, [ANS] x+[ANS] y $)$\nb. The linear transformation $T_2: R^3 \\rightarrow R^3$ is given by:\n$T_2(x, y, z)=(x+1 z, 2x+y, 1 y+z)$.\nFind $T_2^{-1}(x, y, z)$.\n$T_2^{-1}(x, y, z)=($ [ANS] x+[ANS] y+[ANS] z, [ANS] x+[ANS] y+[ANS] z, [ANS] x+[ANS] y+[ANS] z $)$\nc. Using $T_1$ from part a, it is given that:\n$T_1(x, y)=(2,-1)$\nFind x and y.\n$x=$ [ANS] $y=$ [ANS]\nd. Using $T_2$ from part b, it is given that:\n$T_2(x, y, z)=(6,-3,-1)$\nFind x, y, and z.\n$x=$ [ANS] $y=$ [ANS] $z=$ [ANS]", "answer_v3": [ "10.5", "-2.5", "-4", "1", "0.333333333333333", "0.333333333333333", "-0.333333333333333", "-0.666666666666667", "0.333333333333333", "0.666666666666667", "0.666666666666667", "-0.333333333333333", "0.333333333333333", "23.5", "-9", "1.33333333333333", "-5.66666666666667", "4.66666666666667" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0508", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate the 3x3 determinant: $\\left| \\begin{array}{ccc} 4 & 1 & 2 \\\\ 4 &-3 &-3 \\\\ 1 & 1 &-2 \\end{array} \\right|=$ [ANS]", "answer_v1": [ "55" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the 3x3 determinant: $\\left| \\begin{array}{ccc}-7 & 7 &-6 \\\\-3 & 8 &-3 \\\\-5 &-3 & 1 \\end{array} \\right|=$ [ANS]", "answer_v2": [ "-161" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the 3x3 determinant: $\\left| \\begin{array}{ccc}-3 & 2 &-4 \\\\ 1 &-5 &-3 \\\\ 5 & 7 & 6 \\end{array} \\right|=$ [ANS]", "answer_v3": [ "-143" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0509", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "2", "keywords": [ "algebra", "matrix operation", "matrix", "determinant" ], "problem_v1": "Given the matrix A=\\left[\\begin{array}{ccc} 3 &2 &2\\cr 3 &-1 &-1\\cr 2 &2 &0 \\end{array}\\right], find its determinant. The determinant is [ANS].", "answer_v1": [ "18" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given the matrix A=\\left[\\begin{array}{ccc}-3 &5 &-2\\cr 0 &5 &-1\\cr-2 &-1 &2 \\end{array}\\right], find its determinant. The determinant is [ANS].", "answer_v2": [ "-37" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given the matrix A=\\left[\\begin{array}{ccc}-1 &2 &-1\\cr 1 &-2 &0\\cr 4 &5 &4 \\end{array}\\right], find its determinant. The determinant is [ANS].", "answer_v3": [ "-13" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0510", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "2", "keywords": [ "algebra", "matrix operation", "matrix" ], "problem_v1": "If A=\\left[\\begin{array}{cc} 2+i &-2-2i \\cr-1 &-i \\cr \\end{array}\\right] find $|A|$=[ANS].", "answer_v1": [ "-1-4i" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If A=\\left[\\begin{array}{cc}-4+4i &4-2i \\cr 1-4i &3-3i \\cr \\end{array}\\right] find $|A|$=[ANS].", "answer_v2": [ "4+42i" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If A=\\left[\\begin{array}{cc}-2+i &-3-i \\cr 3-3i &-4+i \\cr \\end{array}\\right] find $|A|$=[ANS].", "answer_v3": [ "19-12i" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0511", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "", "keywords": [ "algebra", "matrix operation", "matrix" ], "problem_v1": "Consider the following matrix \\left[\\begin{array}{ccc} 3 &0 &1\\cr 0 &1 &2\\cr-2 &0 &-2 \\end{array}\\right].\n(a) Find its determinant. [ANS]\n(b) Does the matrix have an inverse? [ANS]", "answer_v1": [ "3*1*-2--2*1*1", "Yes" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [ "Yes", "No" ] ], "problem_v2": "Consider the following matrix \\left[\\begin{array}{ccc}-5 &0 &5\\cr 0 &-4 &-2\\cr 5 &0 &-2 \\end{array}\\right].\n(a) Find its determinant. [ANS]\n(b) Does the matrix have an inverse? [ANS]", "answer_v2": [ "-5*-4*-2-5*-4*5", "Yes" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [ "Yes", "No" ] ], "problem_v3": "Consider the following matrix \\left[\\begin{array}{ccc}-2 &0 &1\\cr 0 &-2 &1\\cr-3 &0 &-2 \\end{array}\\right].\n(a) Find its determinant. [ANS]\n(b) Does the matrix have an inverse? [ANS]", "answer_v3": [ "-2*-2*-2--3*-2*1", "Yes" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [ "Yes", "No" ] ] }, { "id": "Linear_algebra_0512", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "3", "keywords": [ "algebra", "matrix operation", "matrix", "determinant" ], "problem_v1": "Given the matrix A=\\left[\\begin{array}{ccc} a &7 &6\\cr a &-4 &7\\cr 3 &-3 &a\\cr \\end{array}\\right], find all values of $a$ that make $|A|=0$. Give your answer as a comma-separated list.\nValues of $a$: [ANS].", "answer_v1": [ "([-3+sqrt(3^2-4*-11*219)]/(2*-11), [-3-sqrt(3^2-4*-11*219)]/(2*-11))" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Given the matrix A=\\left[\\begin{array}{ccc} a &1 &9\\cr a &-8 &4\\cr 9 &-3 &a\\cr \\end{array}\\right], find all values of $a$ that make $|A|=0$. Give your answer as a comma-separated list.\nValues of $a$: [ANS].", "answer_v2": [ "([+15+sqrt((-15)^2-4*-9*684)]/(2*-9), [+15-sqrt((-15)^2-4*-9*684)]/(2*-9))" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Given the matrix A=\\left[\\begin{array}{ccc} a &3 &6\\cr a &-7 &5\\cr 2 &-3 &a\\cr \\end{array}\\right], find all values of $a$ that make $|A|=0$. Give your answer as a comma-separated list.\nValues of $a$: [ANS].", "answer_v3": [ "([+3+sqrt((-3)^2-4*-10*114)]/(2*-10), [+3-sqrt((-3)^2-4*-10*114)]/(2*-10))" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0513", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "2", "keywords": [ "matrix", "determinant" ], "problem_v1": "Find the determinant of the matrix M=\\left[\\begin{array}{ccccc} 2 &0 &0 &1 &0\\cr 1 &0 &2 &0 &0\\cr 0 &-1 &0 &0 &-1\\cr 0 &0 &0 &1 &-1\\cr 0 &1 &-2 &0 &0 \\end{array}\\right]. $\\det(M)=$ [ANS].", "answer_v1": [ "2*2*-1*1*1+1*1*-1*-1*-2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the matrix M=\\left[\\begin{array}{ccccc}-3 &0 &0 &3 &0\\cr-2 &0 &-1 &0 &0\\cr 0 &3 &0 &0 &-1\\cr 0 &0 &0 &-2 &-1\\cr 0 &-3 &1 &0 &0 \\end{array}\\right]. $\\det(M)=$ [ANS].", "answer_v2": [ "-3*-1*-1*-2*-3+3*-2*-1*3*1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the matrix M=\\left[\\begin{array}{ccccc}-1 &0 &0 &1 &0\\cr-2 &0 &-2 &0 &0\\cr 0 &-1 &0 &0 &2\\cr 0 &0 &0 &3 &3\\cr 0 &-2 &-1 &0 &0 \\end{array}\\right]. $\\det(M)=$ [ANS].", "answer_v3": [ "-1*-2*2*3*-2+1*-2*3*-1*-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0514", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "3", "keywords": [], "problem_v1": "Determine all minors and cofactors of A=\\left[\\begin{array}{cc} 5 &2\\cr 2 &4 \\end{array}\\right].\n$M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$C=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v1": [ "4", "2", "2", "5", "4", "-2", "-2", "5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Determine all minors and cofactors of A=\\left[\\begin{array}{cc}-8 &8\\cr-7 &-3 \\end{array}\\right].\n$M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$C=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v2": [ "-3", "-7", "8", "-8", "-3", "7", "-8", "-8" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Determine all minors and cofactors of A=\\left[\\begin{array}{cc}-4 &2\\cr-4 &1 \\end{array}\\right].\n$M=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}\n$C=$ \\begin {array}{cc} [ANS] & [ANS] \\\\ [ANS] & [ANS] \\end{array}", "answer_v3": [ "1", "-4", "2", "-4", "1", "4", "-2", "-4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0515", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "2", "keywords": [], "problem_v1": "Find the derivative of the function f(x)=\\det \\left[\\begin{array}{ccccc} 5 &2 &2 &4 &-4\\cr-3 &0 &1 &-2 &3\\cr-4 &0 &0 &-1 &-7\\cr x &-2 &5 &-1 &-3\\cr-1 &0 &0 &0 &-9\\cr \\end{array}\\right].\n$f'(x)=$ [ANS].", "answer_v1": [ "-2*1*-1*-9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of the function f(x)=\\det \\left[\\begin{array}{ccccc}-8 &8 &-7 &-3 &8\\cr-3 &0 &-3 &1 &-8\\cr 3 &0 &0 &-6 &-6\\cr x &1 &-6 &-3 &1\\cr-7 &0 &0 &0 &-8\\cr \\end{array}\\right].\n$f'(x)=$ [ANS].", "answer_v2": [ "-8*-3*-6*-8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of the function f(x)=\\det \\left[\\begin{array}{ccccc}-4 &2 &-4 &1 &-6\\cr-3 &0 &8 &7 &-6\\cr-4 &0 &0 &1 &9\\cr x &2 &-7 &-4 &3\\cr 4 &0 &0 &0 &7\\cr \\end{array}\\right].\n$f'(x)=$ [ANS].", "answer_v3": [ "-2*8*1*7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0516", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "3", "keywords": [], "problem_v1": "Find $k$ such that the following matrix $M$ is singular. M=\\left[\\begin{array}{ccc} 2 &1 &1\\cr 4 &1 &0\\cr-2+k &0 &1\\cr \\end{array}\\right] $k=$ [ANS]", "answer_v1": [ "0" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $k$ such that the following matrix $M$ is singular. M=\\left[\\begin{array}{ccc}-4 &5 &-3\\cr 4 &-1 &1\\cr 2+k &-9 &5\\cr \\end{array}\\right] $k=$ [ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $k$ such that the following matrix $M$ is singular. M=\\left[\\begin{array}{ccc}-2 &1 &-2\\cr 0 &-3 &-1\\cr-16+k &-4 &-6\\cr \\end{array}\\right] $k=$ [ANS]", "answer_v3": [ "12" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0517", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "3", "keywords": [ "algebra" ], "problem_v1": "The determinant of the matrix A=\\left[\\begin{array}{rrr} 5 & 2 & 2 \\\\ 4 &-4 &-3 \\\\ 1 & 1 &-2 \\\\ \\end{array}\\right] is [ANS], and its inverse is A^{-1}=\\left[\\begin{array}{rrr} b_{11} & b_{12} & b_{13} \\\\ b_{21} & b_{22} & b_{23} \\\\ b_{31} & b_{32} & b_{33} \\\\ \\end{array}\\right], where $b_{11}$=[ANS], $b_{12}$=[ANS], $b_{13}$=[ANS], $b_{21}$=[ANS], $b_{22}$=[ANS], $b_{23}$=[ANS], $b_{31}$=[ANS], $b_{32}$=[ANS], $b_{33}$=[ANS].", "answer_v1": [ "81", "0.135802469135802", "0.0740740740740741", "0.0246913580246914", "0.0617283950617284", "-0.148148148148148", "0.283950617283951", "0.0987654320987654", "-0.037037037037037", "-0.345679012345679" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "The determinant of the matrix A=\\left[\\begin{array}{rrr}-8 & 8 &-7 \\\\-3 & 8 &-3 \\\\-6 &-3 & 1 \\\\ \\end{array}\\right] is [ANS], and its inverse is A^{-1}=\\left[\\begin{array}{rrr} b_{11} & b_{12} & b_{13} \\\\ b_{21} & b_{22} & b_{23} \\\\ b_{31} & b_{32} & b_{33} \\\\ \\end{array}\\right], where $b_{11}$=[ANS], $b_{12}$=[ANS], $b_{13}$=[ANS], $b_{21}$=[ANS], $b_{22}$=[ANS], $b_{23}$=[ANS], $b_{31}$=[ANS], $b_{32}$=[ANS], $b_{33}$=[ANS].", "answer_v2": [ "-223", "0.00448430493273543", "-0.0582959641255605", "-0.143497757847534", "-0.0941704035874439", "0.224215246636771", "0.0134529147982063", "-0.255605381165919", "0.322869955156951", "0.179372197309417" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "The determinant of the matrix A=\\left[\\begin{array}{rrr}-4 & 2 &-4 \\\\ 1 &-6 &-3 \\\\ 6 & 8 & 7 \\\\ \\end{array}\\right] is [ANS], and its inverse is A^{-1}=\\left[\\begin{array}{rrr} b_{11} & b_{12} & b_{13} \\\\ b_{21} & b_{22} & b_{23} \\\\ b_{31} & b_{32} & b_{33} \\\\ \\end{array}\\right], where $b_{11}$=[ANS], $b_{12}$=[ANS], $b_{13}$=[ANS], $b_{21}$=[ANS], $b_{22}$=[ANS], $b_{23}$=[ANS], $b_{31}$=[ANS], $b_{32}$=[ANS], $b_{33}$=[ANS].", "answer_v3": [ "-154", "0.116883116883117", "0.298701298701299", "0.194805194805195", "0.162337662337662", "0.025974025974026", "0.103896103896104", "-0.285714285714286", "-0.285714285714286", "-0.142857142857143" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0518", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Let A=\\left[\\begin{array}{rrr} 5 & 2 & 2 \\\\ 4 &-4 &-3 \\\\ \\end{array}\\right], \\quad B=\\left[\\begin{array}{rr} 1 & 1 \\\\-2 & 0 \\\\ 3 &-4 \\\\ \\end{array}\\right]. Then C=AB=\\left[\\begin{array}{rrr} c_{11} & c_{12} \\\\ c_{21} & c_{22} \\\\ \\end{array}\\right], and D=BA=\\left[\\begin{array}{rrr} d_{11} & d_{12} & d_{13} \\\\ d_{21} & d_{22} & d_{23} \\\\ d_{31} & d_{32} & d_{33} \\\\ \\end{array}\\right], where $c_{11}$=[ANS], $c_{12}$=[ANS], $c_{21}$=[ANS], $c_{22}$=[ANS], and $d_{11}$=[ANS], $d_{12}$=[ANS], $d_{13}$=[ANS], $d_{21}$=[ANS], $d_{22}$=[ANS], $d_{23}$=[ANS], $d_{31}$=[ANS], $d_{32}$=[ANS], $d_{33}$=[ANS]. The determinant of $D$ is [ANS].", "answer_v1": [ "7", "-3", "3", "16", "9", "-2", "-1", "-10", "-4", "-4", "-1", "22", "18", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let A=\\left[\\begin{array}{rrr}-8 & 8 &-7 \\\\-3 & 8 &-3 \\\\ \\end{array}\\right], \\quad B=\\left[\\begin{array}{rr}-6 &-3 \\\\ 1 &-8 \\\\ 3 &-1 \\\\ \\end{array}\\right]. Then C=AB=\\left[\\begin{array}{rrr} c_{11} & c_{12} \\\\ c_{21} & c_{22} \\\\ \\end{array}\\right], and D=BA=\\left[\\begin{array}{rrr} d_{11} & d_{12} & d_{13} \\\\ d_{21} & d_{22} & d_{23} \\\\ d_{31} & d_{32} & d_{33} \\\\ \\end{array}\\right], where $c_{11}$=[ANS], $c_{12}$=[ANS], $c_{21}$=[ANS], $c_{22}$=[ANS], and $d_{11}$=[ANS], $d_{12}$=[ANS], $d_{13}$=[ANS], $d_{21}$=[ANS], $d_{22}$=[ANS], $d_{23}$=[ANS], $d_{31}$=[ANS], $d_{32}$=[ANS], $d_{33}$=[ANS]. The determinant of $D$ is [ANS].", "answer_v2": [ "35", "-33", "17", "-52", "57", "-72", "51", "16", "-56", "17", "-21", "16", "-18", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let A=\\left[\\begin{array}{rrr}-4 & 2 &-4 \\\\ 1 &-6 &-3 \\\\ \\end{array}\\right], \\quad B=\\left[\\begin{array}{rr} 6 & 8 \\\\ 7 &-6 \\\\-4 &-5 \\\\ \\end{array}\\right]. Then C=AB=\\left[\\begin{array}{rrr} c_{11} & c_{12} \\\\ c_{21} & c_{22} \\\\ \\end{array}\\right], and D=BA=\\left[\\begin{array}{rrr} d_{11} & d_{12} & d_{13} \\\\ d_{21} & d_{22} & d_{23} \\\\ d_{31} & d_{32} & d_{33} \\\\ \\end{array}\\right], where $c_{11}$=[ANS], $c_{12}$=[ANS], $c_{21}$=[ANS], $c_{22}$=[ANS], and $d_{11}$=[ANS], $d_{12}$=[ANS], $d_{13}$=[ANS], $d_{21}$=[ANS], $d_{22}$=[ANS], $d_{23}$=[ANS], $d_{31}$=[ANS], $d_{32}$=[ANS], $d_{33}$=[ANS]. The determinant of $D$ is [ANS].", "answer_v3": [ "6", "-24", "-24", "59", "-16", "-36", "-48", "-34", "50", "-10", "11", "22", "31", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0519", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Computing determinants", "level": "2", "keywords": [ "algebra", "linear algebra", "matrices", "determinants" ], "problem_v1": "Find the determinant of the following $2 \\times 2$ matrix. \\begin{bmatrix} \\frac{7}{8} & \\frac{1}{5} \\thinspace \\\\ - \\frac{1}{4} & \\frac{6}{7} \\thinspace \\end{bmatrix} Answer: [ANS]", "answer_v1": [ "7/8*6/7+1/20" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the following $2 \\times 2$ matrix. \\begin{bmatrix} \\frac{1}{2} & \\frac{1}{5} \\thinspace \\\\ - \\frac{1}{4} & \\frac{10}{11} \\thinspace \\end{bmatrix} Answer: [ANS]", "answer_v2": [ "1/2*10/11+1/20" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the following $2 \\times 2$ matrix. \\begin{bmatrix} \\frac{3}{4} & \\frac{1}{5} \\thinspace \\\\ - \\frac{1}{4} & \\frac{7}{8} \\thinspace \\end{bmatrix} Answer: [ANS]", "answer_v3": [ "3/4*7/8+1/20" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0520", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "3", "keywords": [ "matrix", "determinant" ], "problem_v1": "If \\det \\left[\\begin{array}{ccc} a &1 &d\\cr b &1 &e\\cr c &1 &f\\cr \\end{array}\\right]=3, \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\det \\left[\\begin{array}{ccc} a &1 &d\\cr b &2 &e\\cr c &3 &f\\cr \\end{array}\\right]=1, then\n$\\det \\left[\\begin{array}{ccc} a &6 &d\\cr b &6 &e\\cr c &6 &f\\cr \\end{array}\\right]=$ [ANS] and\n$\\det \\left[\\begin{array}{ccc} a &1 &d\\cr b &0 &e\\cr c &-1 &f\\cr \\end{array}\\right]=$ [ANS].", "answer_v1": [ "18", "5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If \\det \\left[\\begin{array}{ccc} a &1 &d\\cr b &1 &e\\cr c &1 &f\\cr \\end{array}\\right]=-5, \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\det \\left[\\begin{array}{ccc} a &1 &d\\cr b &2 &e\\cr c &3 &f\\cr \\end{array}\\right]=5, then\n$\\det \\left[\\begin{array}{ccc} a &3 &d\\cr b &3 &e\\cr c &3 &f\\cr \\end{array}\\right]=$ [ANS] and\n$\\det \\left[\\begin{array}{ccc} a &2 &d\\cr b &5 &e\\cr c &8 &f\\cr \\end{array}\\right]=$ [ANS].", "answer_v2": [ "-15", "20" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If \\det \\left[\\begin{array}{ccc} a &1 &d\\cr b &1 &e\\cr c &1 &f\\cr \\end{array}\\right]=-2, \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\det \\left[\\begin{array}{ccc} a &1 &d\\cr b &2 &e\\cr c &3 &f\\cr \\end{array}\\right]=1, then\n$\\det \\left[\\begin{array}{ccc} a &4 &d\\cr b &4 &e\\cr c &4 &f\\cr \\end{array}\\right]=$ [ANS] and\n$\\det \\left[\\begin{array}{ccc} a &-3 &d\\cr b &-4 &e\\cr c &-5 &f\\cr \\end{array}\\right]=$ [ANS].", "answer_v3": [ "-8", "3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0521", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "3", "keywords": [ "matrix", "determinant" ], "problem_v1": "Find the determinant of the $n \\times n$ matrix $A$ with $8$ 's on the diagonal, $1$ 's above the diagonal, and $0$ 's below the diagonal.\n$\\det(A)=$ [ANS].", "answer_v1": [ "8^n" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the $n \\times n$ matrix $A$ with $2$ 's on the diagonal, $1$ 's above the diagonal, and $0$ 's below the diagonal.\n$\\det(A)=$ [ANS].", "answer_v2": [ "2^n" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the $n \\times n$ matrix $A$ with $4$ 's on the diagonal, $1$ 's above the diagonal, and $0$ 's below the diagonal.\n$\\det(A)=$ [ANS].", "answer_v3": [ "4^n" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0522", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "4", "keywords": [ "matrix", "determinant" ], "problem_v1": "If the determinant of a $5 \\times 5$ matrix $A$ is $\\det(A)=7$, and the matrix $B$ is obtained from $A$ by multiplying the third row by $6$, then\n$\\det(B)=$ [ANS].", "answer_v1": [ "42" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If the determinant of a $3 \\times 3$ matrix $A$ is $\\det(A)=10$, and the matrix $B$ is obtained from $A$ by multiplying the second column by $3$, then\n$\\det(B)=$ [ANS].", "answer_v2": [ "30" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If the determinant of a $3 \\times 3$ matrix $A$ is $\\det(A)=7$, and the matrix $B$ is obtained from $A$ by multiplying the second row by $4$, then\n$\\det(B)=$ [ANS].", "answer_v3": [ "28" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0523", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "3", "keywords": [ "algebra" ], "problem_v1": "The determinant of the matrix A=\\left[\\begin{array}{rrrrr} 0 &-2 & 4 & 0 & 0 \\\\ 2 &-1 & 3 & 0 & 0 \\\\ 0 &-4 & 0 & 0 & 0 \\\\ 1 &-4 &-2 & 5 &-3 \\\\ 1 & 0 & 0 & 0 & 2 \\\\ \\end{array}\\right] is [ANS]. Hint: Find a good row or column and expand by minors.", "answer_v1": [ "-320" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The determinant of the matrix A=\\left[\\begin{array}{rrrrr} 0 &-6 &-3 & 0 & 0 \\\\-7 &-6 & 3 & 0 & 0 \\\\ 0 & 8 & 0 & 0 & 0 \\\\-6 &-1 & 1 &-8 &-3 \\\\-3 & 6 &-8 & 0 & 8 \\\\ \\end{array}\\right] is [ANS]. Hint: Find a good row or column and expand by minors.", "answer_v2": [ "-10752" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The determinant of the matrix A=\\left[\\begin{array}{rrrrr} 0 & 9 & 1 & 0 & 0 \\\\-4 & 1 &-4 & 0 & 0 \\\\ 0 &-6 & 0 & 0 & 0 \\\\ 6 &-5 & 7 &-4 &-3 \\\\ 8 &-9 &-6 & 0 & 2 \\\\ \\end{array}\\right] is [ANS]. Hint: Find a good row or column and expand by minors.", "answer_v3": [ "-192" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0524", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "2", "keywords": [ "determinant" ], "problem_v1": "Find the determinant of the matrix A=\\left [\\matrix {5 & 6 & & \\cdots & 6 \\\\ 6 & 5 & 6 & \\cdots & 6 \\\\ & & \\ddots & & \\\\ 6 & \\cdots & 6 & 5 & 6 \\\\ 6 & \\cdots & & 6 & 5 \\\\}\\right] \\in \\mathbb{R}^{79\\times 79}. $\\det(A)=$ [ANS]", "answer_v1": [ "473" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the determinant of the matrix A=\\left [\\matrix {-8 &-7 & & \\cdots &-7 \\\\-7 &-8 &-7 & \\cdots &-7 \\\\ & & \\ddots & & \\\\-7 & \\cdots &-7 &-8 &-7 \\\\-7 & \\cdots & &-7 &-8 \\\\}\\right] \\in \\mathbb{R}^{97\\times 97}. $\\det(A)=$ [ANS]", "answer_v2": [ "-680" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the determinant of the matrix A=\\left [\\matrix {-4 &-3 & & \\cdots &-3 \\\\-3 &-4 &-3 & \\cdots &-3 \\\\ & & \\ddots & & \\\\-3 & \\cdots &-3 &-4 &-3 \\\\-3 & \\cdots & &-3 &-4 \\\\}\\right] \\in \\mathbb{R}^{80\\times 80}. $\\det(A)=$ [ANS]", "answer_v3": [ "241" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0525", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "2", "keywords": [ "linear algebra", "elementery matrix", "determinant" ], "problem_v1": "Consider the following Gauss elimination:\nA \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &0 &1\\cr 0 &1 &0 \\end{array}\\right]}_{E_1} A \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &5 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_2} E_1 A \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &2 \\end{array}\\right]}_{E_3} E_2E_1 A \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &2\\cr 0 &1 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_4} E_3E_2E_1 A=\\left[\\begin{array}{ccc}-2 &3 &-4\\cr 0 &-1 &-2\\cr 0 &0 &-1 \\end{array}\\right] What is the determinant of $A$? $\\det(A)=$ [ANS]", "answer_v1": [ "0.2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the following Gauss elimination:\nA \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &-7 \\end{array}\\right]}_{E_1} A \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &-8 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_2} E_1 A \\to \\underbrace{\\left[\\begin{array}{ccc} 0 &1 &0\\cr 1 &0 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_3} E_2E_1 A \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &8\\cr 0 &1 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_4} E_3E_2E_1 A=\\left[\\begin{array}{ccc} 1 &-8 &3\\cr 0 &-1 &6\\cr 0 &0 &-6 \\end{array}\\right] What is the determinant of $A$? $\\det(A)=$ [ANS]", "answer_v2": [ "-0.107143" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the following Gauss elimination:\nA \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &-4 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_1} A \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &2\\cr 0 &1 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_2} E_1 A \\to \\underbrace{\\left[\\begin{array}{ccc} 1 &0 &0\\cr 0 &1 &0\\cr 0 &0 &-4 \\end{array}\\right]}_{E_3} E_2E_1 A \\to \\underbrace{\\left[\\begin{array}{ccc} 0 &1 &0\\cr 1 &0 &0\\cr 0 &0 &1 \\end{array}\\right]}_{E_4} E_3E_2E_1 A=\\left[\\begin{array}{ccc} 7 &-6 &-4\\cr 0 &-5 &-9\\cr 0 &0 &1 \\end{array}\\right] What is the determinant of $A$? $\\det(A)=$ [ANS]", "answer_v3": [ "2.1875" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0526", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "2", "keywords": [ "determinants", "matrix similarity", "similar", "matrix" ], "problem_v1": "Consider the matrix\nA=\\left[\\begin{array}{ccc} 5-x &-401 &-15 \\pi \\\\ 0 & 4-x &-4\\\\ 0 & 1 & 0-x \\end{array} \\right] and let $B$ be a matrix similar to $A$, i.e., $B$ is of the form $S^{-1} A S$ for some nonsingular matrix $S$. Find all possible values of $x$ so that $\\det(B)=0$. Separate multiple values by a comma. Answer: [ANS]", "answer_v1": [ "(5, 2)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Consider the matrix\nA=\\left[\\begin{array}{ccc}-8-x &-457 &-15 \\pi \\\\ 0 &-8-x & 8\\\\ 0 &-4 & 4-x \\end{array} \\right] and let $B$ be a matrix similar to $A$, i.e., $B$ is of the form $S^{-1} A S$ for some nonsingular matrix $S$. Find all possible values of $x$ so that $\\det(B)=0$. Separate multiple values by a comma. Answer: [ANS]", "answer_v2": [ "(-8, -4, 0)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Consider the matrix\nA=\\left[\\begin{array}{ccc}-4-x &-771 &-15 \\pi \\\\ 0 & 3-x &-1\\\\ 0 &-18 &-4-x \\end{array} \\right] and let $B$ be a matrix similar to $A$, i.e., $B$ is of the form $S^{-1} A S$ for some nonsingular matrix $S$. Find all possible values of $x$ so that $\\det(B)=0$. Separate multiple values by a comma. Answer: [ANS]", "answer_v3": [ "(-4, 5, -6)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0527", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "3", "keywords": [ "algebra", "linear algebra", "matrices", "determinants" ], "problem_v1": "Evaluate the following $3 \\times 3$ determinant. Use the properties of determinants to your advantage. \\begin{vmatrix} 3 & 1 & 1 \\\\ 4 & 2 &-1 \\\\-2 & 2 & 5 \\end{vmatrix} Answer: [ANS]", "answer_v1": [ "30" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the following $3 \\times 3$ determinant. Use the properties of determinants to your advantage. \\begin{vmatrix}-5 & 4 &-2 \\\\ 2 & 4 &-1 \\\\-4 & 2 & 5 \\end{vmatrix} Answer: [ANS]", "answer_v2": [ "-174" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the following $3 \\times 3$ determinant. Use the properties of determinants to your advantage. \\begin{vmatrix}-2 & 1 &-2 \\\\ 3 & 2 & 1 \\\\-1 & 2 & 5 \\end{vmatrix} Answer: [ANS]", "answer_v3": [ "-48" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0528", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Properties", "level": "3", "keywords": [ "matrix' 'determinant' 'invertible" ], "problem_v1": "If $A=\\left[\\begin{array}{cc} 5 &2\\cr 4 &2 \\end{array}\\right]$ then $\\det(A)$=[ANS]\nIs $A$ invertible? [ANS] [ANS]", "answer_v1": [ "2", "Yes" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [] ], "problem_v2": "If $A=\\left[\\begin{array}{cc}-8 &-7\\cr-3 &8 \\end{array}\\right]$ then $\\det(A)$=[ANS]\nIs $A$ invertible? [ANS] [ANS]", "answer_v2": [ "-85", "Yes" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [] ], "problem_v3": "If $A=\\left[\\begin{array}{cc}-4 &-4\\cr 1 &2 \\end{array}\\right]$ then $\\det(A)$=[ANS]\nIs $A$ invertible? [ANS] [ANS]", "answer_v3": [ "-4", "Yes" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [] ] }, { "id": "Linear_algebra_0529", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Applications", "level": "3", "keywords": [], "problem_v1": "Let $A=\\left[\\begin{array}{ccc} 2 &0 &1\\cr 2 &-1 &-1\\cr 1 &0 &-1 \\end{array}\\right]$.\n(a) Find the determinant of $A$. ${\\rm det} (A)=$ [ANS],\n(b) Find the matrix of cofactors of $A$. $C=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(c) Find the adjoint of $A$. ${\\rm adj}(A)=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(d) Find the inverse of $A$. $A^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v1": [ "1", "1", "1", "0", "-3", "0", "1", "4", "-2", "1", "0", "1", "1", "-3", "4", "1", "0", "-2", "0.333333", "0", "0.333333", "0.333333", "-1", "1.33333", "0.333333", "0", "-0.666667" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $A=\\left[\\begin{array}{ccc}-2 &2 &-3\\cr-1 &3 &-1\\cr-3 &-2 &0 \\end{array}\\right]$.\n(a) Find the determinant of $A$. ${\\rm det} (A)=$ [ANS],\n(b) Find the matrix of cofactors of $A$. $C=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(c) Find the adjoint of $A$. ${\\rm adj}(A)=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(d) Find the inverse of $A$. $A^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v2": [ "-2", "3", "11", "6", "-9", "-10", "7", "1", "-4", "-2", "6", "7", "3", "-9", "1", "11", "-10", "-4", "0.0869565", "-0.26087", "-0.304348", "-0.130435", "0.391304", "-0.0434783", "-0.478261", "0.434783", "0.173913" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $A=\\left[\\begin{array}{ccc}-2 &0 &-1\\cr 0 &-3 &-1\\cr 3 &2 &3 \\end{array}\\right]$.\n(a) Find the determinant of $A$. ${\\rm det} (A)=$ [ANS],\n(b) Find the matrix of cofactors of $A$. $C=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(c) Find the adjoint of $A$. ${\\rm adj}(A)=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}\n(d) Find the inverse of $A$. $A^{-1}=$ \\begin {array}{ccc} [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\\\ [ANS] & [ANS] & [ANS] \\end{array}", "answer_v3": [ "-7", "-3", "9", "-2", "-3", "4", "-3", "-2", "6", "-7", "-2", "-3", "-3", "-3", "-2", "9", "4", "6", "-1.4", "-0.4", "-0.6", "-0.6", "-0.6", "-0.4", "1.8", "0.8", "1.2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Linear_algebra_0530", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Applications", "level": "3", "keywords": [ "algebra", "Cramer's rule", "solving system of equations", "System of Equations", "Matrix" ], "problem_v1": "Use Cramer's rule to find the value of $y$ in the solution of the following system: \\begin{array}{r} 3x+y+z=6 \\\\[1ex] 2x-2y-2z=36 \\\\[1ex] x+y-z=0 \\end{array}\n$y=$ [ANS]", "answer_v1": [ "-9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Cramer's rule to find the value of $z$ in the solution of the following system: \\begin{array}{r}-5x+5y-4z=128 \\\\[1ex]-2x+5y-2z=70 \\\\[1ex]-3x-2y+z=40 \\end{array}\n$z=$ [ANS]", "answer_v2": [ "-2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Cramer's rule to find the value of $x$ in the solution of the following system: \\begin{array}{r}-2x+y-2z=35 \\\\[1ex] x-3y-2z=35 \\\\[1ex] 3x+5y+4z=-121 \\end{array}\n$x=$ [ANS]", "answer_v3": [ "-12" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0531", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Applications", "level": "3", "keywords": [ "algebra", "Cramer's rule", "solving system of equations", "Matrix" ], "problem_v1": "Use Cramer's rule to solve the system \\begin{array}{l} 2x_1+3x_2-5x_3=-3 \\\\ x_1+x_2-x_3=0 \\\\ 2x_2+x_3=1 \\\\ \\end{array} $x_1=$ [ANS], $x_2=$ [ANS], and $x_3=$ [ANS]", "answer_v1": [ "1", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Use Cramer's rule to solve the system \\begin{array}{l} 2x_1+3x_2-5x_3=12 \\\\ x_1+x_2-x_3=2 \\\\ 2x_2+x_3=2 \\\\ \\end{array} $x_1=$ [ANS], $x_2=$ [ANS], and $x_3=$ [ANS]", "answer_v2": [ "-2", "2", "-2" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Use Cramer's rule to solve the system \\begin{array}{l} 2x_1+3x_2-5x_3=6 \\\\ x_1+x_2-x_3=1 \\\\ 2x_2+x_3=1 \\\\ \\end{array} $x_1=$ [ANS], $x_2=$ [ANS], and $x_3=$ [ANS]", "answer_v3": [ "-1", "1", "-1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Linear_algebra_0532", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Applications", "level": "2", "keywords": [], "problem_v1": "Find the volume of the region defined by the vectors. $\\langle \\left[\\begin{array}{c} 5\\cr 4\\cr 1 \\end{array}\\right], \\left[\\begin{array}{c} 2\\cr-4\\cr 1 \\end{array}\\right], \\left[\\begin{array}{c} 2\\cr-3\\cr-2 \\end{array}\\right] \\rangle$ Volume=[ANS]", "answer_v1": [ "81" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the region defined by the vectors. $\\langle \\left[\\begin{array}{c}-8\\cr-3\\cr-6 \\end{array}\\right], \\left[\\begin{array}{c} 8\\cr 8\\cr-3 \\end{array}\\right], \\left[\\begin{array}{c}-7\\cr-3\\cr 1 \\end{array}\\right] \\rangle$ Volume=[ANS]", "answer_v2": [ "223" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the region defined by the vectors. $\\langle \\left[\\begin{array}{c}-4\\cr 1\\cr 6 \\end{array}\\right], \\left[\\begin{array}{c} 2\\cr-6\\cr 8 \\end{array}\\right], \\left[\\begin{array}{c}-4\\cr-3\\cr 7 \\end{array}\\right] \\rangle$ Volume=[ANS]", "answer_v3": [ "154" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0533", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations' 'system' 'cramer" ], "problem_v1": "Let\nA=\\left[\\begin{array}{rrr} 3 & 12 & 26\\\\ 1 & 7 & 18\\\\ 5 & 20 & 45 \\end{array} \\right]\n(a) Compute $\\det(A)=$ [ANS]\n(b) Use Cramer's rule to solve the following system\n\\left\\lbrace \\begin{array}{rrrrrrrr}& 3 \\,x_1&+& 12 \\, x_2&+& 26 \\, x_3&=&-1\\\\& \\,x_1&+& 7 \\, x_2&+& 18 \\, x_3&=&2\\\\& 5 \\,x_1&+& 20 \\, x_2&+& 45 \\, x_3&=&-3\\\\ \\end{array} \\right. $x_1=$ [ANS]\n$x_2=$ [ANS]\n$x_3=$ [ANS]", "answer_v1": [ "15", "-6.46666666666667", "3.26666666666667", "-0.8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let\nA=\\left[\\begin{array}{rrr}-5 &-10 & 4\\\\ 6 & 17 & 4\\\\ 3 & 6 &-3 \\end{array} \\right]\n(a) Compute $\\det(A)=$ [ANS]\n(b) Use Cramer's rule to solve the following system\n\\left\\lbrace \\begin{array}{rrrrrrrr}&-5 \\, x_1&-&10 \\,x_2&+& 4 \\, x_3&=&1\\\\& 6 \\,x_1&+& 17 \\, x_2&+& 4 \\, x_3&=&-5\\\\& 3 \\,x_1&+& 6 \\, x_2&-&3 \\,x_3&=&2\\\\ \\end{array} \\right. $x_1=$ [ANS]\n$x_2=$ [ANS]\n$x_3=$ [ANS]", "answer_v2": [ "15", "-17.4", "6.86666666666667", "-4.33333333333333" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let\nA=\\left[\\begin{array}{rrr}-2 &-6 &-29\\\\ 1 & 6 & 29\\\\ 4 & 12 & 56 \\end{array} \\right]\n(a) Compute $\\det(A)=$ [ANS]\n(b) Use Cramer's rule to solve the following system\n\\left\\lbrace \\begin{array}{rrrrrrrr}&-2 \\, x_1&-&6 \\,x_2&-&29 \\,x_3&=&4\\\\& \\,x_1&+& 6 \\, x_2&+& 29 \\, x_3&=&-3\\\\& 4 \\,x_1&+& 12 \\, x_2&+& 56 \\, x_3&=&-2\\\\ \\end{array} \\right. $x_1=$ [ANS]\n$x_2=$ [ANS]\n$x_3=$ [ANS]", "answer_v3": [ "12", "-1", "14.1666666666667", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0534", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Applications", "level": "3", "keywords": [ "linear equations' 'system' 'cramer" ], "problem_v1": "Solve the system using Cramer's Rule.\n$\\begin{array}{rrrrrrrr}& 5 \\,x&+& 20 \\, y&+& 44 \\, z&=&-1\\\\& 2 \\,x&+& 11 \\, y&+& 27 \\, z&=&2\\\\& 5 \\,x&+& 20 \\, y&+& 45 \\, z&=&-3\\\\ \\end{array}$\n$det=$ [ANS]\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v1": [ "15", "-10.8666666666667", "7.06666666666667", "-2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Solve the system using Cramer's Rule.\n$\\begin{array}{rrrrrrrr}&-8 \\, x&-&16 \\,y&+& 7 \\, z&=&1\\\\& 8 \\,x&+& 21 \\, y&+& 2 \\, z&=&-5\\\\& 3 \\,x&+& 6 \\, y&-&3 \\,z&=&2\\\\ \\end{array}$\n$det=$ [ANS]\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v2": [ "15", "-26.8666666666667", "10.6", "-6.33333333333333" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Solve the system using Cramer's Rule.\n$\\begin{array}{rrrrrrrr}&-4 \\, x&-&12 \\,y&-&57 \\,z&=&4\\\\& 2 \\,x&+& 9 \\, y&+& 43 \\, z&=&-3\\\\& 4 \\,x&+& 12 \\, y&+& 56 \\, z&=&-2\\\\ \\end{array}$\n$det=$ [ANS]\n$x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v3": [ "12", "-0.5", "9.33333333333333", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Linear_algebra_0535", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Applications", "level": "3", "keywords": [ "matrix", "determinant", "area", "triangle" ], "problem_v1": "Find the area of the triangle with vertices $(3, 1)$, $(10, 3)$, and $(2, 6)$.\nArea=[ANS].", "answer_v1": [ "18.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the triangle with vertices $(-5, 5)$, $(-1, 4)$, and $(-2, 10)$.\nArea=[ANS].", "answer_v2": [ "11.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the triangle with vertices $(-2, 1)$, $(3,-1)$, and $(-3, 9)$.\nArea=[ANS].", "answer_v3": [ "19" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Linear_algebra_0536", "subject": "Linear_algebra", "topic": "Determinants", "subtopic": "Applications", "level": "3", "keywords": [ "matrix", "determinant", "area", "parallelogram" ], "problem_v1": "Find the area of the parallelogram with vertices at $(-2,-2),$ $(4, 0),$ $(1, 4),$ and $(7, 6).$\nArea=[ANS].", "answer_v1": [ "30" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the parallelogram with vertices at $(5,-2),$ $(-5, 9),$ $(-4,-6),$ and $(-14, 5).$\nArea=[ANS].", "answer_v2": [ "139" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the parallelogram with vertices at $(-3,-2),$ $(-8, 1),$ $(-9,-1),$ and $(-14, 2).$\nArea=[ANS].", "answer_v3": [ "13" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] } ]