id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-001801 | Linear Algebra: Vectors — Linear Combinations | 1 | State any required conditions first: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-20&11\\-6&-10\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-28\\8\end{pmatrix}... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-20\\cdot(-28)+11\\cdot(8)=648$.",
"Step 2: Second component: $-6\\cdot(-28)+-10\\cdot(8)=88$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(648,88)^T$ here.",
"robustness_analysis": "Sens... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001802 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Answer using clear logical steps: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}6&9\\-17&-23\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-21\\13\end{pmatrix}.$$
... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $6\\cdot(-21)+9\\cdot(13)=-9$.",
"Step 2: Second component: $-17\\cdot(-21)+-23\\cdot(13)=58$.",
"Final step: Therefore $A\\mat... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-9,58)^T$ here.",
"robustness_analysis": "Robustness note: The column vie... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001803 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Derive the result step-by-step: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}24&-8\\-29&-16\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}0\\8\end{pmatrix}.$$
(a... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $24\\cdot(0)+-8\\cdot(8)=-64$.",
"Step 2: Second component: $-29\\cdot(0)+-16\\cdot(8)=-128$.",
"Final step: Therefore $A\\math... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-64,-128)^T$ here.",
"robustness_analysis": "Generality note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001804 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Solve and sanity-check: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-9&-24\\-24&-14\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}25\\-9\end{pmatrix... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-9\\\\-24\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-24\\\\-14\\end{pmatrix}... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-9,-474)^T$ here.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001805 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Give an answer and a quick verification: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}27&-27\\-15&-13\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-14\\9\end{pma... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}27\\\\-15\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-27\\\\-13\\end{pmatrix}... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-621,93)^T$ here.",
"robustness_analysis": "Gen... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001806 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Compute the requested quantity: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-30&-28\\-6&-18\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-29\\-26\en... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-30\\\\-6\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-28\\\\-18\\end{pmatrix}... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(1598,642)^T$ here.",
"robustness_analysis": "Robustness note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001807 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Give an answer and a quick verification: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}4&27\\-8&-14\end{pmatrix},\qquad \mathbf{v}=\begin{pm... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}4\\\\-8\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}27\\\\-14\\end{pmatrix}$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-447,54)^T$ here.",
"robustness_analysis": "Generality note: The column v... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001808 | Linear Algebra: Vectors — Linear Combinations | 1 | Give a theorem-based solution: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}19&17\\27&4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-5\\2... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}19\\\\27\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}17\\\\4\\end{pmatrix}$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(245,-55)^T$ here.",
"robustness_analysis": "Robustness note: The column view ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001809 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Problem: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-30&19\\6&3\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-19\\6\end{pmatrix}.$$
(a) Compute $A... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-30\\\\6\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}19\\\\3\\end{pmatrix}$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(684,-96)^T$ here.",
"robustness_analysis": "If the problem were perturbed... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001810 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Proceed methodically: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-2&-5\\-21&-13\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-4\\-5\end{pmatrix}.$$
(a) Comput... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-2\\\\-21\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-5\\\\-13\\end{pmatrix}$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(33,149)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The column vie... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001811 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Find the exact value: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-8&-10\\9&-25\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20\\-17\end{pmatrix}.$$
(a) Compu... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-8\\cdot(-20)+-10\\cdot(-17)=330$.",
"Step 2: Second component: $9\\cdot(-20)+-25\\cdot(-17)=245$.",
"Final step: Therefore $A... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(330,245)^T$ here.",
"robustness_analysis": "Rob... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001812 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Answer using clear logical steps: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-13&24\\-14&14\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20\\-18\... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-13\\cdot(-20)+24\\cdot(-18)=-172$.",
"Step 2: Second component: $-14\\cdot(-20)+14\\cdot(-18)=28$.",
"Final step: Therefore $... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-172,28)^T$ here.",
"robustness_analysis": "If ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001813 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | State any required conditions first: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}21&26\\-1&-22\end{pmatrix},\qquad \mathbf{v}=\begin{pmatr... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $21\\cdot(-20)+26\\cdot(13)=-82$.",
"Step 2: Second component: $-1\\cdot(-20)+-22\\cdot(13)=-266$.",
"Final step: Therefore $A\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-82,-266)^T$ here.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001814 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Solve and justify each step: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-24&22\\1&-21\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}9\\11\end{pmatri... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-24\\\\1\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}22\\\\-21\\end{pmatrix}$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(26,-222)^T$ here.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001815 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Track quantifiers carefully: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-24&-17\\-1&-17\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-2\\-24\end{pm... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-24\\cdot(-2)+-17\\cdot(-24)=456$.",
"Step 2: Second component: $-1\\cdot(-2)+-17\\cdot(-24)=410$.",
"Final step: Therefore $A... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(456,410)^T$ here.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001816 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Compute the requested quantity: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-11&-30\\18&12\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\-16\end... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-11\\cdot(-24)+-30\\cdot(-16)=744$.",
"Step 2: Second component: $18\\cdot(-24)+12\\cdot(-16)=-624$.",
"Final step: Therefore ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(744,-624)^T$ here.",
"robustness_analysis": "Ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001817 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Write the solution set clearly: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}6&-25\\-6&-12\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}1\\6\end{pmatrix}.$$
(a)... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $6\\cdot(1)+-25\\cdot(6)=-144$.",
"Step 2: Second component: $-6\\cdot(1)+-12\\cdot(6)=-78$.",
"Final step: Therefore $A\\mathb... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-144,-78)^T$ here.",
"robustness_analysis": "Generality note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001818 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Make each step logically reversible (or explain if not): Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}23&-26\\25&23\end{pmatrix},\qquad \mathbf{v}=\b... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}23\\\\25\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-26\\\\23\\end{pmatrix}$.... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(1320,102)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The co... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001819 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Give reasoning, not just computation: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-7&1\\-10&-26\end{pmatrix},\qquad \mathbf{v}=\begin{pmat... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-7\\cdot(-27)+1\\cdot(2)=191$.",
"Step 2: Second component: $-10\\cdot(-27)+-26\\cdot(2)=218$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(191,218)^T$ here.",
"robustness_analysis": "If the problem were perturbed... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001820 | Linear Algebra: Vectors — Linear Combinations | 1 | Work this out carefully: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-7&12\\-4&-11\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}30\\-20\end{pmatrix... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-7\\cdot(30)+12\\cdot(-20)=-450$.",
"Step 2: Second component: $-4\\cdot(30)+-11\\cdot(-20)=100$.",
"Final step: Therefore $A\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-450,100)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The co... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001821 | Linear Algebra: Vectors — Linear Combinations | 1 | Keep the final answer in boxed form: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}6&17\\14&-23\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-11\\12\... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $6\\cdot(-11)+17\\cdot(12)=138$.",
"Step 2: Second component: $14\\cdot(-11)+-23\\cdot(12)=-430$.",
"Final step: Therefore $A\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(138,-430)^T$ here.",
"robustness_analysis": "Se... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001822 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Show all reasoning: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}7&-20\\-12&-6\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\22\end{pm... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $7\\cdot(-24)+-20\\cdot(22)=-608$.",
"Step 2: Second component: $-12\\cdot(-24)+-6\\cdot(22)=156$.",
"Final step: Therefore $A\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-608,156)^T$ here.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001823 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Give a fully justified solution: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}7&-11\\8&-28\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-27\\27\end{pmatrix}.$$
... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}7\\\\8\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-11\\\\-28\\end{pmatrix}$."... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-486,-972)^T$ here.",
"robustness_analysis": "I... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001824 | Linear Algebra: Vectors — Linear Combinations | 1 | Explain each transformation: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}28&19\\-24&-24\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-7\\1\end{pmatrix}.$$
(a) ... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}28\\\\-24\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}19\\\\-24\\end{pmatrix}$... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-177,144)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001825 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Explain each transformation: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}10&-3\\-4&4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}23\\-13... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}10\\\\-4\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-3\\\\4\\end{pmatrix}$.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(269,-144)^T$ here.",
"robustness_analysis": "If the prob... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001826 | Linear Algebra: Vectors — Linear Combinations | 1 | Give a fully justified solution: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-15&-10\\26&-29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}19\\-26\end{pmatrix}.$... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-15\\cdot(19)+-10\\cdot(-26)=-25$.",
"Step 2: Second component: $26\\cdot(19)+-29\\cdot(-26)=1248$.",
"Final step: Therefore $... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-25,1248)^T$ here.",
"robustness_analysis": "Se... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001827 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Provide both a computational and a conceptual explanation: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}5&-3\\12&-30\end{pmatrix},\qquad \mathbf{v}=\b... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $5\\cdot(1)+-3\\cdot(-16)=53$.",
"Step 2: Second component: $12\\cdot(1)+-30\\cdot(-16)=492$.",
"Final step: Therefore $A\\math... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(53,492)^T$ here.",
"robustness_analysis": "If the problem were perturbed:... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001828 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Track units/moduli carefully: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}29&2\\-24&-29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}28\\7\end{pmatrix}.$$
(a) ... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $29\\cdot(28)+2\\cdot(7)=826$.",
"Step 2: Second component: $-24\\cdot(28)+-29\\cdot(7)=-875$.",
"Final step: Therefore $A\\mat... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(826,-875)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001829 | Linear Algebra: Vectors — Linear Combinations | 1 | Provide a rigorous solution: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}4&-10\\22&-24\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}23\\15\end{pmatrix}.$$
(a) ... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}4\\\\22\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-10\\\\-24\\end{pmatrix}$.... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-58,146)^T$ here.",
"robustness_analysis": "If ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001830 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Show all reasoning: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-24&12\\16&-26\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-14\\-25\end{pmatrix}.$... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-24\\\\16\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}12\\\\-26\\end{pmatrix}$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(36,426)^T$ here.",
"robustness_analysis": "Robustness note: The column view gen... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001831 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | State any required conditions first: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-18&28\\22&-12\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}10\\19... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-18\\cdot(10)+28\\cdot(19)=352$.",
"Step 2: Second component: $22\\cdot(10)+-12\\cdot(19)=-8$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(352,-8)^T$ here.",
"robustness_analysis": "Robustness note: The column vi... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001832 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Indicate where a theorem is used: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}0&18\\13&25\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}21... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $0\\cdot(21)+18\\cdot(-10)=-180$.",
"Step 2: Second component: $13\\cdot(21)+25\\cdot(-10)=23$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-180,23)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The col... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001833 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Proceed methodically: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-9&-2\\-17&-7\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}12\\6\end{pmatrix}.$$
(a) Compute ... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-9\\\\-17\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-2\\\\-7\\end{pmatrix}$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-120,-246)^T$ here.",
"robustness_analysis": "If the pro... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001834 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Solve and justify each step: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-6&2\\7&-12\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}30\\-14\end{pmatr... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-6\\\\7\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}2\\\\-12\\end{pmatrix}$.",... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-208,378)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001835 | Linear Algebra: Vectors — Linear Combinations | 1 | Find the exact value: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}9&24\\-8&-19\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\8\end{pm... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $9\\cdot(-24)+24\\cdot(8)=-24$.",
"Step 2: Second component: $-8\\cdot(-24)+-19\\cdot(8)=40$.",
"Final step: Therefore $A\\math... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-24,40)^T$ here.",
"robustness_analysis": "If t... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001836 | Linear Algebra: Vectors — Linear Combinations | 1 | Warm-up: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-19&-18\\-29&-27\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-26\\-26\end{pmatrix}.$$
(a) Compute $A\math... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-19\\cdot(-26)+-18\\cdot(-26)=962$.",
"Step 2: Second component: $-29\\cdot(-26)+-27\\cdot(-26)=1456$.",
"Final step: Therefor... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(962,1456)^T$ here.",
"robustness_analysis": "Ro... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001837 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Checkpoint: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}15&-8\\0&27\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\15\end{pmatrix}.$$
(a) Compute $A\mathbf{... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $15\\cdot(-24)+-8\\cdot(15)=-480$.",
"Step 2: Second component: $0\\cdot(-24)+27\\cdot(15)=405$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-480,405)^T$ here.",
"robustness_analysis": "If the problem were perturbed: The... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001838 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Checkpoint: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-22&-11\\-8&9\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-1\\-24\end{pmatrix}.$$
(a) Com... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-22\\cdot(-1)+-11\\cdot(-24)=286$.",
"Step 2: Second component: $-8\\cdot(-1)+9\\cdot(-24)=-208$.",
"Final step: Therefore $A\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(286,-208)^T$ here.",
"robustness_analysis": "Sensitivity... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001839 | Linear Algebra: Vectors — Linear Combinations | 1 | Find the exact value: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}11&8\\4&-3\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}7\\16\end{pmatrix}.$$
(a)... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $11\\cdot(7)+8\\cdot(16)=205$.",
"Step 2: Second component: $4\\cdot(7)+-3\\cdot(16)=-20$.",
"Final step: Therefore $A\\mathbf{... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(205,-20)^T$ here.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001840 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Prompt: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-10&3\\18&-24\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-28\\22\end{pmatrix}.$$
(... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-10\\\\18\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}3\\\\-24\\end{pmatrix}$.... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(346,-1032)^T$ here.",
"robustness_analysis": "G... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001841 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Task: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-10&-20\\-17&23\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}18\\-15\end{pmatrix}.$$
(a) Compute ... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-10\\cdot(18)+-20\\cdot(-15)=120$.",
"Step 2: Second component: $-17\\cdot(18)+23\\cdot(-15)=-651$.",
"Final step: Therefore $... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(120,-651)^T$ here.",
"robustness_analysis": "Robustness note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001842 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Explain why your operations are valid: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}10&-19\\0&28\end{pmatrix},\qquad \mathbf{v}=\begin{pmat... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $10\\cdot(-26)+-19\\cdot(25)=-735$.",
"Step 2: Second component: $0\\cdot(-26)+28\\cdot(25)=700$.",
"Final step: Therefore $A\\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-735,700)^T$ here.",
"robustness_analysis": "If the problem were perturbe... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001843 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Problem: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}4&-1\\17&-6\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-12\\-10\end{pmatrix}.$$
(a) Compute... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $4\\cdot(-12)+-1\\cdot(-10)=-38$.",
"Step 2: Second component: $17\\cdot(-12)+-6\\cdot(-10)=-144$.",
"Final step: Therefore $A\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-38,-144)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001844 | Linear Algebra: Vectors — Linear Combinations | 1 | Solve and include a self-check: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-12&-5\\-1&30\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}17... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-12\\cdot(17)+-5\\cdot(13)=-269$.",
"Step 2: Second component: $-1\\cdot(17)+30\\cdot(13)=373$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-269,373)^T$ here.",
"robustness_analysis": "Robustness note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001845 | Linear Algebra: Vectors — Linear Combinations | 1 | Derive the result step-by-step: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-11&-6\\10&11\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}8\\0\end{pmat... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-11\\\\10\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-6\\\\11\\end{pmatrix}$.... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-88,80)^T$ here.",
"robustness_analysis": "Gene... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001846 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Work this out carefully: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}12&-22\\-9&-14\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}28\\11\end{pmatrix}.$$
(a) Com... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $12\\cdot(28)+-22\\cdot(11)=94$.",
"Step 2: Second component: $-9\\cdot(28)+-14\\cdot(11)=-406$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(94,-406)^T$ here.",
"robustness_analysis": "Generality note: The column view ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001847 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve with verification: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}25&-4\\-27&13\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}16\\-2\end{pmatrix}... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}25\\\\-27\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-4\\\\13\\end{pmatrix}$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(408,-458)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001848 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Solve and justify each step: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-22&-29\\-22&-30\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\-20\end{pmatrix}.$$
... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-22\\\\-22\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-29\\\\-30\\end{pmatrix... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(1108,1128)^T$ here.",
"robustness_analysis": "G... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001849 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Provide both a computational and a conceptual explanation: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}29&-25\\30&-8\end{pmatrix},\qquad \mathbf{v}=\begin{pmatri... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}29\\\\30\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-25\\\\-8\\end{pmatrix}$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(145,-368)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001850 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Derive the result step-by-step: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-23&-5\\-9&-9\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}4\\-17\end{pm... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-23\\\\-9\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-5\\\\-9\\end{pmatrix}$.... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-7,117)^T$ here.",
"robustness_analysis": "If the problem were perturbed:... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001851 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Problem: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-12&9\\-9&-10\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-4\\-9\end{pmatrix}.$$
(a) Compute ... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-12\\cdot(-4)+9\\cdot(-9)=-33$.",
"Step 2: Second component: $-9\\cdot(-4)+-10\\cdot(-9)=126$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-33,126)^T$ here.",
"robustness_analysis": "Rob... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001852 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Use two approaches if possible: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-24&30\\27&18\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}4\\-4\end{pma... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-24\\cdot(4)+30\\cdot(-4)=-216$.",
"Step 2: Second component: $27\\cdot(4)+18\\cdot(-4)=36$.",
"Final step: Therefore $A\\math... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-216,36)^T$ here.",
"robustness_analysis": "Rob... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001853 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Compute the requested quantity: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}23&22\\2&30\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}19\\-20\end{pm... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}23\\\\2\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}22\\\\30\\end{pmatrix}$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-3,-562)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The column vi... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001854 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Task: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-9&-12\\26&2\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-8\\-19\end{pmatrix}.$$
(a) Compute $A\mathbf{v}$ b... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-9\\\\26\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-12\\\\2\\end{pmatrix}$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(300,-246)^T$ here.",
"robustness_analysis": "Generality note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001855 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Solve and include a self-check: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-6&-28\\20&28\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}15\\9\end{pmatrix}.$$
(a... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-6\\cdot(15)+-28\\cdot(9)=-342$.",
"Step 2: Second component: $20\\cdot(15)+28\\cdot(9)=552$.",
"Final step: Therefore $A\\mat... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-342,552)^T$ here.",
"robustness_analysis": "Sensitivity... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001856 | Linear Algebra: Vectors — Linear Combinations | 1 | Solve and sanity-check: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-13&-20\\-15&-20\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-1\\-26\end{pmatri... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-13\\\\-15\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-20\\\\-20\\end{pmatrix... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(533,535)^T$ here.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001857 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Indicate where a theorem is used: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-5&-10\\-24&4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}14\\23\end{... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-5\\cdot(14)+-10\\cdot(23)=-300$.",
"Step 2: Second component: $-24\\cdot(14)+4\\cdot(23)=-244$.",
"Final step: Therefore $A\\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-300,-244)^T$ here.",
"robustness_analysis": "R... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001858 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Find the exact value: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-26&-16\\22&-7\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-29\\-12\end{pmatrix}... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-26\\cdot(-29)+-16\\cdot(-12)=946$.",
"Step 2: Second component: $22\\cdot(-29)+-7\\cdot(-12)=-554$.",
"Final step: Therefore ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(946,-554)^T$ here.",
"robustness_analysis": "If the problem were perturbe... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001859 | Linear Algebra: Vectors — Linear Combinations | 1 | Show all reasoning: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-10&23\\20&30\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-15\\-6\end{pm... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-10\\cdot(-15)+23\\cdot(-6)=12$.",
"Step 2: Second component: $20\\cdot(-15)+30\\cdot(-6)=-480$.",
"Final step: Therefore $A\\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(12,-480)^T$ here.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001860 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Challenge: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-23&-22\\-10&-11\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}29\\20\end{pmatrix}.$$
(a) Com... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-23\\cdot(29)+-22\\cdot(20)=-1107$.",
"Step 2: Second component: $-10\\cdot(29)+-11\\cdot(20)=-510$.",
"Final step: Therefore ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-1107,-510)^T$ here.",
"robustness_analysis": "Robustness note: The column view... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001861 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Checkpoint: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-15&-13\\16&-22\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-23\\-2\end{pmatrix}.$$
(a) C... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-15\\cdot(-23)+-13\\cdot(-2)=371$.",
"Step 2: Second component: $16\\cdot(-23)+-22\\cdot(-2)=-324$.",
"Final step: Therefore $... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(371,-324)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The co... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001862 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Show all reasoning: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}24&-6\\-18&-4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\24\end{pmatrix}.$$
... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $24\\cdot(-24)+-6\\cdot(24)=-720$.",
"Step 2: Second component: $-18\\cdot(-24)+-4\\cdot(24)=336$.",
"Final step: Therefore $A\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-720,336)^T$ here.",
"robustness_analysis": "Robustness note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001863 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Where appropriate, name the theorem you use: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}29&-29\\-6&-28\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}29\\\\-6\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-29\\\\-28\\end{pmatrix}$... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(290,212)^T$ here.",
"robustness_analysis": "Rob... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001864 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Checkpoint: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-18&23\\-18&-16\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-6\\-24\end{pmatrix}... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-18\\cdot(-6)+23\\cdot(-24)=-444$.",
"Step 2: Second component: $-18\\cdot(-6)+-16\\cdot(-24)=492$.",
"Final step: Therefore $... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-444,492)^T$ here.",
"robustness_analysis": "Ro... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001865 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Start by stating any domain restrictions: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-14&-30\\-29&-25\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-14\\\\-29\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-30\\\\-25\\end{pmatrix... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-922,-1167)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001866 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Make each step logically reversible (or explain if not): Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}6&15\\-7&11\end{pmatrix},\qquad \mathbf{v}=\beg... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}6\\\\-7\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}15\\\\11\\end{pmatrix}$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(393,368)^T$ here.",
"robustness_analysis": "Generality note: The column v... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001867 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Provide a rigorous solution: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-7&27\\-5&5\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}29\\-15\end{pmatr... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-7\\cdot(29)+27\\cdot(-15)=-608$.",
"Step 2: Second component: $-5\\cdot(29)+5\\cdot(-15)=-220$.",
"Final step: Therefore $A\\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-608,-220)^T$ here.",
"robustness_analysis": "Robustness note: The column view ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001868 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Use two approaches if possible: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}19&28\\5&-6\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}23\\... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}19\\\\5\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}28\\\\-6\\end{pmatrix}$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(577,85)^T$ here.",
"robustness_analysis": "Robustness note: The column vi... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001869 | Linear Algebra: Vectors — Linear Combinations | 1 | Try to avoid pattern-matching; explain why: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-24&28\\-13&-4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-24\\\\-13\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}28\\\\-4\\end{pmatrix}$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-184,322)^T$ here.",
"robustness_analysis": "Generality note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001870 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Explain each transformation: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}11&-4\\22&-2\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}12\\-26\end{pmatrix}.$$
(a) ... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}11\\\\22\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-4\\\\-2\\end{pmatrix}$."... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(236,316)^T$ here.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001871 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Do not skip justification steps: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-4&-28\\5&29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-1\\-18\end{pmatrix}.$$
... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-4\\cdot(-1)+-28\\cdot(-18)=508$.",
"Step 2: Second component: $5\\cdot(-1)+29\\cdot(-18)=-527$.",
"Final step: Therefore $A\\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(508,-527)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The column v... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001872 | Linear Algebra: Vectors — Linear Combinations | 1 | Track quantifiers carefully: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-22&28\\-8&-13\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-29\\-8\end{pm... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-22\\\\-8\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}28\\\\-13\\end{pmatrix}$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(414,336)^T$ here.",
"robustness_analysis": "If the problem were perturbed: The ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001873 | Linear Algebra: Vectors — Linear Combinations | 1 | Explain why your operations are valid: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-12&-26\\-11&18\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-12\\\\-11\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-26\\\\18\\end{pmatrix}... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-410,670)^T$ here.",
"robustness_analysis": "If the prob... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001874 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Explain what is being counted/optimized: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-2&-7\\-16&10\end{pmatrix},\qquad \mathbf{v}=\begin{p... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-2\\\\-16\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-7\\\\10\\end{pmatrix}$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-5,-106)^T$ here.",
"robustness_analysis": "Generality note: The column view ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001875 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Show all reasoning: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}29&26\\6&-23\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}28\\29\end{pmatrix}.$$
(a) Compute $A... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $29\\cdot(28)+26\\cdot(29)=1566$.",
"Step 2: Second component: $6\\cdot(28)+-23\\cdot(29)=-499$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(1566,-499)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001876 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Solve and include a self-check: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-2&-23\\-12&-28\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\-28\end{pmatrix}.$... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-2\\\\-12\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-23\\\\-28\\end{pmatrix}... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(692,1072)^T$ here.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001877 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Work carefully and justify each inference: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}3&-11\\-7&12\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}3\\\\-7\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-11\\\\12\\end{pmatrix}$."... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-292,408)^T$ here.",
"robustness_analysis": "If the problem were perturbed: The... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001878 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve with verification: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-26&-25\\9&16\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}19\\-1\end{pmatrix}... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-26\\cdot(19)+-25\\cdot(-1)=-469$.",
"Step 2: Second component: $9\\cdot(19)+16\\cdot(-1)=155$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-469,155)^T$ here.",
"robustness_analysis": "Se... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001879 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve and include a self-check: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-19&28\\8&22\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-12... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-19\\\\8\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}28\\\\22\\end{pmatrix}$."... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-472,-646)^T$ here.",
"robustness_analysis": "Robustness... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001880 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Where appropriate, name the theorem you use: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}16&15\\-25&26\end{pmatrix},\qquad \mathbf{v}=\beg... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $16\\cdot(11)+15\\cdot(-4)=116$.",
"Step 2: Second component: $-25\\cdot(11)+26\\cdot(-4)=-379$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(116,-379)^T$ here.",
"robustness_analysis": "Generality note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001881 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Track quantifiers carefully: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}20&24\\11&-7\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}16\\5\end{pmatrix... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $20\\cdot(16)+24\\cdot(5)=440$.",
"Step 2: Second component: $11\\cdot(16)+-7\\cdot(5)=141$.",
"Final step: Therefore $A\\mathb... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(440,141)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The col... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001882 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Provide a rigorous solution: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-17&25\\15&-30\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}7\\-... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-17\\\\15\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}25\\\\-30\\end{pmatrix}$... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-494,555)^T$ here.",
"robustness_analysis": "Generality note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001883 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Keep the final answer in boxed form: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}8&-17\\-1&25\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}22\\29\en... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}8\\\\-1\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-17\\\\25\\end{pmatrix}$."... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-317,703)^T$ here.",
"robustness_analysis": "Ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001884 | Linear Algebra: Vectors — Linear Combinations | 1 | Start by stating any domain restrictions: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-14&-25\\12&-2\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-1... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-14\\cdot(-13)+-25\\cdot(22)=-368$.",
"Step 2: Second component: $12\\cdot(-13)+-2\\cdot(22)=-200$.",
"Final step: Therefore $... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-368,-200)^T$ here.",
"robustness_analysis": "Generality note: The column view ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001885 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Give a theorem-based solution: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}19&-28\\4&-4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}10\\... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}19\\\\4\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-28\\\\-4\\end{pmatrix}$."... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-174,-12)^T$ here.",
"robustness_analysis": "Robustness note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001886 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Be explicit about assumptions: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-27&2\\19&18\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-21\... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-27\\\\19\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}2\\\\18\\end{pmatrix}$."... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(559,-471)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001887 | Linear Algebra: Vectors — Linear Combinations | 1 | Compute the requested quantity: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-9&-21\\30&0\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-21\\4\end{pma... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-9\\\\30\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-21\\\\0\\end{pmatrix}$."... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(105,-630)^T$ here.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001888 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve and sanity-check: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}5&-24\\-10&-19\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-7\\-23\end{pmatrix}... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $5\\cdot(-7)+-24\\cdot(-23)=517$.",
"Step 2: Second component: $-10\\cdot(-7)+-19\\cdot(-23)=507$.",
"Final step: Therefore $A\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(517,507)^T$ here.",
"robustness_analysis": "Robustness note: The column view ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001889 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Provide both a computational and a conceptual explanation: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-10&-24\\20&-4\end{pmatrix},\qquad \mathbf{v}... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-10\\cdot(3)+-24\\cdot(22)=-558$.",
"Step 2: Second component: $20\\cdot(3)+-4\\cdot(22)=-28$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-558,-28)^T$ here.",
"robustness_analysis": "Generality note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001890 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Be explicit about assumptions: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}13&-3\\29&16\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}8\\4\end{pmatr... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $13\\cdot(8)+-3\\cdot(4)=92$.",
"Step 2: Second component: $29\\cdot(8)+16\\cdot(4)=296$.",
"Final step: Therefore $A\\mathbf{v... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(92,296)^T$ here.",
"robustness_analysis": "If the problem were perturbed:... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001891 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Solve and sanity-check: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}28&21\\15&-20\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}5\\14\end{pmatrix}.$$
(a) Comput... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $28\\cdot(5)+21\\cdot(14)=434$.",
"Step 2: Second component: $15\\cdot(5)+-20\\cdot(14)=-205$.",
"Final step: Therefore $A\\mat... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(434,-205)^T$ here.",
"robustness_analysis": "Generality note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001892 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Indicate where a theorem is used: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}19&11\\-16&10\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-8\\-17\end{pmatrix}.$$... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}19\\\\-16\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}11\\\\10\\end{pmatrix}$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-339,-42)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001893 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Show all reasoning: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-1&2\\-26&8\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-19\\7\end{pmatrix}.$$
(a)... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-1\\cdot(-19)+2\\cdot(7)=33$.",
"Step 2: Second component: $-26\\cdot(-19)+8\\cdot(7)=550$.",
"Final step: Therefore $A\\mathb... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(33,550)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The colu... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001894 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Track units/moduli carefully: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-29&29\\-7&-27\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-10\\-15\end{... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-29\\\\-7\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}29\\\\-27\\end{pmatrix}$... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-145,475)^T$ here.",
"robustness_analysis": "Robustness note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001895 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Use two approaches if possible: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}8&24\\-1&-29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-21\\-29\end{pmatrix}.$$
... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $8\\cdot(-21)+24\\cdot(-29)=-864$.",
"Step 2: Second component: $-1\\cdot(-21)+-29\\cdot(-29)=862$.",
"Final step: Therefore $A... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-864,862)^T$ here.",
"robustness_analysis": "If the problem were perturbed: The... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001896 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Give an answer and a quick verification: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-25&-26\\13&-30\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}4... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-25\\cdot(4)+-26\\cdot(16)=-516$.",
"Step 2: Second component: $13\\cdot(4)+-30\\cdot(16)=-428$.",
"Final step: Therefore $A\\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-516,-428)^T$ here.",
"robustness_analysis": "Generality note: The column view ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001897 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Be explicit about assumptions: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}10&-2\\-9&-25\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}25\\5\end{pmatrix}.$$
(a)... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}10\\\\-9\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-2\\\\-25\\end{pmatrix}$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(240,-350)^T$ here.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001898 | Linear Algebra: Vectors — Linear Combinations | 1 | Explain why your operations are valid: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}14&-4\\4&-12\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-9\\-3... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}14\\\\4\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-4\\\\-12\\end{pmatrix}$."... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-6,324)^T$ here.",
"robustness_analysis": "If t... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001899 | Linear Algebra: Vectors — Linear Combinations | 1 | Give an answer and a quick verification: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-30&-23\\-24&-9\end{pmatrix},\qquad \mathbf{v}=\begin... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-30\\cdot(-12)+-23\\cdot(-21)=843$.",
"Step 2: Second component: $-24\\cdot(-12)+-9\\cdot(-21)=477$.",
"Final step: Therefore ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(843,477)^T$ here.",
"robustness_analysis": "Robustness note: The column view ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001900 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Carefully track domains: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-20&4\\-3&11\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-14\\15\end{pmatrix}.$$
(a) Comp... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-20\\cdot(-14)+4\\cdot(15)=340$.",
"Step 2: Second component: $-3\\cdot(-14)+11\\cdot(15)=207$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(340,207)^T$ here.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
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