id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-001701 | Linear Algebra: Linear Maps — Column Interpretation | 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.
$$A=\begin{pmatrix}-27&5\\-28&17\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}25\... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-27\\cdot(25)+5\\cdot(-14)=-745$.",
"Step 2: Second component: $-28\\cdot(25)+17\\cdot(-14)=-938$.",
"Final step: Therefore $A... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-745,-938)^T$ here.",
"robustness_analysis": "If the problem were perturb... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001702 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Exercise: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-3&16\\-15&-25\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-7\\-21\end{pmatrix}.$$... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-3\\cdot(-7)+16\\cdot(-21)=-315$.",
"Step 2: Second component: $-15\\cdot(-7)+-25\\cdot(-21)=630$.",
"Final step: Therefore $A... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-315,630)^T$ here.",
"robustness_analysis": "If the problem were perturbe... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001703 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Answer with a short justification: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-28&-9\\26&-1\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}19\\-11\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 $\\mathbf{a}_1=\\begin{pmatrix}-28\\\\26\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-9\\\\-1\\end{pmatrix}$.... | {
"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 $(-433,505)^T$ 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\... | 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-001704 | Linear Algebra: Vectors — Linear Combinations | 1 | Give reasoning, not just computation: 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.
$$A=\begin{pmatrix}13&-29\\26&-15\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}30\\-1... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}13\\\\26\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-29\\\\-15\\end{pmatrix}$... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(883,1035)^T$ here.",
"robustness_analysis": "Se... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001705 | Linear Algebra: Vectors — Linear Combinations | 1 | Where appropriate, name the theorem you use: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-2&13\\-5&-6\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}16\\-16\end{p... | [
{
"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: $-2\\cdot(16)+13\\cdot(-16)=-240$.",
"Step 2: Second component: $-5\\cdot(16)+-6\\cdot(-16)=16$.",
"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 $(-240,16)^T$ here.",
"robustness_analysis": "Robustness 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-001706 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 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.
$$A=\begin{pmatrix}20&11\\22&-8\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20\\-20\end{pmatrix}.$$
(a) Com... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}20\\\\22\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}11\\\\-8\\end{pmatrix}$."... | {
"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 $(-620,-280)^T$ 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. |
math-001707 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 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.
$$A=\begin{pmatrix}7&11\\27&24\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}2\\2... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $7\\cdot(2)+11\\cdot(27)=311$.",
"Step 2: Second component: $27\\cdot(2)+24\\cdot(27)=702$.",
"Final step: Therefore $A\\mathbf... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(311,702)^T$ here.",
"robustness_analysis": "If the problem were perturbed: The ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001708 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Try to avoid pattern-matching; explain why: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}30&-2\\7&21\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: $30\\cdot(-12)+-2\\cdot(23)=-406$.",
"Step 2: Second component: $7\\cdot(-12)+21\\cdot(23)=399$.",
"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 $(-406,399)^T$ 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\... | 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-001709 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Proceed methodically: 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.
$$A=\begin{pmatrix}11&19\\-15&-29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}23\\-23\end... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $11\\cdot(23)+19\\cdot(-23)=-184$.",
"Step 2: Second component: $-15\\cdot(23)+-29\\cdot(-23)=322$.",
"Final step: Therefore $A... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-184,322)^T$ here.",
"robustness_analysis": "If the problem were perturbed: The... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001710 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Write the solution set clearly: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}16&6\\18&-4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}8\\-23\end{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: $16\\cdot(8)+6\\cdot(-23)=-10$.",
"Step 2: Second component: $18\\cdot(8)+-4\\cdot(-23)=236$.",
"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 $(-10,236)^T$ 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\... | 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-001711 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 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.
$$A=\begin{pmatrix}-26&-30\\7&-29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-16\\15\end{... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-26\\cdot(-16)+-30\\cdot(15)=-34$.",
"Step 2: Second component: $7\\cdot(-16)+-29\\cdot(15)=-547$.",
"Final step: Therefore $A... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-34,-547)^T$ here.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001712 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Solve with verification: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}8&16\\-19&28\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20\\-30\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: $8\\cdot(-20)+16\\cdot(-30)=-640$.",
"Step 2: Second component: $-19\\cdot(-20)+28\\cdot(-30)=-460$.",
"Final step: Therefore $... | {
"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 $(-640,-460)^T$ 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\... | 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-001713 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Problem: 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.
$$A=\begin{pmatrix}-3&13\\13&10\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}13\\8\end{pmatrix}.$$
(a) Compute $A\mathbf{v}$ ... | [
{
"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: $-3\\cdot(13)+13\\cdot(8)=65$.",
"Step 2: Second component: $13\\cdot(13)+10\\cdot(8)=249$.",
"Final step: Therefore $A\\mathbf... | {
"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 $(65,249)^T$ here.",
"robustness_analysis": "Robu... | [
{
"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-001714 | Linear Algebra: Vectors — Linear Combinations | 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.
$$A=\begin{pmatrix}17&-17\\-15&5\end{pmatrix},\qquad \mathbf{v}=\begin{pmat... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}17\\\\-15\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-17\\\\5\\end{pmatrix}$.... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-663,305)^T$ here.",
"robustness_analysis": "If... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001715 | Linear Algebra: Vectors — Linear Combinations | 1 | Solve and justify each step: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-25&4\\20&29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}17\\-6\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: $-25\\cdot(17)+4\\cdot(-6)=-449$.",
"Step 2: Second component: $20\\cdot(17)+29\\cdot(-6)=166$.",
"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 $(-449,166)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001716 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Provide both a computational and a conceptual explanation: 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.
$$A=\begin{pmatrix}29&3\\8&-13\end{pmatrix},\qquad \ma... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}29\\\\8\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}3\\\\-13\\end{pmatrix}$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-362,232)^T$ here.",
"robustness_analysis": "Robustness note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001717 | Linear Algebra: Vectors — Linear Combinations | 1 | Solve (and briefly cross-validate): Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-12&22\\29&-6\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-27\\-12\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 $\\mathbf{a}_1=\\begin{pmatrix}-12\\\\29\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}22\\\\-6\\end{pmatrix}$.... | {
"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 $(60,-711)^T$ 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\... | 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-001718 | Linear Algebra: Vectors — Linear Combinations | 1 | Track units/moduli carefully: 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.
$$A=\begin{pmatrix}-13&2\\30&17\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-26\\... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-13\\\\30\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}2\\\\17\\end{pmatrix}$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(308,-1035)^T$ here.",
"robustness_analysis": "If the problem were perturb... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001719 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Carefully track domains: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-10&9\\-6&30\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\0\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 $\\mathbf{a}_1=\\begin{pmatrix}-10\\\\-6\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}9\\\\30\\end{pmatrix}$."... | {
"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 $(240,144)^T$ 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\... | 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-001720 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve and include a self-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.
$$A=\begin{pmatrix}-30&9\\-28&18\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-4\\15\end{p... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-30\\cdot(-4)+9\\cdot(15)=255$.",
"Step 2: Second component: $-28\\cdot(-4)+18\\cdot(15)=382$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(255,382)^T$ here.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001721 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Make each step logically reversible (or explain if not): Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-8&11\\11&1\end{pmatrix},\qquad \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: $-8\\cdot(17)+11\\cdot(-5)=-191$.",
"Step 2: Second component: $11\\cdot(17)+1\\cdot(-5)=182$.",
"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 $(-191,182)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001722 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Prompt: 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.
$$A=\begin{pmatrix}30&-6\\-29&-20\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}5\\-28\end{pmatrix}.$$
(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 $\\mathbf{a}_1=\\begin{pmatrix}30\\\\-29\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-6\\\\-20\\end{pmatrix}$... | {
"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 $(318,415)^T$ 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\... | 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-001723 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Determine the requested 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.
$$A=\begin{pmatrix}0&17\\-18&20\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}25\\-25\end{pma... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}0\\\\-18\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}17\\\\20\\end{pmatrix}$."... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-425,-950)^T$ here.",
"robustness_analysis": "Generality note: The column view ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001724 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Derive the result step-by-step: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-1&25\\28&29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-14\\-11\end{pmatrix}.$$
... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-1\\cdot(-14)+25\\cdot(-11)=-261$.",
"Step 2: Second component: $28\\cdot(-14)+29\\cdot(-11)=-711$.",
"Final step: Therefore $... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-261,-711)^T$ here.",
"robustness_analysis": "Sensitivit... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001725 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve with verification: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-4&-14\\-22&29\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-10\\-30... | [
{
"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 $\\mathbf{a}_2=\\begin{pmatrix}-14\\\\29\\end{pmatrix}$... | {
"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 $(460,-650)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001726 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Prompt: 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.
$$A=\begin{pmatrix}-21&6\\-14&-4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}15\\4\end{pmatrix}.$$
(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 $\\mathbf{a}_1=\\begin{pmatrix}-21\\\\-14\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}6\\\\-4\\end{pmatrix}$.... | {
"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 $(-291,-226)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001727 | Linear Algebra: Vectors — Linear Combinations | 1 | Checkpoint: 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.
$$A=\begin{pmatrix}20&14\\-12&-3\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-13\\-26\end{pmatrix}.$$
(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: $20\\cdot(-13)+14\\cdot(-26)=-624$.",
"Step 2: Second component: $-12\\cdot(-13)+-3\\cdot(-26)=234$.",
"Final step: Therefore $... | {
"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 $(-624,234)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001728 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 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.
$$A=\begin{pmatrix}23&16\\-5&27\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}19\\-27\end{pma... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}23\\\\-5\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}16\\\\27\\end{pmatrix}$."... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(5,-824)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The column vie... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001729 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Warm-up: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-25&10\\-18&-22\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-6\\-24\end{pmatrix}.$$
(a) Compute $A\mathbf... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-25\\cdot(-6)+10\\cdot(-24)=-90$.",
"Step 2: Second component: $-18\\cdot(-6)+-22\\cdot(-24)=636$.",
"Final step: Therefore $A... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-90,636)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The col... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001730 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Warm-up: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}11&-6\\2&21\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-23\\-19\end{pmatrix}.$$
(... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}11\\\\2\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-6\\\\21\\end{pmatrix}$.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-139,-445)^T$ here.",
"robustness_analysis": "R... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001731 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Indicate where a theorem is used: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-4&-5\\29&26\end{pmatrix},\qquad \mathbf{v}=\begin{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: $-4\\cdot(-22)+-5\\cdot(4)=68$.",
"Step 2: Second component: $29\\cdot(-22)+26\\cdot(4)=-534$.",
"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 $(68,-534)^T$ 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\... | 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-001732 | Linear Algebra: Vectors — Linear Combinations | 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.
$$A=\begin{pmatrix}-13&22\\-24&-28\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-26... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-13\\cdot(-26)+22\\cdot(7)=492$.",
"Step 2: Second component: $-24\\cdot(-26)+-28\\cdot(7)=428$.",
"Final step: Therefore $A\\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(492,428)^T$ here.",
"robustness_analysis": "Robustness note: The column v... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001733 | Linear Algebra: Vectors — Linear Combinations | 1 | Show all reasoning: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}12&-26\\12&23\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-9\\-3\end{pmatrix}.$$
(a) Compute $... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $12\\cdot(-9)+-26\\cdot(-3)=-30$.",
"Step 2: Second component: $12\\cdot(-9)+23\\cdot(-3)=-177$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-30,-177)^T$ here.",
"robustness_analysis": "Se... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001734 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Provide both a computational and a conceptual explanation: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}21&20\\12&22\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 $\\mathbf{a}_1=\\begin{pmatrix}21\\\\12\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}20\\\\22\\end{pmatrix}$."... | {
"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 $(-38,158)^T$ 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\... | 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-001735 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 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.
$$A=\begin{pmatrix}18&9\\-13&2\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-11\\28\end{pmatrix}.$$
(a) Compu... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}18\\\\-13\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}9\\\\2\\end{pmatrix}$.",... | {
"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 $(54,199)^T$ here.",
"robustness_analysis": "Robu... | [
{
"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-001736 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Question: 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.
$$A=\begin{pmatrix}-9&4\\8&-28\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20\\-20\end{pmatrix}.$$
... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}-9\\\\8\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}4\\\\-28\\end{pmatrix}$.",... | {
"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 $(100,400)^T$ 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\... | 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-001737 | Linear Algebra: Matrices — Action on Basis Vectors | 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.
$$A=\begin{pmatrix}3&20\\17&-10\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-26\\-24\end{pma... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $3\\cdot(-26)+20\\cdot(-24)=-558$.",
"Step 2: Second component: $17\\cdot(-26)+-10\\cdot(-24)=-202$.",
"Final step: Therefore $... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-558,-202)^T$ here.",
"robustness_analysis": "Sensitivit... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001738 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Complete the analysis: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-10&-24\\1&17\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}27\\1\end{pmatrix}.$$... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-10\\cdot(27)+-24\\cdot(1)=-294$.",
"Step 2: Second component: $1\\cdot(27)+17\\cdot(1)=44$.",
"Final step: Therefore $A\\math... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-294,44)^T$ here.",
"robustness_analysis": "If ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001739 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Start by stating any domain restrictions: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}13&-15\\-18&17\end{pmatrix},\qquad \mathbf{v}=\begin{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: $13\\cdot(-26)+-15\\cdot(-8)=-218$.",
"Step 2: Second component: $-18\\cdot(-26)+17\\cdot(-8)=332$.",
"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 $(-218,332)^T$ 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-001740 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Explain why your operations are valid: 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.
$$A=\begin{pmatrix}-25&-14\\23&-13\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}3\\2... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-25\\\\23\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-14\\\\-13\\end{pmatrix}... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-411,-243)^T$ here.",
"robustness_analysis": "Generality note: The column view ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001741 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Give a fully justified solution: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-2&13\\-27&24\end{pmatrix},\qquad \mathbf{v}=\begin{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: $-2\\cdot(-30)+13\\cdot(-6)=-18$.",
"Step 2: Second component: $-27\\cdot(-30)+24\\cdot(-6)=666$.",
"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 $(-18,666)^T$ 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-001742 | Linear Algebra: Vectors — Linear Combinations | 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.
$$A=\begin{pmatrix}-8&-23\\14&-20\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}2... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-8\\\\14\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-23\\\\-20\\end{pmatrix}$... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-452,68)^T$ here.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001743 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Give reasoning, not just computation: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-21&28\\-26&17\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}13\\3... | [
{
"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: $-21\\cdot(13)+28\\cdot(3)=-189$.",
"Step 2: Second component: $-26\\cdot(13)+17\\cdot(3)=-287$.",
"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 $(-189,-287)^T$ 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\... | 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-001744 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Explain what is being counted/optimized: 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.
$$A=\begin{pmatrix}9&19\\12&20\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}18\\-19\end{pmatri... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}9\\\\12\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}19\\\\20\\end{pmatrix}$.",... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-199,-164)^T$ here.",
"robustness_analysis": "S... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001745 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Solve and justify each step: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-13&-16\\26&-19\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-3\\12\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: $-13\\cdot(-3)+-16\\cdot(12)=-153$.",
"Step 2: Second component: $26\\cdot(-3)+-19\\cdot(12)=-306$.",
"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 $(-153,-306)^T$ 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-001746 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 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.
$$A=\begin{pmatrix}-2&-29\\-28&-20\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-28\\-30\end{pmat... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-2\\\\-28\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-29\\\\-20\\end{pmatrix}... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(926,1384)^T$ here.",
"robustness_analysis": "Se... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001747 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Write the solution set clearly: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}23&-13\\-18&17\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}13\\22\end{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 $\\mathbf{a}_1=\\begin{pmatrix}23\\\\-18\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-13\\\\17\\end{pmatrix}$... | {
"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 $(13,140)^T$ here.",
"robustness_analysis": "If the problem were perturbed: The c... | [
{
"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-001748 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Answer using clear logical steps: 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.
$$A=\begin{pmatrix}-12&9\\-5&-6\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-28\\-29\end... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-12\\cdot(-28)+9\\cdot(-29)=75$.",
"Step 2: Second component: $-5\\cdot(-28)+-6\\cdot(-29)=314$.",
"Final step: Therefore $A\\... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(75,314)^T$ here.",
"robustness_analysis": "If the problem were perturbed: The c... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001749 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Work carefully and justify each inference: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-28&-3\\-7&-9\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-4\\1\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 $\\mathbf{a}_1=\\begin{pmatrix}-28\\\\-7\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-3\\\\-9\\end{pmatrix}$.... | {
"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 $(109,19)^T$ 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\... | 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-001750 | 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.
$$A=\begin{pmatrix}-6&16\\1&0\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}23\... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-6\\cdot(23)+16\\cdot(-12)=-330$.",
"Step 2: Second component: $1\\cdot(23)+0\\cdot(-12)=23$.",
"Final step: Therefore $A\\mat... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-330,23)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The column vi... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001751 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Where appropriate, name the theorem you use: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-13&6\\-4&19\end{pmatrix},\qquad \mathbf{v}=\begi... | [
{
"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: $-13\\cdot(-10)+6\\cdot(2)=142$.",
"Step 2: Second component: $-4\\cdot(-10)+19\\cdot(2)=78$.",
"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 $(142,78)^T$ 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-001752 | Linear Algebra: Linear Maps — Column Interpretation | 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.
$$A=\begin{pmatrix}26&6\\-19&-4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}27\\-16\end{pmatrix}.$$
(a) Comp... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}26\\\\-19\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}6\\\\-4\\end{pmatrix}$."... | {
"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 $(606,-449)^T$ 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\... | 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-001753 | Linear Algebra: Matrices — Action on Basis Vectors | 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.
$$A=\begin{pmatrix}-8&23\\-11&17\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}11\\12\end{pmatrix}.$$
(a) Compute $A\mathbf... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}-8\\\\-11\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}23\\\\17\\end{pmatrix}$.... | {
"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 $(188,83)^T$ 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\... | 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-001754 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Answer with a short justification: 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.
$$A=\begin{pmatrix}-4&-28\\25&20\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}6\\-5\end{pmatrix}.$$
... | [
{
"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: $-4\\cdot(6)+-28\\cdot(-5)=116$.",
"Step 2: Second component: $25\\cdot(6)+20\\cdot(-5)=50$.",
"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 $(116,50)^T$ 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-001755 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Carefully track domains: 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.
$$A=\begin{pmatrix}-11&-25\\-4&26\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}0\\-2\en... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-11\\\\-4\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-25\\\\26\\end{pmatrix}$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(50,-52)^T$ here.",
"robustness_analysis": "Robustness note: The column view gen... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001756 | Linear Algebra: Vectors — Linear Combinations | 1 | Be explicit about assumptions: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}3&-24\\-4&-17\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-5\\29\end{pmatrix}.$$
(a... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}3\\\\-4\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-24\\\\-17\\end{pmatrix}$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-711,-473)^T$ here.",
"robustness_analysis": "Robustness... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001757 | Linear Algebra: Vectors — Linear Combinations | 1 | Explain what is being counted/optimized: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}18&22\\-13&-2\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 $\\mathbf{a}_1=\\begin{pmatrix}18\\\\-13\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}22\\\\-2\\end{pmatrix}$.... | {
"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 $(370,-295)^T$ 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. |
math-001758 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Solve and then verify: 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.
$$A=\begin{pmatrix}1&-1\\30&21\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-22\\-8\end{pmatrix}.$$... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}1\\\\30\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-1\\\\21\\end{pmatrix}$.",... | {
"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 $(-14,-828)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001759 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Prompt: 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.
$$A=\begin{pmatrix}-4&-10\\-6&2\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}10\\-1\end{pmatrix}.$$
(a) Compute $A\mathbf{v}$ ... | [
{
"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: $-4\\cdot(10)+-10\\cdot(-1)=-30$.",
"Step 2: Second component: $-6\\cdot(10)+2\\cdot(-1)=-62$.",
"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 $(-30,-62)^T$ 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-001760 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve and include a self-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.
$$A=\begin{pmatrix}19&1\\16&18\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20\\11\end{pma... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $19\\cdot(-20)+1\\cdot(11)=-369$.",
"Step 2: Second component: $16\\cdot(-20)+18\\cdot(11)=-122$.",
"Final step: Therefore $A\\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-369,-122)^T$ here.",
"robustness_analysis": "If the problem were perturb... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001761 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Provide a rigorous solution: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-10&16\\29&19\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 $\\mathbf{a}_1=\\begin{pmatrix}-10\\\\29\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}16\\\\19\\end{pmatrix}$.... | {
"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 $(-116,271)^T$ 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\... | 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-001762 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve and include a self-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.
$$A=\begin{pmatrix}-2&-16\\-10&4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-3\\-25\end{... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-2\\\\-10\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-16\\\\4\\end{pmatrix}$.... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(406,-70)^T$ here.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001763 | Linear Algebra: Vectors — Linear Combinations | 1 | Explain what is being counted/optimized: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-23&-22\\-10&10\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}4\... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}-23\\\\-10\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-22\\\\10\\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 $(-202,10)^T$ 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\... | 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-001764 | Linear Algebra: Linear Maps — Column Interpretation | 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.
$$A=\begin{pmatrix}-13&10\\2&30\end{pmatrix},\qquad \mathbf{v}=\... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-13\\cdot(-24)+10\\cdot(-17)=142$.",
"Step 2: Second component: $2\\cdot(-24)+30\\cdot(-17)=-558$.",
"Final step: Therefore $A... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(142,-558)^T$ here.",
"robustness_analysis": "If the problem were perturbe... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001765 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Carefully track domains: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}17&-12\\-23&2\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}29\\13\end{pmatrix}.$$
(a) Comp... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $17\\cdot(29)+-12\\cdot(13)=337$.",
"Step 2: Second component: $-23\\cdot(29)+2\\cdot(13)=-641$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(337,-641)^T$ here.",
"robustness_analysis": "Generality note: The column view g... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001766 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Determine the requested value: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-4&-9\\-25&-8\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-25... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}-4\\\\-25\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-9\\\\-8\\end{pmatrix}$.... | {
"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 $(370,865)^T$ 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-001767 | 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.
$$A=\begin{pmatrix}-24&-14\\6&-4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20\\-17\end{pmatrix}.$$... | [
{
"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: $-24\\cdot(-20)+-14\\cdot(-17)=718$.",
"Step 2: Second component: $6\\cdot(-20)+-4\\cdot(-17)=-52$.",
"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 $(718,-52)^T$ 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\... | 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-001768 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 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.
$$A=\begin{pmatrix}14&-13\\23&28\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-27\\-23\end{pmatrix}.$$
(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: $14\\cdot(-27)+-13\\cdot(-23)=-79$.",
"Step 2: Second component: $23\\cdot(-27)+28\\cdot(-23)=-1265$.",
"Final step: Therefore ... | {
"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 $(-79,-1265)^T$ 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-001769 | 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.
$$A=\begin{pmatrix}3&-15\\6&22\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-3\\-19\end{pmatrix}.$$
(a) Compu... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}3\\\\6\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-15\\\\22\\end{pmatrix}$.",... | {
"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 $(276,-436)^T$ 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-001770 | Linear Algebra: Vectors — Linear Combinations | 1 | Start by stating any domain restrictions: 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.
$$A=\begin{pmatrix}0&15\\-26&5\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-14\\-22\end{pmat... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $0\\cdot(-14)+15\\cdot(-22)=-330$.",
"Step 2: Second component: $-26\\cdot(-14)+5\\cdot(-22)=254$.",
"Final step: Therefore $A\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-330,254)^T$ here.",
"robustness_analysis": "Ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001771 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Explain what is being counted/optimized: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-18&-15\\-11&-23\end{pmatrix},\qquad \mathbf{v}=\begi... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}-18\\\\-11\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-15\\\\-23\\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 $(-168,-407)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001772 | Linear Algebra: Matrices — Action on Basis Vectors | 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.
$$A=\begin{pmatrix}24&-5\\-20&4\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-3\\-5\end{pmatrix}.$$
(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 $\\mathbf{a}_1=\\begin{pmatrix}24\\\\-20\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-5\\\\4\\end{pmatrix}$."... | {
"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 $(-47,40)^T$ 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-001773 | Linear Algebra: Vectors — Linear Combinations | 1 | Complete the analysis: 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.
$$A=\begin{pmatrix}-13&-13\\25&7\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}18\\-10\end{pmatrix}.... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-13\\cdot(18)+-13\\cdot(-10)=-104$.",
"Step 2: Second component: $25\\cdot(18)+7\\cdot(-10)=380$.",
"Final step: Therefore $A\... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-104,380)^T$ here.",
"robustness_analysis": "Ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001774 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Give a theorem-based solution: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-15&-9\\27&-23\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-1\\5\end{pmatrix}.$$
(a... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-15\\cdot(-1)+-9\\cdot(5)=-30$.",
"Step 2: Second component: $27\\cdot(-1)+-23\\cdot(5)=-142$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-30,-142)^T$ here.",
"robustness_analysis": "Robustness note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001775 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Give an answer and a quick verification: Find $A\mathbf{v}$, then rewrite it explicitly as $x\mathbf{a}_1+y\mathbf{a}_2$ where $\mathbf{a}_i$ are columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-9&20\\-25&-9\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-22... | [
{
"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: $-9\\cdot(-22)+20\\cdot(27)=738$.",
"Step 2: Second component: $-25\\cdot(-22)+-9\\cdot(27)=307$.",
"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 $(738,307)^T$ here.",
"robustness_analysis": "Robustness 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\... | 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-001776 | Linear Algebra: Vectors — Linear Combinations | 1 | Checkpoint: 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.
$$A=\begin{pmatrix}-19&3\\-21&-19\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}12\\-18\end{pmatrix}.$$
(a) Com... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}-19\\\\-21\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}3\\\\-19\\end{pmatrix}$... | {
"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 $(-282,90)^T$ here.",
"robustness_analysis": "If the probl... | [
{
"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-001777 | Linear Algebra: Linear Maps — Column Interpretation | 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.
$$A=\begin{pmatrix}12&-28\\2&3\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}28\\4\end{pmatrix}.$$
(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 $\\mathbf{a}_1=\\begin{pmatrix}12\\\\2\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-28\\\\3\\end{pmatrix}$.",... | {
"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 $(224,68)^T$ 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-001778 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Task: 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.
$$A=\begin{pmatrix}15&-5\\11&8\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-27\\-6\end{pmatrix}.$$
(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: $15\\cdot(-27)+-5\\cdot(-6)=-375$.",
"Step 2: Second component: $11\\cdot(-27)+8\\cdot(-6)=-345$.",
"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 $(-375,-345)^T$ 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-001779 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Answer with a short justification: 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.
$$A=\begin{pmatrix}28&-20\\-19&25\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}9\\20\end... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $28\\cdot(9)+-20\\cdot(20)=-148$.",
"Step 2: Second component: $-19\\cdot(9)+25\\cdot(20)=329$.",
"Final step: Therefore $A\\ma... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-148,329)^T$ here.",
"robustness_analysis": "Se... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001780 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Challenge: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}16&-14\\-15&-13\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}10\\26\end{pmatrix}.$$
(a) Comp... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}16\\\\-15\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-14\\\\-13\\end{pmatrix}... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-204,-488)^T$ here.",
"robustness_analysis": "R... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001781 | Linear Algebra: Vectors — Linear Combinations | 1 | Give a theorem-based solution: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}27&-26\\24&21\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-23\\-20\end{pmatrix}.$$
... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}27\\\\24\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-26\\\\21\\end{pmatrix}$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-101,-972)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Key idea: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001782 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Question: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-5&16\\-28&7\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-12\\11\end{pmatrix}.$$
... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-5\\\\-28\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}16\\\\7\\end{pmatrix}$."... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(236,413)^T$ here.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001783 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Track quantifiers carefully: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-8&0\\11&-15\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-26\\7\end{pmatri... | [
{
"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: $-8\\cdot(-26)+0\\cdot(7)=208$.",
"Step 2: Second component: $11\\cdot(-26)+-15\\cdot(7)=-391$.",
"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 $(208,-391)^T$ 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\... | 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-001784 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Start by stating any domain restrictions: 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.
$$A=\begin{pmatrix}5&-18\\-20&-18\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-18\\1\end{pma... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $5\\cdot(-18)+-18\\cdot(1)=-108$.",
"Step 2: Second component: $-20\\cdot(-18)+-18\\cdot(1)=342$.",
"Final step: Therefore $A\\... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-108,342)^T$ here.",
"robustness_analysis": "Generality note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001785 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Give a theorem-based solution: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}7&18\\-23&9\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}18\\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 $\\mathbf{a}_1=\\begin{pmatrix}7\\\\-23\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}18\\\\9\\end{pmatrix}$.",... | {
"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 $(378,-288)^T$ 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\... | 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-001786 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Work carefully and justify each inference: 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.
$$A=\begin{pmatrix}25&18\\6&-27\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}20... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}25\\\\6\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}18\\\\-27\\end{pmatrix}$."... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(392,282)^T$ here.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001787 | Linear Algebra: Vectors — Linear Combinations | 1 | Solve and then verify: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-16&6\\9&12\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-22\\-24\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 $\\mathbf{a}_1=\\begin{pmatrix}-16\\\\9\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}6\\\\12\\end{pmatrix}$.",... | {
"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 $(208,-486)^T$ 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\... | 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-001788 | Linear Algebra: Matrices — Action on Basis Vectors | 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.
$$A=\begin{pmatrix}26&2\\2&13\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}9\\-23\end{pmatrix}.$$
(a) Compute $A\m... | [
{
"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 $\\mathbf{a}_1=\\begin{pmatrix}26\\\\2\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}2\\\\13\\end{pmatrix}$.",
... | {
"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 $(188,-281)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001789 | Linear Algebra: Vectors — Linear Combinations | 1 | Answer using clear logical steps: 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.
$$A=\begin{pmatrix}-5&-26\\10&-14\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-15\\-9\en... | [
{
"method_name": "Column Combination",
"approach": "Matrix-vector multiplication equals $x$ times the first column plus $y$ times the second column.",
"steps": [
"Step 1: Columns are $\\mathbf{a}_1=\\begin{pmatrix}-5\\\\10\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-26\\\\-14\\end{pmatrix}$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(309,-24)^T$ here.",
"robustness_analysis": "Robustness note: The column view ge... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Remember: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001790 | Linear Algebra: Linear Maps — Column Interpretation | 1 | Checkpoint: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-22&25\\-14&12\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-21\\-2\end{pmatrix}.$$
(a) Compute $A\math... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-22\\cdot(-21)+25\\cdot(-2)=412$.",
"Step 2: Second component: $-14\\cdot(-21)+12\\cdot(-2)=270$.",
"Final step: Therefore $A\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(412,270)^T$ here.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001791 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Solve and justify each step: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}15&-7\\21&-16\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-4\\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 $\\mathbf{a}_1=\\begin{pmatrix}15\\\\21\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-7\\\\-16\\end{pmatrix}$.... | {
"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 $(-109,-196)^T$ 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. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001792 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 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.
$$A=\begin{pmatrix}1&-10\\-13&7\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-30\\19\end{p... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $1\\cdot(-30)+-10\\cdot(19)=-220$.",
"Step 2: Second component: $-13\\cdot(-30)+7\\cdot(19)=523$.",
"Final step: Therefore $A\\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-220,523)^T$ here.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Takeaway: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001793 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Where appropriate, name the theorem you use: Evaluate $A\mathbf{v}$ and provide both a computational and a conceptual (column-space) explanation:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}3&14\\-5&-27\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-20\\20\end{p... | [
{
"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: $3\\cdot(-20)+14\\cdot(20)=220$.",
"Step 2: Second component: $-5\\cdot(-20)+-27\\cdot(20)=-440$.",
"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 $(220,-440)^T$ 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-001794 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Try to avoid pattern-matching; explain why: 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.
$$A=\begin{pmatrix}6&6\\16&5\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}2\\5\... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $6\\cdot(2)+6\\cdot(5)=42$.",
"Step 2: Second component: $16\\cdot(2)+5\\cdot(5)=57$.",
"Final step: Therefore $A\\mathbf{v}=\\... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(42,57)^T$ here.",
"robustness_analysis": "If the problem... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. (Here the result is $\boxed{\begin{pmatrix}$.) |
math-001795 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | State any required conditions first: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}-26&22\\-6&13\end{pmatrix},\qquad \mathbf{v}=\begin{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 $\\mathbf{a}_1=\\begin{pmatrix}-26\\\\-6\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}22\\\\13\\end{pmatrix}$.... | {
"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 $(1330,505)^T$ 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\... | 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-001796 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 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.
$$A=\begin{pmatrix}-22&-29\\21&0\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}-24\\2\end{p... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $-22\\cdot(-24)+-29\\cdot(2)=470$.",
"Step 2: Second component: $21\\cdot(-24)+0\\cdot(2)=-504$.",
"Final step: Therefore $A\\m... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(470,-504)^T$ here.",
"robustness_analysis": "Sensitivity analysis: The co... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001797 | Linear Algebra: Coordinate-Free Meaning of $A\mathbf{v}$ | 1 | Complete the analysis: Compute $A\mathbf{v}$ in two ways: (i) row-by-column, (ii) as a linear combination of columns. Then explain why both match:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}15&-25\\19&-24\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}7\\13\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 $\\mathbf{a}_1=\\begin{pmatrix}15\\\\19\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-25\\\\-24\\end{pmatrix}$... | {
"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 $(-220,-179)^T$ here.",
"robustness_analysis": "S... | [
{
"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-001798 | Linear Algebra: Matrices — Action on Basis Vectors | 1 | Make each step logically reversible (or explain if not): 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.
$$A=\begin{pmatrix}26&20\\-15&19\end{pmatrix},\qquad \ma... | [
{
"method_name": "Row-by-Column",
"approach": "Each output entry is the dot product of a row of $A$ with $\\mathbf{v}$.",
"steps": [
"Step 1: First component: $26\\cdot(0)+20\\cdot(-5)=-100$.",
"Step 2: Second component: $-15\\cdot(0)+19\\cdot(-5)=-95$.",
"Final step: Therefore $A\\mat... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\begin{pmatrix}$.\nRow-by-column multiplication and column-combination multiplication are equivalent definitions of $A\\mathbf{v}$. Both yield $(-100,-95)^T$ here.",
"robustness_analysis": "Robustness note: The column ... | [
{
"error_description": "Did entrywise (Hadamard) multiplication.",
"why_plausible": "Elementwise multiplication is familiar from vectors.",
"why_wrong": "Matrix-vector multiplication is defined via dot products/linear combinations; entrywise multiplication does not represent a linear map $\\mathbb{R}^2\... | Core principle: Think of $A\mathbf{v}$ as a linear combination of columns of $A$ with weights from $\mathbf{v}$; this interpretation explains why the computational dot-product rule works. |
math-001799 | Linear Algebra: Vectors — Linear Combinations | 1 | Give an answer and a quick verification: Matrix-vector multiplication: compute $A\mathbf{v}$ and interpret the result as a linear combination of columns:
Compute $A\mathbf{v}$ and interpret the result as a linear combination of columns.
$$A=\begin{pmatrix}22&-30\\16&2\end{pmatrix},\qquad \mathbf{v}=\begin{pmatrix}20\\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 $\\mathbf{a}_1=\\begin{pmatrix}22\\\\16\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-30\\\\2\\end{pmatrix}$."... | {
"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 $(230,334)^T$ 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. |
math-001800 | Linear Algebra: Matrix Multiplication — Row/Column Rules | 1 | Try to avoid pattern-matching; explain why: 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&-23\\-23&-19\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 $\\mathbf{a}_1=\\begin{pmatrix}7\\\\-23\\end{pmatrix}$ and $\\mathbf{a}_2=\\begin{pmatrix}-23\\\\-19\\end{pmatrix}$... | {
"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 $(585,915)^T$ here.",
"robustness_analysis": "Sen... | [
{
"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 $\boxed{\begin{pmatrix}$.) |
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