# Question 7568: Basis-Image Columns and Two Matrix-Vector Routes

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0014-07568 · q-20261012-advance-20261016-r03-linear-transformation-0018



## Original question and givens

Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[5, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).


- G001 — Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[5, 1], [2, -1]], v=[1, -1]. Give (first column M*e1, second column M*e2, Mv, v1*column1+v2*column2).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-linear-transformation](https://ocme.evemisslab.com/en/mko/mko-linear-transformation/)





## Step-by-step derivation

S001 · impliesBefore

M=[[5, 1], [2, -1]];v=[1, -1]

After

declared R2 matrix model;finite values are not a test of an arbitrary function

Conditions: none

The declared matrix model Mv gives a linear map on all R² by distributivity.The listed finite evaluations do not prove an arbitrary function linear.The zero map is linear;nonzero values or invertibility are not required.

Step identifiers and dependenciesRule: declared_linear_model_scope · Uses: G001





S002 · calculationBefore

Mv[0] term0 := mul(5, 1)

After

Mv[0] term0 = 5

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Mv[0] term1 := mul(1, -1)

After

Mv[0] term1 = -1

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

Mv[0] row sum := sum(5, -1)

After

Mv[0] row sum = 4

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

Mv[1] term0 := mul(2, 1)

After

Mv[1] term0 = 2

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

Mv[1] term1 := mul(-1, -1)

After

Mv[1] term1 = 1

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

Mv[1] row sum := sum(2, 1)

After

Mv[1] row sum = 3

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

M=[[5, 1], [2, -1]];e1=[1, 0];e2=[0, 1]

After

M e1=[5, 2];M e2=[1, -1]

Conditions: none

The first standard basis vector selects the first column,and the second selects the second column.Do not substitute rows for basis images.

Step identifiers and dependenciesRule: basis_images_are_columns · Uses: G001, S007





S009 · calculationBefore

v0 column1[0] := mul(1, 5)

After

v0 column1[0] = 5

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

v0 column1[1] := mul(1, 2)

After

v0 column1[1] = 2

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

v1 column2[0] := mul(-1, 1)

After

v1 column2[0] = -1

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

v1 column2[1] := mul(-1, -1)

After

v1 column2[1] = 1

Conditions: none

Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

basis reconstruction[0] := sum(5, -1)

After

basis reconstruction[0] = 4

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

basis reconstruction[1] := sum(2, 1)

After

basis reconstruction[1] = 3

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

v=[1, -1];Mv=[4, 3]

After

column reconstruction=[4, 3]

Conditions: none

First compute Mv by row times column vector,then reconstruct it from the basis images.The two routes do not reuse a proposed answer.

Step identifiers and dependenciesRule: basis_reconstruction_audit · Uses: G001, S014









## Reverse checks


- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Recompute basis reconstruction[1] with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute basis reconstruction[0] with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute v1 column2[1] with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute v1 column2[0] with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute v0 column1[1] with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute v0 column1[0] with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Recompute Mv[1] row sum with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute Mv[1] term1 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute Mv[1] term0 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute Mv[0] row sum with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute Mv[0] term1 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Mv[0] term0 with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

([5, 2], [1, -1], [4, 3], [4, 3])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 978230b55df576ab3b63804dd75036343319c61687e08836e2ae4c9508b740ae
- candidate record SHA-256: cebd7bf30be96056333f970217302d78093b584c1c277d9a7e5277eb7d07554f
- semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
- question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r03-linear-transformation-0018.json)