Question 7563: Basis-Image Columns and Two Matrix-Vector Routes

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=[[0, 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=[[0, 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

Step-by-step derivation

M=[[0, 1], [2, -1]];v=[1, -1]

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

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

Mv[0] term0 = 0

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

Mv[0] term1 = -1

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

Mv[0] row sum = -1

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

Mv[1] term0 = 2

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

Mv[1] term1 = 1

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

Mv[1] row sum = 3

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

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

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

v0 column1[0] = 0

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

v0 column1[1] = 2

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

v1 column2[0] = -1

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

v1 column2[1] = 1

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

basis reconstruction[0] = -1

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

basis reconstruction[1] = 3

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

column reconstruction=[-1, 3]

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

([0, 2], [1, -1], [-1, 3], [-1, 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.