Question 7591: Matrix-Model Scalar Routes and Defect

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=[[1, 0], [0, 2]], v=[3, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).

• 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=[[1, 0], [0, 2]], v=[3, 1], c=-1. Give (Mv, M(cv), cMv, difference of the last two sides).

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=[[1, 0], [0, 2]];v=[3, 1]

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

Mv[0] term0 := mul(1, 3)

Mv[0] term0 = 3

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

Mv[0] term1 = 0

Mv[0] row sum := sum(3, 0)

Mv[0] row sum = 3

Mv[1] term0 := mul(0, 3)

Mv[1] term0 = 0

Mv[1] term1 := mul(2, 1)

Mv[1] term1 = 2

Mv[1] row sum := sum(0, 2)

Mv[1] row sum = 2

cv[0] := mul(-1, 3)

cv[0] = -3

cv[1] := mul(-1, 1)

cv[1] = -1

M(cv)[0] term0 := mul(1, -3)

M(cv)[0] term0 = -3

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

M(cv)[0] term1 = 0

M(cv)[0] row sum := sum(-3, 0)

M(cv)[0] row sum = -3

M(cv)[1] term0 := mul(0, -3)

M(cv)[1] term0 = 0

M(cv)[1] term1 := mul(2, -1)

M(cv)[1] term1 = -2

M(cv)[1] row sum := sum(0, -2)

M(cv)[1] row sum = -2

cMv[0] := mul(-1, 3)

cMv[0] = -3

cMv[1] := mul(-1, 2)

cMv[1] = -2

linear scalar residual[0] := sub(-3, -3)

linear scalar residual[0] = 0

linear scalar residual[1] := sub(-2, -2)

linear scalar residual[1] = 0

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

left=[-3, -2];right=[-3, -2];residual=[0, 0]

Reverse checks

• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute linear scalar residual[1] with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute linear scalar residual[0] with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute cMv[1] with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute cMv[0] with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute M(cv)[1] row sum with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute M(cv)[1] term1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute M(cv)[1] term0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute M(cv)[0] row sum with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute M(cv)[0] term1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute M(cv)[0] term0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute cv[1] with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute cv[0] with exact rational arithmetic, not rounded decimals.
• 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

([3, 2], [-3, -2], [-3, -2], [0, 0])

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.