Question 7514: Matrix-Model Additivity Values 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, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, 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, 1], [0, 2]], u=[1, 0], v=[0, 1]. Give (Mu, Mv, M(u+v), Mu+Mv, 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, 1], [0, 2]];v=[0, 1]

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

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

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(0, 0)

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

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

Mu[0] term0 = 1

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

Mu[0] term1 = 0

Mu[0] row sum := sum(1, 0)

Mu[0] row sum = 1

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

Mu[1] term0 = 0

Mu[1] term1 := mul(2, 0)

Mu[1] term1 = 0

Mu[1] row sum := sum(0, 0)

Mu[1] row sum = 0

u+v[0] := sum(1, 0)

u+v[0] = 1

u+v[1] := sum(0, 1)

u+v[1] = 1

M(u+v)[0] term0 := mul(1, 1)

M(u+v)[0] term0 = 1

M(u+v)[0] term1 := mul(1, 1)

M(u+v)[0] term1 = 1

M(u+v)[0] row sum := sum(1, 1)

M(u+v)[0] row sum = 2

M(u+v)[1] term0 := mul(0, 1)

M(u+v)[1] term0 = 0

M(u+v)[1] term1 := mul(2, 1)

M(u+v)[1] term1 = 2

M(u+v)[1] row sum := sum(0, 2)

M(u+v)[1] row sum = 2

Mu+Mv[0] := sum(1, 1)

Mu+Mv[0] = 2

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

Mu+Mv[1] = 2

linear additivity residual[0] := sub(2, 2)

linear additivity residual[0] = 0

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

linear additivity residual[1] = 0

u=[1, 0];v=[0, 1]

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

Reverse checks

• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute linear additivity residual[1] with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute linear additivity residual[0] with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Mu+Mv[1] with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Mu+Mv[0] with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute M(u+v)[1] row sum with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute M(u+v)[1] term1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute M(u+v)[1] term0 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute M(u+v)[0] row sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute M(u+v)[0] term1 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute M(u+v)[0] term0 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute u+v[1] with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute u+v[0] with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Mu[1] row sum with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Mu[1] term1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Mu[1] term0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Mu[0] row sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Mu[0] term1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Mu[0] term0 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

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