# Question 7506: Matrix-Model Additivity Values and Defect [Return to catalogue page 38](https://ocme.evemisslab.com/en/questions/page/38/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r03-linear-transformation-foundations/#q-20261012-advance-20261016-r03-linear-transformation-0031) 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. R0014-07506 · q-20261012-advance-20261016-r03-linear-transformation-0031 ## 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=[[-7, 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=[[-7, 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](https://ocme.evemisslab.com/en/mko/mko-linear-transformation/) ## Step-by-step derivation S001 · impliesBefore M=[[-7, 1], [0, 2]];v=[0, 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(-7, 0) After Mv[0] term0 = 0 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(0, 1) After Mv[0] row sum = 1 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(0, 0) After Mv[1] term0 = 0 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(2, 1) After Mv[1] term1 = 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, S005 S007 · calculationBefore Mv[1] row sum := sum(0, 2) After Mv[1] row sum = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Mu[0] term0 := mul(-7, 1) After Mu[0] term0 = -7 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, S007 S009 · calculationBefore Mu[0] term1 := mul(1, 0) After Mu[0] term1 = 0 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 Mu[0] row sum := sum(-7, 0) After Mu[0] row sum = -7 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Mu[1] term0 := mul(0, 1) After Mu[1] term0 = 0 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 Mu[1] term1 := mul(2, 0) After Mu[1] term1 = 0 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 Mu[1] row sum := sum(0, 0) After Mu[1] row sum = 0 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore u+v[0] := sum(1, 0) After u+v[0] = 1 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore u+v[1] := sum(0, 1) After u+v[1] = 1 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore M(u+v)[0] term0 := mul(-7, 1) After M(u+v)[0] term0 = -7 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, S015 S017 · calculationBefore M(u+v)[0] term1 := mul(1, 1) After M(u+v)[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, S016 S018 · calculationBefore M(u+v)[0] row sum := sum(-7, 1) After M(u+v)[0] row sum = -6 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore M(u+v)[1] term0 := mul(0, 1) After M(u+v)[1] term0 = 0 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, S018 S020 · calculationBefore M(u+v)[1] term1 := mul(2, 1) After M(u+v)[1] term1 = 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, S019 S021 · calculationBefore M(u+v)[1] row sum := sum(0, 2) After M(u+v)[1] row sum = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Mu+Mv[0] := sum(-7, 1) After Mu+Mv[0] = -6 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Mu+Mv[1] := sum(0, 2) After Mu+Mv[1] = 2 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore linear additivity residual[0] := sub(-6, -6) After linear additivity residual[0] = 0 Conditions: none Subtract the second operand from the first in the requested order,retaining the residual sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore linear additivity residual[1] := sub(2, 2) After linear additivity residual[1] = 0 Conditions: none Subtract the second operand from the first in the requested order,retaining the residual sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore u=[1, 0];v=[0, 1] After left=[-6, 2];right=[-6, 2];residual=[0, 0] Conditions: none Evaluate both routes component by component for these inputs.The general matrix additivity law comes from distributivity,not extrapolating an arbitrary function from one zero residual. Step identifiers and dependenciesRule: matrix_additivity_value_audit · Uses: G001, S025 ## 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 ([-7, 0], [1, 2], [-6, 2], [-6, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: a5184ca87ce143be72969cbea68d50149b86cf37863dab2543a65989e340f09e - candidate record SHA-256: d2df784addc4c4058c8589f1c5d36e5ab7b67f9358663071e4734e0492bd6343 - 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-0031.json)