# Question 13011: Compare Both Sides of Additivity for the Supplied Affine Map [Return to catalogue page 66](https://ocme.evemisslab.com/en/questions/page/66/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261027-r05-general-linear-map-foundations/#q-20261014-advance-20261027-r05-general-linear-map-0061) 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. R0025-13011 · q-20261014-advance-20261027-r05-general-linear-map-0061 ## Original question and givens Use the complete rectangular matrix A=[[-2, 0, 3], [-2, 2, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[1, 0], u=[1, -2, 0], v=[5, 1, 0]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?). - G001 — Use the complete rectangular matrix A=[[-2, 0, 3], [-2, 2, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[1, 0], u=[1, -2, 0], v=[5, 1, 0]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?). ## 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-general-linear-map](https://ocme.evemisslab.com/en/mko/mko-general-linear-map/) ## Step-by-step derivation S001 · calculationBefore u plus v coordinate0 first scaled := mul(1, 1) After u plus v coordinate0 first scaled = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore u plus v coordinate0 second scaled := mul(1, 5) After u plus v coordinate0 second scaled = 5 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore u plus v coordinate0 sum := sum(1, 5) After u plus v coordinate0 sum = 6 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore u plus v coordinate1 first scaled := mul(1, -2) After u plus v coordinate1 first scaled = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore u plus v coordinate1 second scaled := mul(1, 1) After u plus v coordinate1 second scaled = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore u plus v coordinate1 sum := sum(-2, 1) After u plus v coordinate1 sum = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore u plus v coordinate2 first scaled := mul(1, 0) After u plus v coordinate2 first scaled = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore u plus v coordinate2 second scaled := mul(1, 0) After u plus v coordinate2 second scaled = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore u plus v coordinate2 sum := sum(0, 0) After u plus v coordinate2 sum = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore matrix of u plus v:matrix=[[-2, 0, 3], [-2, 2, 5]];input=[6, -1, 0] After R^3->R^2;each output is its original row dot input Conditions: none Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates. Step identifiers and dependenciesRule: rectangular_linear_map_source_target_roles · Uses: G001, S009 S011 · calculationBefore matrix of u plus v row0 column0 term := mul(-2, 6) After matrix of u plus v row0 column0 term = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore matrix of u plus v row0 column1 term := mul(0, -1) After matrix of u plus v row0 column1 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore matrix of u plus v row0 column2 term := mul(3, 0) After matrix of u plus v row0 column2 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore matrix of u plus v row0 sum := sum(-12, 0, 0) After matrix of u plus v row0 sum = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore matrix of u plus v row1 column0 term := mul(-2, 6) After matrix of u plus v row1 column0 term = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore matrix of u plus v row1 column1 term := mul(2, -1) After matrix of u plus v row1 column1 term = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore matrix of u plus v row1 column2 term := mul(5, 0) After matrix of u plus v row1 column2 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore matrix of u plus v row1 sum := sum(-12, -2, 0) After matrix of u plus v row1 sum = -14 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore matrix of u:matrix=[[-2, 0, 3], [-2, 2, 5]];input=[1, -2, 0] After R^3->R^2;each output is its original row dot input Conditions: none Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates. Step identifiers and dependenciesRule: rectangular_linear_map_source_target_roles · Uses: G001, S018 S020 · calculationBefore matrix of u row0 column0 term := mul(-2, 1) After matrix of u row0 column0 term = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore matrix of u row0 column1 term := mul(0, -2) After matrix of u row0 column1 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore matrix of u row0 column2 term := mul(3, 0) After matrix of u row0 column2 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore matrix of u row0 sum := sum(-2, 0, 0) After matrix of u row0 sum = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore matrix of u row1 column0 term := mul(-2, 1) After matrix of u row1 column0 term = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore matrix of u row1 column1 term := mul(2, -2) After matrix of u row1 column1 term = -4 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore matrix of u row1 column2 term := mul(5, 0) After matrix of u row1 column2 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore matrix of u row1 sum := sum(-2, -4, 0) After matrix of u row1 sum = -6 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore matrix of v:matrix=[[-2, 0, 3], [-2, 2, 5]];input=[5, 1, 0] After R^3->R^2;each output is its original row dot input Conditions: none Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates. Step identifiers and dependenciesRule: rectangular_linear_map_source_target_roles · Uses: G001, S027 S029 · calculationBefore matrix of v row0 column0 term := mul(-2, 5) After matrix of v row0 column0 term = -10 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore matrix of v row0 column1 term := mul(0, 1) After matrix of v row0 column1 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore matrix of v row0 column2 term := mul(3, 0) After matrix of v row0 column2 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore matrix of v row0 sum := sum(-10, 0, 0) After matrix of v row0 sum = -10 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore matrix of v row1 column0 term := mul(-2, 5) After matrix of v row1 column0 term = -10 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore matrix of v row1 column1 term := mul(2, 1) After matrix of v row1 column1 term = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore matrix of v row1 column2 term := mul(5, 0) After matrix of v row1 column2 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore matrix of v row1 sum := sum(-10, 2, 0) After matrix of v row1 sum = -8 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore affine lhs coordinate0 := sum(-12, 1) After affine lhs coordinate0 = -11 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore T u coordinate0 := sum(-2, 1) After T u coordinate0 = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore T v coordinate0 := sum(-10, 1) After T v coordinate0 = -9 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore affine rhs coordinate0 := sum(-1, -9) After affine rhs coordinate0 = -10 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore additivity signed residual0 := sub(-11, -10) After additivity signed residual0 = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore affine lhs coordinate1 := sum(-14, 0) After affine lhs coordinate1 = -14 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore T u coordinate1 := sum(-6, 0) After T u coordinate1 = -6 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore T v coordinate1 := sum(-8, 0) After T v coordinate1 = -8 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore affine rhs coordinate1 := sum(-6, -8) After affine rhs coordinate1 = -14 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore additivity signed residual1 := sub(-14, -14) After additivity signed residual1 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · impliesBefore bias=[1, 0];T(u+v)=[-11, -14];T(u)+T(v)=[-10, -14] After lhs-rhs=[-1, 0]=-bias;this instance additive=No Conditions: none The left side has one bias and the right side two.The matrix-linear parts cancel,leaving negative bias with its sign retained.This is an exact witness for the declared affine formula,not a sample proof of an arbitrary black box. Step identifiers and dependenciesRule: affine_one_bias_versus_two_biases · Uses: G001, S046 ## Reverse checks - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Recompute additivity signed residual1 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute affine rhs coordinate1 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute T v coordinate1 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute T u coordinate1 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute affine lhs coordinate1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute additivity signed residual0 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute affine rhs coordinate0 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute T v coordinate0 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute T u coordinate0 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute affine lhs coordinate0 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute matrix of v row1 sum with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute matrix of v row1 column2 term with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute matrix of v row1 column1 term with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute matrix of v row1 column0 term with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute matrix of v row0 sum with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute matrix of v row0 column2 term with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute matrix of v row0 column1 term with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute matrix of v row0 column0 term with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Recompute matrix of u row1 sum with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute matrix of u row1 column2 term with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute matrix of u row1 column1 term with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute matrix of u row1 column0 term with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute matrix of u row0 sum with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute matrix of u row0 column2 term with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute matrix of u row0 column1 term with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute matrix of u row0 column0 term with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute matrix of u plus v row1 sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute matrix of u plus v row1 column2 term with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute matrix of u plus v row1 column1 term with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute matrix of u plus v row1 column0 term with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute matrix of u plus v row0 sum with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute matrix of u plus v row0 column2 term with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute matrix of u plus v row0 column1 term with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute matrix of u plus v row0 column0 term with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute u plus v coordinate2 sum with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute u plus v coordinate2 second scaled with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute u plus v coordinate2 first scaled with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute u plus v coordinate1 sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute u plus v coordinate1 second scaled with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute u plus v coordinate1 first scaled with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute u plus v coordinate0 sum with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute u plus v coordinate0 second scaled with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute u plus v coordinate0 first scaled with exact rational arithmetic, not rounded decimals. ## Result ([1, -2, 0], [5, 1, 0], [1, 0], [-11, -14], [-10, -14], [-1, 0], No) 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: ef64005b81d303883cf84a6d43c25fdae1d1445b91b5007b9a5cff8480e6370e - candidate record SHA-256: b089d8e8d318ffd9e0b49f9207a4674269e386cf338f3f3834804e6432c0550e - semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5 - question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261027-r05-general-linear-map-0061.json)