# Question 13055: The Matrix Formula Preserves a Whole Linear Combination [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-0080) 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. R0025-13055 · q-20261014-advance-20261027-r05-general-linear-map-0080 ## Original question and givens Use the complete rectangular matrix A=[[-8, 1, -4], [2, 8, 1]] 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. u=[1, 2, 1], v=[-1, 1, -2], alpha=2, beta=1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate). - G001 — Use the complete rectangular matrix A=[[-8, 1, -4], [2, 8, 1]] 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. u=[1, 2, 1], v=[-1, 1, -2], alpha=2, beta=1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate). ## 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 input combination coordinate0 first scaled := mul(2, 1) After input combination coordinate0 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 S002 · calculationBefore input combination coordinate0 second scaled := mul(1, -1) After input combination coordinate0 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, S001 S003 · calculationBefore input combination coordinate0 sum := sum(2, -1) After input combination coordinate0 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, S002 S004 · calculationBefore input combination coordinate1 first scaled := mul(2, 2) After input combination coordinate1 first scaled = 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, S003 S005 · calculationBefore input combination coordinate1 second scaled := mul(1, 1) After input combination 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 input combination coordinate1 sum := sum(4, 1) After input combination coordinate1 sum = 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, S005 S007 · calculationBefore input combination coordinate2 first scaled := mul(2, 1) After input combination coordinate2 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, S006 S008 · calculationBefore input combination coordinate2 second scaled := mul(1, -2) After input combination coordinate2 second 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, S007 S009 · calculationBefore input combination coordinate2 sum := sum(2, -2) After input combination 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 combined image:matrix=[[-8, 1, -4], [2, 8, 1]];input=[1, 5, 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 combined image row0 column0 term := mul(-8, 1) After combined image row0 column0 term = -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, S010 S012 · calculationBefore combined image row0 column1 term := mul(1, 5) After combined image row0 column1 term = 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, S011 S013 · calculationBefore combined image row0 column2 term := mul(-4, 0) After combined image 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 combined image row0 sum := sum(-8, 5, 0) After combined image row0 sum = -3 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 combined image row1 column0 term := mul(2, 1) After combined image 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, S014 S016 · calculationBefore combined image row1 column1 term := mul(8, 5) After combined image row1 column1 term = 40 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 combined image row1 column2 term := mul(1, 0) After combined image 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 combined image row1 sum := sum(2, 40, 0) After combined image row1 sum = 42 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 u image:matrix=[[-8, 1, -4], [2, 8, 1]];input=[1, 2, 1] 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 u image row0 column0 term := mul(-8, 1) After u image row0 column0 term = -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, S019 S021 · calculationBefore u image row0 column1 term := mul(1, 2) After u image row0 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, S020 S022 · calculationBefore u image row0 column2 term := mul(-4, 1) After u image row0 column2 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, S021 S023 · calculationBefore u image row0 sum := sum(-8, 2, -4) After u image 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, S022 S024 · calculationBefore u image row1 column0 term := mul(2, 1) After u image 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 u image row1 column1 term := mul(8, 2) After u image row1 column1 term = 16 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 u image row1 column2 term := mul(1, 1) After u image row1 column2 term = 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, S025 S027 · calculationBefore u image row1 sum := sum(2, 16, 1) After u image row1 sum = 19 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 v image:matrix=[[-8, 1, -4], [2, 8, 1]];input=[-1, 1, -2] 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 v image row0 column0 term := mul(-8, -1) After v image row0 column0 term = 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, S028 S030 · calculationBefore v image row0 column1 term := mul(1, 1) After v image row0 column1 term = 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, S029 S031 · calculationBefore v image row0 column2 term := mul(-4, -2) After v image row0 column2 term = 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, S030 S032 · calculationBefore v image row0 sum := sum(8, 1, 8) After v image row0 sum = 17 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 v image row1 column0 term := mul(2, -1) After v image 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, S032 S034 · calculationBefore v image row1 column1 term := mul(8, 1) After v image row1 column1 term = 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, S033 S035 · calculationBefore v image row1 column2 term := mul(1, -2) After v image row1 column2 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, S034 S036 · calculationBefore v image row1 sum := sum(-2, 8, -2) After v image row1 sum = 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, S035 S037 · calculationBefore output combination coordinate0 first scaled := mul(2, -10) After output combination coordinate0 first scaled = -20 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 output combination coordinate0 second scaled := mul(1, 17) After output combination coordinate0 second scaled = 17 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 output combination coordinate0 sum := sum(-20, 17) After output combination coordinate0 sum = -3 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 output combination coordinate1 first scaled := mul(2, 19) After output combination coordinate1 first scaled = 38 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 output combination coordinate1 second scaled := mul(1, 4) After output combination coordinate1 second scaled = 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, S040 S042 · calculationBefore output combination coordinate1 sum := sum(38, 4) After output combination coordinate1 sum = 42 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 · impliesBefore matrix formula;alpha=2;beta=1;lhs=[-3, 42];rhs=[-3, 42] After all row dot products distribute over all real linear combinations;instance equality=Yes Conditions: none Every output coordinate is a finite sum distributing over arbitrary real scalars.All rows together give a universal linearity certificate;the full coordinate calculation also checks this instance. Step identifiers and dependenciesRule: matrix_combination_universal_coordinate_certificate · Uses: G001, S042 ## Reverse checks - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Recompute output combination coordinate1 sum with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute output combination coordinate1 second scaled with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute output combination coordinate1 first scaled with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute output combination coordinate0 sum with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute output combination coordinate0 second scaled with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute output combination coordinate0 first scaled with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute v image row1 sum with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute v image row1 column2 term with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute v image row1 column1 term with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute v image row1 column0 term with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute v image row0 sum with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute v image row0 column2 term with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute v image row0 column1 term with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute v image 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 u image row1 sum with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute u image row1 column2 term with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute u image row1 column1 term with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute u image row1 column0 term with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute u image row0 sum with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute u image row0 column2 term with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute u image row0 column1 term with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute u image 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 combined image row1 sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute combined image row1 column2 term with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute combined image row1 column1 term with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute combined image row1 column0 term with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute combined image row0 sum with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute combined image row0 column2 term with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute combined image row0 column1 term with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute combined image 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 input combination coordinate2 sum with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute input combination coordinate2 second scaled with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute input combination coordinate2 first scaled with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute input combination coordinate1 sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute input combination coordinate1 second scaled with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute input combination coordinate1 first scaled with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute input combination coordinate0 sum with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute input combination coordinate0 second scaled with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute input combination coordinate0 first scaled with exact rational arithmetic, not rounded decimals. ## Result (2, 1, [1, 5, 0], [-3, 42], [-3, 42], Yes, matrix_formula_universal_certificate) 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: 45fb20dcff681793b4acc866b6663d59a964687da072eff891d990c4045fbdb7 - candidate record SHA-256: 76a80712f3829ae59499bcb36e75f031afa8496692e23957c14ca9cf3a7a7c74 - 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-0080.json)