# Question 32872: Both Values in a Matrix Scalar Identity [Return to catalogue page 165](https://ocme.evemisslab.com/en/questions/page/165/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261006-advance-20261007-r02-linear-transformation-foundations/#q-20261006-advance-20261007-r02-linear-transformation-0097) 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. R0064-32872 · q-20261006-advance-20261007-r02-linear-transformation-0097 ## Original question and givens Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[9, 22] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector. - G001 — Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[9, 22] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector. ## 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 source roles = [qf-linear-map-homogeneity-values, homogeneity_values, reals, []] After derived = [[VECTOR_EQUALS_VECTOR_STRING]] Conditions: none Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status. Step identifiers and dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001 S002 · calculationBefore Homogeneity ORIGINAL c times u ORIGINAL component0 := mul(-2, 9) After Homogeneity ORIGINAL c times u ORIGINAL component0 = -18 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(-2, 22) After Homogeneity ORIGINAL c times u ORIGINAL component1 = -44 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(9, -18) After Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = -162 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(2, -44) After Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = -88 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(-162, -88) After Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = -250 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Homogeneity LEFT M of scaled input, 0, [9, 2], [-18, -44]] After derived = [[-162, -88], -250] Conditions: none In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors. Step identifiers and dependenciesRule: LinearMap_EACH_row_by_column_component_ALL_products · Uses: G001, S006 S008 · calculationBefore Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -18) After Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 36 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(22, -44) After Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = -968 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(36, -968) After Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = -932 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 22], [-18, -44]] After derived = [[36, -968], -932] Conditions: none In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors. Step identifiers and dependenciesRule: LinearMap_EACH_row_by_column_component_ALL_products · Uses: G001, S010 S012 · calculationBefore Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(9, 9) After Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 81 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(2, 22) After Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 44 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(81, 44) After Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 125 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore source roles = [Homogeneity RIGHT M of u, 0, [9, 2], [9, 22]] After derived = [[81, 44], 125] Conditions: none In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors. Step identifiers and dependenciesRule: LinearMap_EACH_row_by_column_component_ALL_products · Uses: G001, S014 S016 · calculationBefore Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 9) After Homogeneity RIGHT M of u ORIGINAL row1 column0 product = -18 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(22, 22) After Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 484 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(-18, 484) After Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 466 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore source roles = [Homogeneity RIGHT M of u, 1, [-2, 22], [9, 22]] After derived = [[-18, 484], 466] Conditions: none In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors. Step identifiers and dependenciesRule: LinearMap_EACH_row_by_column_component_ALL_products · Uses: G001, S018 S020 · calculationBefore Homogeneity RIGHT c times image ORIGINAL component0 := mul(-2, 125) After Homogeneity RIGHT c times image ORIGINAL component0 = -250 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Homogeneity RIGHT c times image ORIGINAL component1 := mul(-2, 466) After Homogeneity RIGHT c times image ORIGINAL component1 = -932 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore source roles = [homogeneity_values, [[9, 2], [-2, 22]], [9, 22], -2] After derived = [[-250, -932], [-250, -932], [-250, -932] = [-250, -932]] Conditions: none Compute BOTH actual vectors completely,then compare.Output[left]=[right],not YesNo/a tuple/a point.Zero/negative scalars and zero vectors are legal.These are values in the explicit matrix model,not finite-sample proof of linearity for an arbitrary function. Step identifiers and dependenciesRule: LinearMap_BOTH_actual_vectors_equality_STRING_NOT_sample_linearity_proof · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S021 · premise_and_result_audit — Recompute Homogeneity RIGHT c times image ORIGINAL component1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Homogeneity RIGHT c times image ORIGINAL component0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S018 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S014 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S010 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S006 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Homogeneity ORIGINAL c times u ORIGINAL component1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Homogeneity ORIGINAL c times u ORIGINAL component0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. ## Result [-250, -932] = [-250, -932] 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: f4f2220b91c68f160ed3b8f0387b6f9ed231806ab442b48080d81bc11ec71819 - candidate record SHA-256: e0a8533daa589947a646941e984c8db052b52c7ea3ee8c5987b002de07976e2f - semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b - question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r02-linear-transformation-0097.json)