# Question 32840: Linearity of an Explicit Affine Map [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-0040) 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. R0064-32840 · q-20261006-advance-20261007-r02-linear-transformation-0040 ## Original question and givens Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-14, 0]. Is F linear on the whole R²? - G001 — Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-14, 0]. Is F linear on the whole R²? ## 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-affine-test, affine_linearity, reals, []] After derived = [[zero_offset_linearity_YesNo_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 Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product := mul(2, 0) After Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product = 0 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 Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product := mul(3, 0) After Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product = 0 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 Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum := sum(0, 0) After Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum = 0 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 · impliesBefore source roles = [Affine ORIGINAL matrix at zero, 0, [2, 3], [0, 0]] After derived = [[0, 0], 0] 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, S004 S006 · calculationBefore Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product := mul(-2, 0) After Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product = 0 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 · calculationBefore Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product := mul(15, 0) After Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product = 0 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, S006 S008 · calculationBefore Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum := sum(0, 0) After Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum = 0 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 · impliesBefore source roles = [Affine ORIGINAL matrix at zero, 1, [-2, 15], [0, 0]] After derived = [[0, 0], 0] 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, S008 S010 · calculationBefore Affine ORIGINAL M0 plus offset ORIGINAL component0 := sum(0, -14) After Affine ORIGINAL M0 plus offset ORIGINAL component0 = -14 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 · calculationBefore Affine ORIGINAL M0 plus offset ORIGINAL component1 := sum(0, 0) After Affine ORIGINAL M0 plus offset ORIGINAL component1 = 0 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, S010 S012 · impliesBefore source roles = [[[2, 3], [-2, 15]], [-14, 0]] After derived = [[-14, 0], No] Conditions: none For the GIVEN whole affine law on R²,F(0)=b.Nonzero b violates necessary linearity;at b0,distributivity makes Mx linear.M may be singular/zero;no determinant/invertibility gate or nonzero-point sampling is needed. Step identifiers and dependenciesRule: LinearMap_AFFINE_whole_form_linear_IFF_zero_offset_NOT_det_gate · Uses: G001, S011 ## Reverse checks - S012 · 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. - S011 · premise_and_result_audit — Recompute Affine ORIGINAL M0 plus offset ORIGINAL component1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Affine ORIGINAL M0 plus offset ORIGINAL component0 with exact rational arithmetic, not rounded decimals. - S009 · 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. - S008 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals. - S005 · 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. - S004 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product 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 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: d37078fa09cc6aa703092b7935b990b9bd95c317a272d883693b70a0248f2017 - candidate record SHA-256: a6883c394735cb2e4a315282921af561b12200cc02ef8940def96b63e556746e - 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-0040.json)