# Question 13030: A Nonzero Affine Translation Fails the Zero-Image Obligation [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-0030) 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. R0025-13030 · q-20261014-advance-20261027-r05-general-linear-map-0030 ## 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. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear. - 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. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear. ## 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 · impliesBefore zero input matrix image:matrix=[[-8, 1, -4], [2, 8, 1]];input=[0, 0, 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 S002 · calculationBefore zero input matrix image row0 column0 term := mul(-8, 0) After zero input matrix image row0 column0 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, S001 S003 · calculationBefore zero input matrix image row0 column1 term := mul(1, 0) After zero input matrix image 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, S002 S004 · calculationBefore zero input matrix image row0 column2 term := mul(-4, 0) After zero input matrix 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, S003 S005 · calculationBefore zero input matrix image row0 sum := sum(0, 0, 0) After zero input matrix image row0 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, S004 S006 · calculationBefore zero input matrix image row1 column0 term := mul(2, 0) After zero input matrix image row1 column0 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, S005 S007 · calculationBefore zero input matrix image row1 column1 term := mul(8, 0) After zero input matrix image row1 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, S006 S008 · calculationBefore zero input matrix image row1 column2 term := mul(1, 0) After zero input matrix 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, S007 S009 · calculationBefore zero input matrix image row1 sum := sum(0, 0, 0) After zero input matrix image row1 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 · calculationBefore T zero coordinate0 plus bias := sum(0, 0) After T zero coordinate0 plus bias = 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, S009 S011 · calculationBefore T zero coordinate1 plus bias := sum(0, 0) After T zero coordinate1 plus bias = 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, S010 S012 · impliesBefore affine map T(v)=A*v+bias;bias=[0, 0];T(0)=[0, 0] After linear=Yes;zero bias leaves the universal matrix formula Conditions: none Linearity requires T(0)=0.A nonzero bias immediately refutes it.With zero bias,matrix distributivity applies to every real linear combination,not merely one passing sample. Step identifiers and dependenciesRule: affine_zero_image_necessary_and_matrix_sufficient · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute T zero coordinate1 plus bias with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute T zero coordinate0 plus bias with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute zero input matrix image row1 sum with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute zero input matrix image row1 column2 term with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute zero input matrix image row1 column1 term with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute zero input matrix image row1 column0 term with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute zero input matrix image row0 sum with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute zero input matrix image row0 column2 term with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute zero input matrix image row0 column1 term with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute zero input matrix image row0 column0 term 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 ([[-8, 1, -4], [2, 8, 1]], [0, 0], [0, 0], Yes, matrix_linear_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: 309203f5c9a2492d6aa12c1d5cf70ac85b8934cca2659197b7d5defeb0bd3444 - candidate record SHA-256: 7a79599df7c3265130f6683303ff9e5d212d95576c2ae71e8fe5cf99f144cc0b - 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-0030.json)