Question 13048: A Nonzero Affine Translation Fails the Zero-Image Obligation

Original question and givens

Use the complete rectangular matrix A=[[10, -2, -5], [0, -10, 0]] 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=[3, -2]. 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=[[10, -2, -5], [0, -10, 0]] 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=[3, -2]. 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

Step-by-step derivation

zero input matrix image:matrix=[[10, -2, -5], [0, -10, 0]];input=[0, 0, 0]

R^3->R^2;each output is its original row dot input

zero input matrix image row0 column0 term := mul(10, 0)

zero input matrix image row0 column0 term = 0

zero input matrix image row0 column1 term := mul(-2, 0)

zero input matrix image row0 column1 term = 0

zero input matrix image row0 column2 term := mul(-5, 0)

zero input matrix image row0 column2 term = 0

zero input matrix image row0 sum := sum(0, 0, 0)

zero input matrix image row0 sum = 0

zero input matrix image row1 column0 term := mul(0, 0)

zero input matrix image row1 column0 term = 0

zero input matrix image row1 column1 term := mul(-10, 0)

zero input matrix image row1 column1 term = 0

zero input matrix image row1 column2 term := mul(0, 0)

zero input matrix image row1 column2 term = 0

zero input matrix image row1 sum := sum(0, 0, 0)

zero input matrix image row1 sum = 0

T zero coordinate0 plus bias := sum(0, 3)

T zero coordinate0 plus bias = 3

T zero coordinate1 plus bias := sum(0, -2)

T zero coordinate1 plus bias = -2

affine map T(v)=A*v+bias;bias=[3, -2];T(0)=[3, -2]

linear=No;zero bias leaves the universal matrix formula

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

([[10, -2, -5], [0, -10, 0]], [3, -2], [3, -2], No, nonzero_translation_is_not_linear)

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.