Question 13078: Evaluate a Possibly Rectangular Real Linear Map

Original question and givens

Use the complete rectangular matrix A=[[-10, -1, 1], [0, 10, 4]] 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. vector=[-2, 0, -1]. Give (matrix,vector,image).

• G001 — Use the complete rectangular matrix A=[[-10, -1, 1], [0, 10, 4]] 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. vector=[-2, 0, -1]. Give (matrix,vector,image).

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

requested matrix image:matrix=[[-10, -1, 1], [0, 10, 4]];input=[-2, 0, -1]

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

requested matrix image row0 column0 term := mul(-10, -2)

requested matrix image row0 column0 term = 20

requested matrix image row0 column1 term := mul(-1, 0)

requested matrix image row0 column1 term = 0

requested matrix image row0 column2 term := mul(1, -1)

requested matrix image row0 column2 term = -1

requested matrix image row0 sum := sum(20, 0, -1)

requested matrix image row0 sum = 19

requested matrix image row1 column0 term := mul(0, -2)

requested matrix image row1 column0 term = 0

requested matrix image row1 column1 term := mul(10, 0)

requested matrix image row1 column1 term = 0

requested matrix image row1 column2 term := mul(4, -1)

requested matrix image row1 column2 term = -4

requested matrix image row1 sum := sum(0, 0, -4)

requested matrix image row1 sum = -4

Reverse checks

• S009 · premise_and_result_audit — Recompute requested matrix image row1 sum with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute requested matrix image row1 column2 term with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute requested matrix image row1 column1 term with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute requested matrix image row1 column0 term with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute requested matrix image row0 sum with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute requested matrix image row0 column2 term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute requested matrix image row0 column1 term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute requested 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, -1, 1], [0, 10, 4]], [-2, 0, -1], [19, -4])

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.