Question 13085: Evaluate a Possibly Rectangular Real Linear Map

Original question and givens

Use the complete rectangular matrix A=[[-3, -1, 0], [2, 3, 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. vector=[-4, 2, -1]. Give (matrix,vector,image).

• G001 — Use the complete rectangular matrix A=[[-3, -1, 0], [2, 3, 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. vector=[-4, 2, -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=[[-3, -1, 0], [2, 3, 0]];input=[-4, 2, -1]

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

requested matrix image row0 column0 term := mul(-3, -4)

requested matrix image row0 column0 term = 12

requested matrix image row0 column1 term := mul(-1, 2)

requested matrix image row0 column1 term = -2

requested matrix image row0 column2 term := mul(0, -1)

requested matrix image row0 column2 term = 0

requested matrix image row0 sum := sum(12, -2, 0)

requested matrix image row0 sum = 10

requested matrix image row1 column0 term := mul(2, -4)

requested matrix image row1 column0 term = -8

requested matrix image row1 column1 term := mul(3, 2)

requested matrix image row1 column1 term = 6

requested matrix image row1 column2 term := mul(0, -1)

requested matrix image row1 column2 term = 0

requested matrix image row1 sum := sum(-8, 6, 0)

requested matrix image row1 sum = -2

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

([[-3, -1, 0], [2, 3, 0]], [-4, 2, -1], [10, -2])

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.