Question 32896: Vector Image under a 2×2 Matrix

Original question and givens

Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-20, 1], find T(v).

• G001 — Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-20, 1], find T(v).

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

Step-by-step derivation

source roles = [qf-linear-map-image, linear_image, reals, []]

derived = [[VECTOR_STRING]]

Image M times ORIGINAL v ORIGINAL row0 column0 product := mul(8, -20)

Image M times ORIGINAL v ORIGINAL row0 column0 product = -160

Image M times ORIGINAL v ORIGINAL row0 column1 product := mul(1, 1)

Image M times ORIGINAL v ORIGINAL row0 column1 product = 1

Image M times ORIGINAL v ORIGINAL row0 two products sum := sum(-160, 1)

Image M times ORIGINAL v ORIGINAL row0 two products sum = -159

source roles = [Image M times ORIGINAL v, 0, [8, 1], [-20, 1]]

derived = [[-160, 1], -159]

Image M times ORIGINAL v ORIGINAL row1 column0 product := mul(-2, -20)

Image M times ORIGINAL v ORIGINAL row1 column0 product = 40

Image M times ORIGINAL v ORIGINAL row1 column1 product := mul(21, 1)

Image M times ORIGINAL v ORIGINAL row1 column1 product = 21

Image M times ORIGINAL v ORIGINAL row1 two products sum := sum(40, 21)

Image M times ORIGINAL v ORIGINAL row1 two products sum = 61

source roles = [Image M times ORIGINAL v, 1, [-2, 21], [-20, 1]]

derived = [[40, 21], 61]

Reverse checks

• 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 Image M times ORIGINAL v ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Image M times ORIGINAL v ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Image M times ORIGINAL v 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 Image M times ORIGINAL v ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Image M times ORIGINAL v ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Image M times ORIGINAL v 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

[-159, 61]

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.