Question 32846: Linearity of an Explicit Affine Map

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. The affine map F(x)=Mx+b has b=[-20, 0]. Is F linear on the whole R²?

• 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. The affine map F(x)=Mx+b has b=[-20, 0]. Is F linear on the whole R²?

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-affine-test, affine_linearity, reals, []]

derived = [[zero_offset_linearity_YesNo_STRING]]

Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product := mul(8, 0)

Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product = 0

Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product := mul(1, 0)

Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product = 0

Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum := sum(0, 0)

Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum = 0

source roles = [Affine ORIGINAL matrix at zero, 0, [8, 1], [0, 0]]

derived = [[0, 0], 0]

Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product := mul(-2, 0)

Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product = 0

Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product := mul(21, 0)

Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product = 0

Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum := sum(0, 0)

Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum = 0

source roles = [Affine ORIGINAL matrix at zero, 1, [-2, 21], [0, 0]]

derived = [[0, 0], 0]

Affine ORIGINAL M0 plus offset ORIGINAL component0 := sum(0, -20)

Affine ORIGINAL M0 plus offset ORIGINAL component0 = -20

Affine ORIGINAL M0 plus offset ORIGINAL component1 := sum(0, 0)

Affine ORIGINAL M0 plus offset ORIGINAL component1 = 0

source roles = [[[8, 1], [-2, 21]], [-20, 0]]

derived = [[-20, 0], No]

Reverse checks

• S012 · 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.
• S011 · premise_and_result_audit — Recompute Affine ORIGINAL M0 plus offset ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Affine ORIGINAL M0 plus offset ORIGINAL component0 with exact rational arithmetic, not rounded decimals.
• 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 Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero 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 Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Affine ORIGINAL matrix at zero 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

No

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.