Question 32857: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-6, 7] and c=2, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-6, 7] and c=2, compute T(cu) and cT(u). Answer as left vector = right vector.

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-homogeneity-values, homogeneity_values, reals, []]

derived = [[VECTOR_EQUALS_VECTOR_STRING]]

Homogeneity ORIGINAL c times u ORIGINAL component0 := mul(2, -6)

Homogeneity ORIGINAL c times u ORIGINAL component0 = -12

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(2, 7)

Homogeneity ORIGINAL c times u ORIGINAL component1 = 14

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(-6, -12)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = 72

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(3, 14)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = 42

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(72, 42)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = 114

source roles = [Homogeneity LEFT M of scaled input, 0, [-6, 3], [-12, 14]]

derived = [[72, 42], 114]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -12)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 24

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(7, 14)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = 98

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(24, 98)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = 122

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 7], [-12, 14]]

derived = [[24, 98], 122]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(-6, -6)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 36

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(3, 7)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 21

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(36, 21)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 57

source roles = [Homogeneity RIGHT M of u, 0, [-6, 3], [-6, 7]]

derived = [[36, 21], 57]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, -6)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = 12

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(7, 7)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 49

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(12, 49)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 61

source roles = [Homogeneity RIGHT M of u, 1, [-2, 7], [-6, 7]]

derived = [[12, 49], 61]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(2, 57)

Homogeneity RIGHT c times image ORIGINAL component0 = 114

Homogeneity RIGHT c times image ORIGINAL component1 := mul(2, 61)

Homogeneity RIGHT c times image ORIGINAL component1 = 122

source roles = [homogeneity_values, [[-6, 3], [-2, 7]], [-6, 7], 2]

derived = [[114, 122], [114, 122], [114, 122] = [114, 122]]

Reverse checks

• S022 · 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.
• S021 · premise_and_result_audit — Recompute Homogeneity RIGHT c times image ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Homogeneity RIGHT c times image ORIGINAL component0 with exact rational arithmetic, not rounded decimals.
• S019 · 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.
• S018 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals.
• S015 · 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.
• S014 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Homogeneity RIGHT M of u ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals.
• S011 · 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.
• S010 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals.
• S007 · 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.
• S006 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Homogeneity ORIGINAL c times u ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Homogeneity ORIGINAL c times u ORIGINAL component0 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

[114, 122] = [114, 122]

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.