Question 32855: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-8, 5] and c=5, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-8, 5] and c=5, 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(5, -8)

Homogeneity ORIGINAL c times u ORIGINAL component0 = -40

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(5, 5)

Homogeneity ORIGINAL c times u ORIGINAL component1 = 25

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(-8, -40)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = 320

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(1, 25)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = 25

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(320, 25)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = 345

source roles = [Homogeneity LEFT M of scaled input, 0, [-8, 1], [-40, 25]]

derived = [[320, 25], 345]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -40)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 80

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(5, 25)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = 125

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(80, 125)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = 205

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 5], [-40, 25]]

derived = [[80, 125], 205]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(-8, -8)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 64

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(1, 5)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 5

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(64, 5)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 69

source roles = [Homogeneity RIGHT M of u, 0, [-8, 1], [-8, 5]]

derived = [[64, 5], 69]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, -8)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = 16

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(5, 5)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 25

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(16, 25)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 41

source roles = [Homogeneity RIGHT M of u, 1, [-2, 5], [-8, 5]]

derived = [[16, 25], 41]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(5, 69)

Homogeneity RIGHT c times image ORIGINAL component0 = 345

Homogeneity RIGHT c times image ORIGINAL component1 := mul(5, 41)

Homogeneity RIGHT c times image ORIGINAL component1 = 205

source roles = [homogeneity_values, [[-8, 1], [-2, 5]], [-8, 5], 5]

derived = [[345, 205], [345, 205], [345, 205] = [345, 205]]

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

[345, 205] = [345, 205]

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.