Question 32869: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[6, 19] and c=4, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[6, 19] and c=4, 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(4, 6)

Homogeneity ORIGINAL c times u ORIGINAL component0 = 24

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(4, 19)

Homogeneity ORIGINAL c times u ORIGINAL component1 = 76

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(6, 24)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = 144

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(3, 76)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = 228

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(144, 228)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = 372

source roles = [Homogeneity LEFT M of scaled input, 0, [6, 3], [24, 76]]

derived = [[144, 228], 372]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, 24)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = -48

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(19, 76)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = 1444

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(-48, 1444)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = 1396

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 19], [24, 76]]

derived = [[-48, 1444], 1396]

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, 19)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 57

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(36, 57)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 93

source roles = [Homogeneity RIGHT M of u, 0, [6, 3], [6, 19]]

derived = [[36, 57], 93]

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(19, 19)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 361

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(-12, 361)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 349

source roles = [Homogeneity RIGHT M of u, 1, [-2, 19], [6, 19]]

derived = [[-12, 361], 349]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(4, 93)

Homogeneity RIGHT c times image ORIGINAL component0 = 372

Homogeneity RIGHT c times image ORIGINAL component1 := mul(4, 349)

Homogeneity RIGHT c times image ORIGINAL component1 = 1396

source roles = [homogeneity_values, [[6, 3], [-2, 19]], [6, 19], 4]

derived = [[372, 1396], [372, 1396], [372, 1396] = [372, 1396]]

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

[372, 1396] = [372, 1396]

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.