Question 32873: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[10, 23] and c=3, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[10, 23] and c=3, 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(3, 10)

Homogeneity ORIGINAL c times u ORIGINAL component0 = 30

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(3, 23)

Homogeneity ORIGINAL c times u ORIGINAL component1 = 69

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(10, 30)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = 300

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(3, 69)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = 207

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(300, 207)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = 507

source roles = [Homogeneity LEFT M of scaled input, 0, [10, 3], [30, 69]]

derived = [[300, 207], 507]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, 30)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = -60

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(23, 69)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = 1587

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(-60, 1587)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = 1527

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 23], [30, 69]]

derived = [[-60, 1587], 1527]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(10, 10)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 100

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(3, 23)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 69

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(100, 69)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 169

source roles = [Homogeneity RIGHT M of u, 0, [10, 3], [10, 23]]

derived = [[100, 69], 169]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 10)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = -20

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(23, 23)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 529

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(-20, 529)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 509

source roles = [Homogeneity RIGHT M of u, 1, [-2, 23], [10, 23]]

derived = [[-20, 529], 509]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(3, 169)

Homogeneity RIGHT c times image ORIGINAL component0 = 507

Homogeneity RIGHT c times image ORIGINAL component1 := mul(3, 509)

Homogeneity RIGHT c times image ORIGINAL component1 = 1527

source roles = [homogeneity_values, [[10, 3], [-2, 23]], [10, 23], 3]

derived = [[507, 1527], [507, 1527], [507, 1527] = [507, 1527]]

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

[507, 1527] = [507, 1527]

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.