Question 32870: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[7, 20] and c=-5, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[7, 20] 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, 7)

Homogeneity ORIGINAL c times u ORIGINAL component0 = -35

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(-5, 20)

Homogeneity ORIGINAL c times u ORIGINAL component1 = -100

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(7, -35)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = -245

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(4, -100)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = -400

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(-245, -400)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = -645

source roles = [Homogeneity LEFT M of scaled input, 0, [7, 4], [-35, -100]]

derived = [[-245, -400], -645]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -35)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 70

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(20, -100)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = -2000

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(70, -2000)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = -1930

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 20], [-35, -100]]

derived = [[70, -2000], -1930]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(7, 7)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 49

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(4, 20)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 80

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(49, 80)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 129

source roles = [Homogeneity RIGHT M of u, 0, [7, 4], [7, 20]]

derived = [[49, 80], 129]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 7)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = -14

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(20, 20)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 400

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(-14, 400)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 386

source roles = [Homogeneity RIGHT M of u, 1, [-2, 20], [7, 20]]

derived = [[-14, 400], 386]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(-5, 129)

Homogeneity RIGHT c times image ORIGINAL component0 = -645

Homogeneity RIGHT c times image ORIGINAL component1 := mul(-5, 386)

Homogeneity RIGHT c times image ORIGINAL component1 = -1930

source roles = [homogeneity_values, [[7, 4], [-2, 20]], [7, 20], -5]

derived = [[-645, -1930], [-645, -1930], [-645, -1930] = [-645, -1930]]

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

[-645, -1930] = [-645, -1930]

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.