Question 32868: Both Values in a Matrix Scalar Identity

Original question and givens

Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[5, 18] and c=-3, compute T(cu) and cT(u). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[5, 18] 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, 5)

Homogeneity ORIGINAL c times u ORIGINAL component0 = -15

Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(-3, 18)

Homogeneity ORIGINAL c times u ORIGINAL component1 = -54

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(5, -15)

Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = -75

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(2, -54)

Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = -108

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(-75, -108)

Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = -183

source roles = [Homogeneity LEFT M of scaled input, 0, [5, 2], [-15, -54]]

derived = [[-75, -108], -183]

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -15)

Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 30

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(18, -54)

Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = -972

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(30, -972)

Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = -942

source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 18], [-15, -54]]

derived = [[30, -972], -942]

Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(5, 5)

Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 25

Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(2, 18)

Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 36

Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(25, 36)

Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 61

source roles = [Homogeneity RIGHT M of u, 0, [5, 2], [5, 18]]

derived = [[25, 36], 61]

Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 5)

Homogeneity RIGHT M of u ORIGINAL row1 column0 product = -10

Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(18, 18)

Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 324

Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(-10, 324)

Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 314

source roles = [Homogeneity RIGHT M of u, 1, [-2, 18], [5, 18]]

derived = [[-10, 324], 314]

Homogeneity RIGHT c times image ORIGINAL component0 := mul(-3, 61)

Homogeneity RIGHT c times image ORIGINAL component0 = -183

Homogeneity RIGHT c times image ORIGINAL component1 := mul(-3, 314)

Homogeneity RIGHT c times image ORIGINAL component1 = -942

source roles = [homogeneity_values, [[5, 2], [-2, 18]], [5, 18], -3]

derived = [[-183, -942], [-183, -942], [-183, -942] = [-183, -942]]

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

[-183, -942] = [-183, -942]

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.