Question 32821: Both Values in a Matrix Additivity Identity

Original question and givens

Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[20, 1] and v=[-20, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[20, 1] and v=[-20, 2], compute T(u+v) and T(u)+T(v). 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-additivity-values, additivity_values, reals, []]

derived = [[VECTOR_EQUALS_VECTOR_STRING]]

Additivity ORIGINAL u plus v ORIGINAL component0 := sum(20, -20)

Additivity ORIGINAL u plus v ORIGINAL component0 = 0

Additivity ORIGINAL u plus v ORIGINAL component1 := sum(1, 2)

Additivity ORIGINAL u plus v ORIGINAL component1 = 3

Additivity LEFT M of sum ORIGINAL row0 column0 product := mul(8, 0)

Additivity LEFT M of sum ORIGINAL row0 column0 product = 0

Additivity LEFT M of sum ORIGINAL row0 column1 product := mul(1, 3)

Additivity LEFT M of sum ORIGINAL row0 column1 product = 3

Additivity LEFT M of sum ORIGINAL row0 two products sum := sum(0, 3)

Additivity LEFT M of sum ORIGINAL row0 two products sum = 3

source roles = [Additivity LEFT M of sum, 0, [8, 1], [0, 3]]

derived = [[0, 3], 3]

Additivity LEFT M of sum ORIGINAL row1 column0 product := mul(-2, 0)

Additivity LEFT M of sum ORIGINAL row1 column0 product = 0

Additivity LEFT M of sum ORIGINAL row1 column1 product := mul(21, 3)

Additivity LEFT M of sum ORIGINAL row1 column1 product = 63

Additivity LEFT M of sum ORIGINAL row1 two products sum := sum(0, 63)

Additivity LEFT M of sum ORIGINAL row1 two products sum = 63

source roles = [Additivity LEFT M of sum, 1, [-2, 21], [0, 3]]

derived = [[0, 63], 63]

Additivity RIGHT M of u ORIGINAL row0 column0 product := mul(8, 20)

Additivity RIGHT M of u ORIGINAL row0 column0 product = 160

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

Additivity RIGHT M of u ORIGINAL row0 column1 product = 1

Additivity RIGHT M of u ORIGINAL row0 two products sum := sum(160, 1)

Additivity RIGHT M of u ORIGINAL row0 two products sum = 161

source roles = [Additivity RIGHT M of u, 0, [8, 1], [20, 1]]

derived = [[160, 1], 161]

Additivity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 20)

Additivity RIGHT M of u ORIGINAL row1 column0 product = -40

Additivity RIGHT M of u ORIGINAL row1 column1 product := mul(21, 1)

Additivity RIGHT M of u ORIGINAL row1 column1 product = 21

Additivity RIGHT M of u ORIGINAL row1 two products sum := sum(-40, 21)

Additivity RIGHT M of u ORIGINAL row1 two products sum = -19

source roles = [Additivity RIGHT M of u, 1, [-2, 21], [20, 1]]

derived = [[-40, 21], -19]

Additivity RIGHT M of v ORIGINAL row0 column0 product := mul(8, -20)

Additivity RIGHT M of v ORIGINAL row0 column0 product = -160

Additivity RIGHT M of v ORIGINAL row0 column1 product := mul(1, 2)

Additivity RIGHT M of v ORIGINAL row0 column1 product = 2

Additivity RIGHT M of v ORIGINAL row0 two products sum := sum(-160, 2)

Additivity RIGHT M of v ORIGINAL row0 two products sum = -158

source roles = [Additivity RIGHT M of v, 0, [8, 1], [-20, 2]]

derived = [[-160, 2], -158]

Additivity RIGHT M of v ORIGINAL row1 column0 product := mul(-2, -20)

Additivity RIGHT M of v ORIGINAL row1 column0 product = 40

Additivity RIGHT M of v ORIGINAL row1 column1 product := mul(21, 2)

Additivity RIGHT M of v ORIGINAL row1 column1 product = 42

Additivity RIGHT M of v ORIGINAL row1 two products sum := sum(40, 42)

Additivity RIGHT M of v ORIGINAL row1 two products sum = 82

source roles = [Additivity RIGHT M of v, 1, [-2, 21], [-20, 2]]

derived = [[40, 42], 82]

Additivity RIGHT images sum ORIGINAL component0 := sum(161, -158)

Additivity RIGHT images sum ORIGINAL component0 = 3

Additivity RIGHT images sum ORIGINAL component1 := sum(-19, 82)

Additivity RIGHT images sum ORIGINAL component1 = 63

source roles = [additivity_values, [[8, 1], [-2, 21]], [20, 1], [-20, 2]]

derived = [[3, 63], [3, 63], [3, 63] = [3, 63]]

Reverse checks

• S030 · 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.
• S029 · premise_and_result_audit — Recompute Additivity RIGHT images sum ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Additivity RIGHT images sum ORIGINAL component0 with exact rational arithmetic, not rounded decimals.
• S027 · 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.
• S026 · premise_and_result_audit — Recompute Additivity RIGHT M of v ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Additivity RIGHT M of v ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Additivity RIGHT M of v ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals.
• S023 · 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.
• S022 · premise_and_result_audit — Recompute Additivity RIGHT M of v ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Additivity RIGHT M of v ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Additivity RIGHT M of v ORIGINAL row0 column0 product 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 Additivity RIGHT M of u ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Additivity RIGHT M of u ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Additivity 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 Additivity RIGHT M of u ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Additivity RIGHT M of u ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Additivity 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 Additivity LEFT M of sum ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Additivity LEFT M of sum ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Additivity LEFT M of sum 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 Additivity LEFT M of sum ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Additivity LEFT M of sum ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Additivity LEFT M of sum ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Additivity ORIGINAL u plus v ORIGINAL component1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Additivity ORIGINAL u plus v 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

[3, 63] = [3, 63]

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.