Question 32805: Both Values in a Matrix Additivity Identity

Original question and givens

Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[4, 1] and v=[-4, 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, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[4, 1] and v=[-4, 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(4, -4)

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(5, 3)

Additivity LEFT M of sum ORIGINAL row1 column1 product = 15

Additivity LEFT M of sum ORIGINAL row1 two products sum := sum(0, 15)

Additivity LEFT M of sum ORIGINAL row1 two products sum = 15

source roles = [Additivity LEFT M of sum, 1, [-2, 5], [0, 3]]

derived = [[0, 15], 15]

Additivity RIGHT M of u ORIGINAL row0 column0 product := mul(-8, 4)

Additivity RIGHT M of u ORIGINAL row0 column0 product = -32

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(-32, 1)

Additivity RIGHT M of u ORIGINAL row0 two products sum = -31

source roles = [Additivity RIGHT M of u, 0, [-8, 1], [4, 1]]

derived = [[-32, 1], -31]

Additivity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 4)

Additivity RIGHT M of u ORIGINAL row1 column0 product = -8

Additivity RIGHT M of u ORIGINAL row1 column1 product := mul(5, 1)

Additivity RIGHT M of u ORIGINAL row1 column1 product = 5

Additivity RIGHT M of u ORIGINAL row1 two products sum := sum(-8, 5)

Additivity RIGHT M of u ORIGINAL row1 two products sum = -3

source roles = [Additivity RIGHT M of u, 1, [-2, 5], [4, 1]]

derived = [[-8, 5], -3]

Additivity RIGHT M of v ORIGINAL row0 column0 product := mul(-8, -4)

Additivity RIGHT M of v ORIGINAL row0 column0 product = 32

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(32, 2)

Additivity RIGHT M of v ORIGINAL row0 two products sum = 34

source roles = [Additivity RIGHT M of v, 0, [-8, 1], [-4, 2]]

derived = [[32, 2], 34]

Additivity RIGHT M of v ORIGINAL row1 column0 product := mul(-2, -4)

Additivity RIGHT M of v ORIGINAL row1 column0 product = 8

Additivity RIGHT M of v ORIGINAL row1 column1 product := mul(5, 2)

Additivity RIGHT M of v ORIGINAL row1 column1 product = 10

Additivity RIGHT M of v ORIGINAL row1 two products sum := sum(8, 10)

Additivity RIGHT M of v ORIGINAL row1 two products sum = 18

source roles = [Additivity RIGHT M of v, 1, [-2, 5], [-4, 2]]

derived = [[8, 10], 18]

Additivity RIGHT images sum ORIGINAL component0 := sum(-31, 34)

Additivity RIGHT images sum ORIGINAL component0 = 3

Additivity RIGHT images sum ORIGINAL component1 := sum(-3, 18)

Additivity RIGHT images sum ORIGINAL component1 = 15

source roles = [additivity_values, [[-8, 1], [-2, 5]], [4, 1], [-4, 2]]

derived = [[3, 15], [3, 15], [3, 15] = [3, 15]]

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, 15] = [3, 15]

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.