Question 32806: Both Values in a Matrix Additivity Identity

Original question and givens

Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[5, 1] and v=[-5, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector.

• G001 — Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[5, 1] and v=[-5, 1], 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(5, -5)

Additivity ORIGINAL u plus v ORIGINAL component0 = 0

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

Additivity ORIGINAL u plus v ORIGINAL component1 = 2

Additivity LEFT M of sum ORIGINAL row0 column0 product := mul(-7, 0)

Additivity LEFT M of sum ORIGINAL row0 column0 product = 0

Additivity LEFT M of sum ORIGINAL row0 column1 product := mul(2, 2)

Additivity LEFT M of sum ORIGINAL row0 column1 product = 4

Additivity LEFT M of sum ORIGINAL row0 two products sum := sum(0, 4)

Additivity LEFT M of sum ORIGINAL row0 two products sum = 4

source roles = [Additivity LEFT M of sum, 0, [-7, 2], [0, 2]]

derived = [[0, 4], 4]

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(6, 2)

Additivity LEFT M of sum ORIGINAL row1 column1 product = 12

Additivity LEFT M of sum ORIGINAL row1 two products sum := sum(0, 12)

Additivity LEFT M of sum ORIGINAL row1 two products sum = 12

source roles = [Additivity LEFT M of sum, 1, [-2, 6], [0, 2]]

derived = [[0, 12], 12]

Additivity RIGHT M of u ORIGINAL row0 column0 product := mul(-7, 5)

Additivity RIGHT M of u ORIGINAL row0 column0 product = -35

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

Additivity RIGHT M of u ORIGINAL row0 column1 product = 2

Additivity RIGHT M of u ORIGINAL row0 two products sum := sum(-35, 2)

Additivity RIGHT M of u ORIGINAL row0 two products sum = -33

source roles = [Additivity RIGHT M of u, 0, [-7, 2], [5, 1]]

derived = [[-35, 2], -33]

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

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

Additivity RIGHT M of u ORIGINAL row1 column1 product := mul(6, 1)

Additivity RIGHT M of u ORIGINAL row1 column1 product = 6

Additivity RIGHT M of u ORIGINAL row1 two products sum := sum(-10, 6)

Additivity RIGHT M of u ORIGINAL row1 two products sum = -4

source roles = [Additivity RIGHT M of u, 1, [-2, 6], [5, 1]]

derived = [[-10, 6], -4]

Additivity RIGHT M of v ORIGINAL row0 column0 product := mul(-7, -5)

Additivity RIGHT M of v ORIGINAL row0 column0 product = 35

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

Additivity RIGHT M of v ORIGINAL row0 column1 product = 2

Additivity RIGHT M of v ORIGINAL row0 two products sum := sum(35, 2)

Additivity RIGHT M of v ORIGINAL row0 two products sum = 37

source roles = [Additivity RIGHT M of v, 0, [-7, 2], [-5, 1]]

derived = [[35, 2], 37]

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

Additivity RIGHT M of v ORIGINAL row1 column0 product = 10

Additivity RIGHT M of v ORIGINAL row1 column1 product := mul(6, 1)

Additivity RIGHT M of v ORIGINAL row1 column1 product = 6

Additivity RIGHT M of v ORIGINAL row1 two products sum := sum(10, 6)

Additivity RIGHT M of v ORIGINAL row1 two products sum = 16

source roles = [Additivity RIGHT M of v, 1, [-2, 6], [-5, 1]]

derived = [[10, 6], 16]

Additivity RIGHT images sum ORIGINAL component0 := sum(-33, 37)

Additivity RIGHT images sum ORIGINAL component0 = 4

Additivity RIGHT images sum ORIGINAL component1 := sum(-4, 16)

Additivity RIGHT images sum ORIGINAL component1 = 12

source roles = [additivity_values, [[-7, 2], [-2, 6]], [5, 1], [-5, 1]]

derived = [[4, 12], [4, 12], [4, 12] = [4, 12]]

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

[4, 12] = [4, 12]

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.