Question 32804: Both Values in a Matrix Additivity Identity

Original question and givens

Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[3, 1] and v=[-3, -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=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[3, 1] and v=[-3, -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(3, -3)

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 = 0

Additivity LEFT M of sum ORIGINAL row0 column0 product := mul(-9, 0)

Additivity LEFT M of sum ORIGINAL row0 column0 product = 0

Additivity LEFT M of sum ORIGINAL row0 column1 product := mul(4, 0)

Additivity LEFT M of sum ORIGINAL row0 column1 product = 0

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

Additivity LEFT M of sum ORIGINAL row0 two products sum = 0

source roles = [Additivity LEFT M of sum, 0, [-9, 4], [0, 0]]

derived = [[0, 0], 0]

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(4, 0)

Additivity LEFT M of sum ORIGINAL row1 column1 product = 0

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

Additivity LEFT M of sum ORIGINAL row1 two products sum = 0

source roles = [Additivity LEFT M of sum, 1, [-2, 4], [0, 0]]

derived = [[0, 0], 0]

Additivity RIGHT M of u ORIGINAL row0 column0 product := mul(-9, 3)

Additivity RIGHT M of u ORIGINAL row0 column0 product = -27

Additivity RIGHT M of u ORIGINAL row0 column1 product := mul(4, 1)

Additivity RIGHT M of u ORIGINAL row0 column1 product = 4

Additivity RIGHT M of u ORIGINAL row0 two products sum := sum(-27, 4)

Additivity RIGHT M of u ORIGINAL row0 two products sum = -23

source roles = [Additivity RIGHT M of u, 0, [-9, 4], [3, 1]]

derived = [[-27, 4], -23]

Additivity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 3)

Additivity RIGHT M of u ORIGINAL row1 column0 product = -6

Additivity RIGHT M of u ORIGINAL row1 column1 product := mul(4, 1)

Additivity RIGHT M of u ORIGINAL row1 column1 product = 4

Additivity RIGHT M of u ORIGINAL row1 two products sum := sum(-6, 4)

Additivity RIGHT M of u ORIGINAL row1 two products sum = -2

source roles = [Additivity RIGHT M of u, 1, [-2, 4], [3, 1]]

derived = [[-6, 4], -2]

Additivity RIGHT M of v ORIGINAL row0 column0 product := mul(-9, -3)

Additivity RIGHT M of v ORIGINAL row0 column0 product = 27

Additivity RIGHT M of v ORIGINAL row0 column1 product := mul(4, -1)

Additivity RIGHT M of v ORIGINAL row0 column1 product = -4

Additivity RIGHT M of v ORIGINAL row0 two products sum := sum(27, -4)

Additivity RIGHT M of v ORIGINAL row0 two products sum = 23

source roles = [Additivity RIGHT M of v, 0, [-9, 4], [-3, -1]]

derived = [[27, -4], 23]

Additivity RIGHT M of v ORIGINAL row1 column0 product := mul(-2, -3)

Additivity RIGHT M of v ORIGINAL row1 column0 product = 6

Additivity RIGHT M of v ORIGINAL row1 column1 product := mul(4, -1)

Additivity RIGHT M of v ORIGINAL row1 column1 product = -4

Additivity RIGHT M of v ORIGINAL row1 two products sum := sum(6, -4)

Additivity RIGHT M of v ORIGINAL row1 two products sum = 2

source roles = [Additivity RIGHT M of v, 1, [-2, 4], [-3, -1]]

derived = [[6, -4], 2]

Additivity RIGHT images sum ORIGINAL component0 := sum(-23, 23)

Additivity RIGHT images sum ORIGINAL component0 = 0

Additivity RIGHT images sum ORIGINAL component1 := sum(-2, 2)

Additivity RIGHT images sum ORIGINAL component1 = 0

source roles = [additivity_values, [[-9, 4], [-2, 4]], [3, 1], [-3, -1]]

derived = [[0, 0], [0, 0], [0, 0] = [0, 0]]

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

[0, 0] = [0, 0]

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.