Question 32814: Both Values in a Matrix Additivity Identity

Original question and givens

Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[13, 1] and v=[-13, 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=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[13, 1] and v=[-13, 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(13, -13)

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(1, 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, [1, 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(14, 2)

Additivity LEFT M of sum ORIGINAL row1 column1 product = 28

Additivity LEFT M of sum ORIGINAL row1 two products sum := sum(0, 28)

Additivity LEFT M of sum ORIGINAL row1 two products sum = 28

source roles = [Additivity LEFT M of sum, 1, [-2, 14], [0, 2]]

derived = [[0, 28], 28]

Additivity RIGHT M of u ORIGINAL row0 column0 product := mul(1, 13)

Additivity RIGHT M of u ORIGINAL row0 column0 product = 13

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(13, 2)

Additivity RIGHT M of u ORIGINAL row0 two products sum = 15

source roles = [Additivity RIGHT M of u, 0, [1, 2], [13, 1]]

derived = [[13, 2], 15]

Additivity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, 13)

Additivity RIGHT M of u ORIGINAL row1 column0 product = -26

Additivity RIGHT M of u ORIGINAL row1 column1 product := mul(14, 1)

Additivity RIGHT M of u ORIGINAL row1 column1 product = 14

Additivity RIGHT M of u ORIGINAL row1 two products sum := sum(-26, 14)

Additivity RIGHT M of u ORIGINAL row1 two products sum = -12

source roles = [Additivity RIGHT M of u, 1, [-2, 14], [13, 1]]

derived = [[-26, 14], -12]

Additivity RIGHT M of v ORIGINAL row0 column0 product := mul(1, -13)

Additivity RIGHT M of v ORIGINAL row0 column0 product = -13

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(-13, 2)

Additivity RIGHT M of v ORIGINAL row0 two products sum = -11

source roles = [Additivity RIGHT M of v, 0, [1, 2], [-13, 1]]

derived = [[-13, 2], -11]

Additivity RIGHT M of v ORIGINAL row1 column0 product := mul(-2, -13)

Additivity RIGHT M of v ORIGINAL row1 column0 product = 26

Additivity RIGHT M of v ORIGINAL row1 column1 product := mul(14, 1)

Additivity RIGHT M of v ORIGINAL row1 column1 product = 14

Additivity RIGHT M of v ORIGINAL row1 two products sum := sum(26, 14)

Additivity RIGHT M of v ORIGINAL row1 two products sum = 40

source roles = [Additivity RIGHT M of v, 1, [-2, 14], [-13, 1]]

derived = [[26, 14], 40]

Additivity RIGHT images sum ORIGINAL component0 := sum(15, -11)

Additivity RIGHT images sum ORIGINAL component0 = 4

Additivity RIGHT images sum ORIGINAL component1 := sum(-12, 40)

Additivity RIGHT images sum ORIGINAL component1 = 28

source roles = [additivity_values, [[1, 2], [-2, 14]], [13, 1], [-13, 1]]

derived = [[4, 28], [4, 28], [4, 28] = [4, 28]]

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, 28] = [4, 28]

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.