Question 13001: Compare Both Sides of Additivity for the Supplied Affine Map

Original question and givens

Use the complete rectangular matrix A=[[-12, -3, -5], [-2, 12, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, -2, -3], v=[-5, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).

• G001 — Use the complete rectangular matrix A=[[-12, -3, -5], [-2, 12, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, -2, -3], v=[-5, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).

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-general-linear-map

Step-by-step derivation

u plus v coordinate0 first scaled := mul(1, 1)

u plus v coordinate0 first scaled = 1

u plus v coordinate0 second scaled := mul(1, -5)

u plus v coordinate0 second scaled = -5

u plus v coordinate0 sum := sum(1, -5)

u plus v coordinate0 sum = -4

u plus v coordinate1 first scaled := mul(1, -2)

u plus v coordinate1 first scaled = -2

u plus v coordinate1 second scaled := mul(1, 1)

u plus v coordinate1 second scaled = 1

u plus v coordinate1 sum := sum(-2, 1)

u plus v coordinate1 sum = -1

u plus v coordinate2 first scaled := mul(1, -3)

u plus v coordinate2 first scaled = -3

u plus v coordinate2 second scaled := mul(1, -2)

u plus v coordinate2 second scaled = -2

u plus v coordinate2 sum := sum(-3, -2)

u plus v coordinate2 sum = -5

matrix of u plus v:matrix=[[-12, -3, -5], [-2, 12, -6]];input=[-4, -1, -5]

R^3->R^2;each output is its original row dot input

matrix of u plus v row0 column0 term := mul(-12, -4)

matrix of u plus v row0 column0 term = 48

matrix of u plus v row0 column1 term := mul(-3, -1)

matrix of u plus v row0 column1 term = 3

matrix of u plus v row0 column2 term := mul(-5, -5)

matrix of u plus v row0 column2 term = 25

matrix of u plus v row0 sum := sum(48, 3, 25)

matrix of u plus v row0 sum = 76

matrix of u plus v row1 column0 term := mul(-2, -4)

matrix of u plus v row1 column0 term = 8

matrix of u plus v row1 column1 term := mul(12, -1)

matrix of u plus v row1 column1 term = -12

matrix of u plus v row1 column2 term := mul(-6, -5)

matrix of u plus v row1 column2 term = 30

matrix of u plus v row1 sum := sum(8, -12, 30)

matrix of u plus v row1 sum = 26

matrix of u:matrix=[[-12, -3, -5], [-2, 12, -6]];input=[1, -2, -3]

R^3->R^2;each output is its original row dot input

matrix of u row0 column0 term := mul(-12, 1)

matrix of u row0 column0 term = -12

matrix of u row0 column1 term := mul(-3, -2)

matrix of u row0 column1 term = 6

matrix of u row0 column2 term := mul(-5, -3)

matrix of u row0 column2 term = 15

matrix of u row0 sum := sum(-12, 6, 15)

matrix of u row0 sum = 9

matrix of u row1 column0 term := mul(-2, 1)

matrix of u row1 column0 term = -2

matrix of u row1 column1 term := mul(12, -2)

matrix of u row1 column1 term = -24

matrix of u row1 column2 term := mul(-6, -3)

matrix of u row1 column2 term = 18

matrix of u row1 sum := sum(-2, -24, 18)

matrix of u row1 sum = -8

matrix of v:matrix=[[-12, -3, -5], [-2, 12, -6]];input=[-5, 1, -2]

R^3->R^2;each output is its original row dot input

matrix of v row0 column0 term := mul(-12, -5)

matrix of v row0 column0 term = 60

matrix of v row0 column1 term := mul(-3, 1)

matrix of v row0 column1 term = -3

matrix of v row0 column2 term := mul(-5, -2)

matrix of v row0 column2 term = 10

matrix of v row0 sum := sum(60, -3, 10)

matrix of v row0 sum = 67

matrix of v row1 column0 term := mul(-2, -5)

matrix of v row1 column0 term = 10

matrix of v row1 column1 term := mul(12, 1)

matrix of v row1 column1 term = 12

matrix of v row1 column2 term := mul(-6, -2)

matrix of v row1 column2 term = 12

matrix of v row1 sum := sum(10, 12, 12)

matrix of v row1 sum = 34

affine lhs coordinate0 := sum(76, 0)

affine lhs coordinate0 = 76

T u coordinate0 := sum(9, 0)

T u coordinate0 = 9

T v coordinate0 := sum(67, 0)

T v coordinate0 = 67

affine rhs coordinate0 := sum(9, 67)

affine rhs coordinate0 = 76

additivity signed residual0 := sub(76, 76)

additivity signed residual0 = 0

affine lhs coordinate1 := sum(26, 0)

affine lhs coordinate1 = 26

T u coordinate1 := sum(-8, 0)

T u coordinate1 = -8

T v coordinate1 := sum(34, 0)

T v coordinate1 = 34

affine rhs coordinate1 := sum(-8, 34)

affine rhs coordinate1 = 26

additivity signed residual1 := sub(26, 26)

additivity signed residual1 = 0

bias=[0, 0];T(u+v)=[76, 26];T(u)+T(v)=[76, 26]

lhs-rhs=[0, 0]=-bias;this instance additive=Yes

Reverse checks

• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Recompute additivity signed residual1 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute affine rhs coordinate1 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute T v coordinate1 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute T u coordinate1 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute affine lhs coordinate1 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute additivity signed residual0 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute affine rhs coordinate0 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute T v coordinate0 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute T u coordinate0 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute affine lhs coordinate0 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute matrix of v row1 sum with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute matrix of v row1 column2 term with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute matrix of v row1 column1 term with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute matrix of v row1 column0 term with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute matrix of v row0 sum with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute matrix of v row0 column2 term with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute matrix of v row0 column1 term with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute matrix of v row0 column0 term with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Recompute matrix of u row1 sum with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute matrix of u row1 column2 term with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute matrix of u row1 column1 term with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute matrix of u row1 column0 term with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute matrix of u row0 sum with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute matrix of u row0 column2 term with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute matrix of u row0 column1 term with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute matrix of u row0 column0 term with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute matrix of u plus v row1 sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute matrix of u plus v row1 column2 term with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute matrix of u plus v row1 column1 term with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute matrix of u plus v row1 column0 term with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute matrix of u plus v row0 sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute matrix of u plus v row0 column2 term with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute matrix of u plus v row0 column1 term with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute matrix of u plus v row0 column0 term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute u plus v coordinate2 sum with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute u plus v coordinate2 second scaled with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute u plus v coordinate2 first scaled with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute u plus v coordinate1 sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute u plus v coordinate1 second scaled with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute u plus v coordinate1 first scaled with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute u plus v coordinate0 sum with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute u plus v coordinate0 second scaled with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute u plus v coordinate0 first scaled with exact rational arithmetic, not rounded decimals.

Result

([1, -2, -3], [-5, 1, -2], [0, 0], [76, 26], [76, 26], [0, 0], Yes)

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.