Question 13021: Compare Both Sides of Additivity for the Supplied Affine Map

Original question and givens

Use the complete rectangular matrix A=[[8, 3, 0], [-2, -8, 3]] 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=[4, 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=[[8, 3, 0], [-2, -8, 3]] 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=[4, 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, 4)

u plus v coordinate0 second scaled = 4

u plus v coordinate0 sum := sum(1, 4)

u plus v coordinate0 sum = 5

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 = 1

matrix of u plus v:matrix=[[8, 3, 0], [-2, -8, 3]];input=[5, -1, 1]

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

matrix of u plus v row0 column0 term := mul(8, 5)

matrix of u plus v row0 column0 term = 40

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(0, 1)

matrix of u plus v row0 column2 term = 0

matrix of u plus v row0 sum := sum(40, -3, 0)

matrix of u plus v row0 sum = 37

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

matrix of u plus v row1 column0 term = -10

matrix of u plus v row1 column1 term := mul(-8, -1)

matrix of u plus v row1 column1 term = 8

matrix of u plus v row1 column2 term := mul(3, 1)

matrix of u plus v row1 column2 term = 3

matrix of u plus v row1 sum := sum(-10, 8, 3)

matrix of u plus v row1 sum = 1

matrix of u:matrix=[[8, 3, 0], [-2, -8, 3]];input=[1, -2, 3]

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

matrix of u row0 column0 term := mul(8, 1)

matrix of u row0 column0 term = 8

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

matrix of u row0 column1 term = -6

matrix of u row0 column2 term := mul(0, 3)

matrix of u row0 column2 term = 0

matrix of u row0 sum := sum(8, -6, 0)

matrix of u row0 sum = 2

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

matrix of u row1 column0 term = -2

matrix of u row1 column1 term := mul(-8, -2)

matrix of u row1 column1 term = 16

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

matrix of u row1 column2 term = 9

matrix of u row1 sum := sum(-2, 16, 9)

matrix of u row1 sum = 23

matrix of v:matrix=[[8, 3, 0], [-2, -8, 3]];input=[4, 1, -2]

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

matrix of v row0 column0 term := mul(8, 4)

matrix of v row0 column0 term = 32

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

matrix of v row0 column1 term = 3

matrix of v row0 column2 term := mul(0, -2)

matrix of v row0 column2 term = 0

matrix of v row0 sum := sum(32, 3, 0)

matrix of v row0 sum = 35

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

matrix of v row1 column0 term = -8

matrix of v row1 column1 term := mul(-8, 1)

matrix of v row1 column1 term = -8

matrix of v row1 column2 term := mul(3, -2)

matrix of v row1 column2 term = -6

matrix of v row1 sum := sum(-8, -8, -6)

matrix of v row1 sum = -22

affine lhs coordinate0 := sum(37, 0)

affine lhs coordinate0 = 37

T u coordinate0 := sum(2, 0)

T u coordinate0 = 2

T v coordinate0 := sum(35, 0)

T v coordinate0 = 35

affine rhs coordinate0 := sum(2, 35)

affine rhs coordinate0 = 37

additivity signed residual0 := sub(37, 37)

additivity signed residual0 = 0

affine lhs coordinate1 := sum(1, 0)

affine lhs coordinate1 = 1

T u coordinate1 := sum(23, 0)

T u coordinate1 = 23

T v coordinate1 := sum(-22, 0)

T v coordinate1 = -22

affine rhs coordinate1 := sum(23, -22)

affine rhs coordinate1 = 1

additivity signed residual1 := sub(1, 1)

additivity signed residual1 = 0

bias=[0, 0];T(u+v)=[37, 1];T(u)+T(v)=[37, 1]

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], [4, 1, -2], [0, 0], [37, 1], [37, 1], [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.