Question 13025: Compare Both Sides of Additivity for the Supplied Affine Map

Original question and givens

Use the complete rectangular matrix A=[[12, 0, 1], [2, -12, -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, 0], v=[-3, 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, 0, 1], [2, -12, -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, 0], v=[-3, 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, -3)

u plus v coordinate0 second scaled = -3

u plus v coordinate0 sum := sum(1, -3)

u plus v coordinate0 sum = -2

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 = 3

u plus v coordinate2 first scaled := mul(1, 0)

u plus v coordinate2 first scaled = 0

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

u plus v coordinate2 second scaled = -2

u plus v coordinate2 sum := sum(0, -2)

u plus v coordinate2 sum = -2

matrix of u plus v:matrix=[[12, 0, 1], [2, -12, -3]];input=[-2, 3, -2]

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

matrix of u plus v row0 column0 term := mul(12, -2)

matrix of u plus v row0 column0 term = -24

matrix of u plus v row0 column1 term := mul(0, 3)

matrix of u plus v row0 column1 term = 0

matrix of u plus v row0 column2 term := mul(1, -2)

matrix of u plus v row0 column2 term = -2

matrix of u plus v row0 sum := sum(-24, 0, -2)

matrix of u plus v row0 sum = -26

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

matrix of u plus v row1 column0 term = -4

matrix of u plus v row1 column1 term := mul(-12, 3)

matrix of u plus v row1 column1 term = -36

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

matrix of u plus v row1 column2 term = 6

matrix of u plus v row1 sum := sum(-4, -36, 6)

matrix of u plus v row1 sum = -34

matrix of u:matrix=[[12, 0, 1], [2, -12, -3]];input=[1, 2, 0]

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(0, 2)

matrix of u row0 column1 term = 0

matrix of u row0 column2 term := mul(1, 0)

matrix of u row0 column2 term = 0

matrix of u row0 sum := sum(12, 0, 0)

matrix of u row0 sum = 12

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(-3, 0)

matrix of u row1 column2 term = 0

matrix of u row1 sum := sum(2, -24, 0)

matrix of u row1 sum = -22

matrix of v:matrix=[[12, 0, 1], [2, -12, -3]];input=[-3, 1, -2]

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

matrix of v row0 column0 term := mul(12, -3)

matrix of v row0 column0 term = -36

matrix of v row0 column1 term := mul(0, 1)

matrix of v row0 column1 term = 0

matrix of v row0 column2 term := mul(1, -2)

matrix of v row0 column2 term = -2

matrix of v row0 sum := sum(-36, 0, -2)

matrix of v row0 sum = -38

matrix of v row1 column0 term := mul(2, -3)

matrix of v row1 column0 term = -6

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

matrix of v row1 column1 term = -12

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

matrix of v row1 column2 term = 6

matrix of v row1 sum := sum(-6, -12, 6)

matrix of v row1 sum = -12

affine lhs coordinate0 := sum(-26, 0)

affine lhs coordinate0 = -26

T u coordinate0 := sum(12, 0)

T u coordinate0 = 12

T v coordinate0 := sum(-38, 0)

T v coordinate0 = -38

affine rhs coordinate0 := sum(12, -38)

affine rhs coordinate0 = -26

additivity signed residual0 := sub(-26, -26)

additivity signed residual0 = 0

affine lhs coordinate1 := sum(-34, 0)

affine lhs coordinate1 = -34

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

T u coordinate1 = -22

T v coordinate1 := sum(-12, 0)

T v coordinate1 = -12

affine rhs coordinate1 := sum(-22, -12)

affine rhs coordinate1 = -34

additivity signed residual1 := sub(-34, -34)

additivity signed residual1 = 0

bias=[0, 0];T(u+v)=[-26, -34];T(u)+T(v)=[-26, -34]

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, 0], [-3, 1, -2], [0, 0], [-26, -34], [-26, -34], [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.