Question 12709: Linearity in the First Slot with the Second Slot Fixed

Original question and givens

Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[-2, 1], alpha=1, beta=-2, fixed y=[3, 4]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).

• G001 — Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[-2, 1], alpha=1, beta=-2, fixed y=[3, 4]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).

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-real-bilinear-form

Step-by-step derivation

fixed real matrix=[[-4, -2], [2, 4]];B(x,y)=sum_i sum_j x_i*A_ij*y_j

distributivity proves real linearity in each slot separately for all vectors and scalars

combined coordinate0 alpha term := mul(1, 1)

combined coordinate0 alpha term = 1

combined coordinate0 beta term := mul(-2, -2)

combined coordinate0 beta term = 4

combined coordinate0 sum := sum(1, 4)

combined coordinate0 sum = 5

combined coordinate1 alpha term := mul(1, 1)

combined coordinate1 alpha term = 1

combined coordinate1 beta term := mul(-2, 1)

combined coordinate1 beta term = -2

combined coordinate1 sum := sum(1, -2)

combined coordinate1 sum = -1

expanding first slot;alpha=1;beta=-2;u=[1, 1];v=[-2, 1];fixed other input=[3, 4]

combined=[5, -1];other slot held fixed

combination lhs:matrix=[[-4, -2], [2, 4]];first slot=[5, -1];second slot=[3, 4]

B(first,second)=first^T*(matrix*second);slot order fixed

combination lhs matrix row0 column0 term := mul(-4, 3)

combination lhs matrix row0 column0 term = -12

combination lhs matrix row0 column1 term := mul(-2, 4)

combination lhs matrix row0 column1 term = -8

combination lhs matrix-times-second row0 := sum(-12, -8)

combination lhs matrix-times-second row0 = -20

combination lhs matrix row1 column0 term := mul(2, 3)

combination lhs matrix row1 column0 term = 6

combination lhs matrix row1 column1 term := mul(4, 4)

combination lhs matrix row1 column1 term = 16

combination lhs matrix-times-second row1 := sum(6, 16)

combination lhs matrix-times-second row1 = 22

combination lhs first-slot coordinate0 term := mul(5, -20)

combination lhs first-slot coordinate0 term = -100

combination lhs first-slot coordinate1 term := mul(-1, 22)

combination lhs first-slot coordinate1 term = -22

combination lhs final first-slot dot sum := sum(-100, -22)

combination lhs final first-slot dot sum = -122

u form:matrix=[[-4, -2], [2, 4]];first slot=[1, 1];second slot=[3, 4]

B(first,second)=first^T*(matrix*second);slot order fixed

u form matrix row0 column0 term := mul(-4, 3)

u form matrix row0 column0 term = -12

u form matrix row0 column1 term := mul(-2, 4)

u form matrix row0 column1 term = -8

u form matrix-times-second row0 := sum(-12, -8)

u form matrix-times-second row0 = -20

u form matrix row1 column0 term := mul(2, 3)

u form matrix row1 column0 term = 6

u form matrix row1 column1 term := mul(4, 4)

u form matrix row1 column1 term = 16

u form matrix-times-second row1 := sum(6, 16)

u form matrix-times-second row1 = 22

u form first-slot coordinate0 term := mul(1, -20)

u form first-slot coordinate0 term = -20

u form first-slot coordinate1 term := mul(1, 22)

u form first-slot coordinate1 term = 22

u form final first-slot dot sum := sum(-20, 22)

u form final first-slot dot sum = 2

v form:matrix=[[-4, -2], [2, 4]];first slot=[-2, 1];second slot=[3, 4]

B(first,second)=first^T*(matrix*second);slot order fixed

v form matrix row0 column0 term := mul(-4, 3)

v form matrix row0 column0 term = -12

v form matrix row0 column1 term := mul(-2, 4)

v form matrix row0 column1 term = -8

v form matrix-times-second row0 := sum(-12, -8)

v form matrix-times-second row0 = -20

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

v form matrix row1 column0 term = 6

v form matrix row1 column1 term := mul(4, 4)

v form matrix row1 column1 term = 16

v form matrix-times-second row1 := sum(6, 16)

v form matrix-times-second row1 = 22

v form first-slot coordinate0 term := mul(-2, -20)

v form first-slot coordinate0 term = 40

v form first-slot coordinate1 term := mul(1, 22)

v form first-slot coordinate1 term = 22

v form final first-slot dot sum := sum(40, 22)

v form final first-slot dot sum = 62

rhs alpha form value := mul(1, 2)

rhs alpha form value = 2

rhs beta form value := mul(-2, 62)

rhs beta form value = -124

combination rhs sum := sum(2, -124)

combination rhs sum = -122

fixed matrix formula;first slot expanded;lhs=-122;rhs=-122

equal=Yes;universal justification is matrix distributivity,not one passing sample

Reverse checks

• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Recompute combination rhs sum with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute rhs beta form value with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute rhs alpha form value with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute v form final first-slot dot sum with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute v form first-slot coordinate1 term with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute v form first-slot coordinate0 term with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute v form matrix-times-second row1 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute v form matrix row1 column1 term with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute v form matrix row1 column0 term with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute v form matrix-times-second row0 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute v form matrix row0 column1 term with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute v form matrix row0 column0 term with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Recompute u form final first-slot dot sum with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute u form first-slot coordinate1 term with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute u form first-slot coordinate0 term with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute u form matrix-times-second row1 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute u form matrix row1 column1 term with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute u form matrix row1 column0 term with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute u form matrix-times-second row0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute u form matrix row0 column1 term with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute u form matrix 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 combination lhs final first-slot dot sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute combination lhs first-slot coordinate1 term with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute combination lhs first-slot coordinate0 term with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute combination lhs matrix-times-second row1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute combination lhs matrix row1 column1 term with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute combination lhs matrix row1 column0 term with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute combination lhs matrix-times-second row0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute combination lhs matrix row0 column1 term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute combination lhs matrix row0 column0 term with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute combined coordinate1 sum with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute combined coordinate1 beta term with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute combined coordinate1 alpha term with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute combined coordinate0 sum with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute combined coordinate0 beta term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute combined coordinate0 alpha term with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(1, -2, [5, -1], -122, -122, 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.