Question 12771: Linearity in the Second Slot with the First Slot Fixed

Original question and givens

Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [-4, -2]]. 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. fixed x=[-2, 3], u=[1, -2], v=[3, 1], alpha=-2, beta=3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).

• G001 — Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [-4, -2]]. 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. fixed x=[-2, 3], u=[1, -2], v=[3, 1], alpha=-2, beta=3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),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=[[8, 3], [-4, -2]];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(-2, 1)

combined coordinate0 alpha term = -2

combined coordinate0 beta term := mul(3, 3)

combined coordinate0 beta term = 9

combined coordinate0 sum := sum(-2, 9)

combined coordinate0 sum = 7

combined coordinate1 alpha term := mul(-2, -2)

combined coordinate1 alpha term = 4

combined coordinate1 beta term := mul(3, 1)

combined coordinate1 beta term = 3

combined coordinate1 sum := sum(4, 3)

combined coordinate1 sum = 7

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

combined=[7, 7];other slot held fixed

combination lhs:matrix=[[8, 3], [-4, -2]];first slot=[-2, 3];second slot=[7, 7]

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

combination lhs matrix row0 column0 term := mul(8, 7)

combination lhs matrix row0 column0 term = 56

combination lhs matrix row0 column1 term := mul(3, 7)

combination lhs matrix row0 column1 term = 21

combination lhs matrix-times-second row0 := sum(56, 21)

combination lhs matrix-times-second row0 = 77

combination lhs matrix row1 column0 term := mul(-4, 7)

combination lhs matrix row1 column0 term = -28

combination lhs matrix row1 column1 term := mul(-2, 7)

combination lhs matrix row1 column1 term = -14

combination lhs matrix-times-second row1 := sum(-28, -14)

combination lhs matrix-times-second row1 = -42

combination lhs first-slot coordinate0 term := mul(-2, 77)

combination lhs first-slot coordinate0 term = -154

combination lhs first-slot coordinate1 term := mul(3, -42)

combination lhs first-slot coordinate1 term = -126

combination lhs final first-slot dot sum := sum(-154, -126)

combination lhs final first-slot dot sum = -280

u form:matrix=[[8, 3], [-4, -2]];first slot=[-2, 3];second slot=[1, -2]

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

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

u form matrix row0 column0 term = 8

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

u form matrix row0 column1 term = -6

u form matrix-times-second row0 := sum(8, -6)

u form matrix-times-second row0 = 2

u form matrix row1 column0 term := mul(-4, 1)

u form matrix row1 column0 term = -4

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

u form matrix row1 column1 term = 4

u form matrix-times-second row1 := sum(-4, 4)

u form matrix-times-second row1 = 0

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

u form first-slot coordinate0 term = -4

u form first-slot coordinate1 term := mul(3, 0)

u form first-slot coordinate1 term = 0

u form final first-slot dot sum := sum(-4, 0)

u form final first-slot dot sum = -4

v form:matrix=[[8, 3], [-4, -2]];first slot=[-2, 3];second slot=[3, 1]

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

v form matrix row0 column0 term := mul(8, 3)

v form matrix row0 column0 term = 24

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

v form matrix row0 column1 term = 3

v form matrix-times-second row0 := sum(24, 3)

v form matrix-times-second row0 = 27

v form matrix row1 column0 term := mul(-4, 3)

v form matrix row1 column0 term = -12

v form matrix row1 column1 term := mul(-2, 1)

v form matrix row1 column1 term = -2

v form matrix-times-second row1 := sum(-12, -2)

v form matrix-times-second row1 = -14

v form first-slot coordinate0 term := mul(-2, 27)

v form first-slot coordinate0 term = -54

v form first-slot coordinate1 term := mul(3, -14)

v form first-slot coordinate1 term = -42

v form final first-slot dot sum := sum(-54, -42)

v form final first-slot dot sum = -96

rhs alpha form value := mul(-2, -4)

rhs alpha form value = 8

rhs beta form value := mul(3, -96)

rhs beta form value = -288

combination rhs sum := sum(8, -288)

combination rhs sum = -280

fixed matrix formula;second slot expanded;lhs=-280;rhs=-280

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

(-2, 3, [7, 7], -280, -280, 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.