Question 12772: 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=[[9, -3], [1, -1]]. 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=[-1, -3], u=[1, -1], v=[-3, 1], alpha=-1, 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=[[9, -3], [1, -1]]. 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=[-1, -3], u=[1, -1], v=[-3, 1], alpha=-1, 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=[[9, -3], [1, -1]];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(-3, -3)

combined coordinate0 beta term = 9

combined coordinate0 sum := sum(-1, 9)

combined coordinate0 sum = 8

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

combined coordinate1 alpha term = 1

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

combined coordinate1 beta term = -3

combined coordinate1 sum := sum(1, -3)

combined coordinate1 sum = -2

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

combined=[8, -2];other slot held fixed

combination lhs:matrix=[[9, -3], [1, -1]];first slot=[-1, -3];second slot=[8, -2]

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

combination lhs matrix row0 column0 term := mul(9, 8)

combination lhs matrix row0 column0 term = 72

combination lhs matrix row0 column1 term := mul(-3, -2)

combination lhs matrix row0 column1 term = 6

combination lhs matrix-times-second row0 := sum(72, 6)

combination lhs matrix-times-second row0 = 78

combination lhs matrix row1 column0 term := mul(1, 8)

combination lhs matrix row1 column0 term = 8

combination lhs matrix row1 column1 term := mul(-1, -2)

combination lhs matrix row1 column1 term = 2

combination lhs matrix-times-second row1 := sum(8, 2)

combination lhs matrix-times-second row1 = 10

combination lhs first-slot coordinate0 term := mul(-1, 78)

combination lhs first-slot coordinate0 term = -78

combination lhs first-slot coordinate1 term := mul(-3, 10)

combination lhs first-slot coordinate1 term = -30

combination lhs final first-slot dot sum := sum(-78, -30)

combination lhs final first-slot dot sum = -108

u form:matrix=[[9, -3], [1, -1]];first slot=[-1, -3];second slot=[1, -1]

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

u form matrix row0 column0 term := mul(9, 1)

u form matrix row0 column0 term = 9

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

u form matrix row0 column1 term = 3

u form matrix-times-second row0 := sum(9, 3)

u form matrix-times-second row0 = 12

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

u form matrix row1 column0 term = 1

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

u form matrix row1 column1 term = 1

u form matrix-times-second row1 := sum(1, 1)

u form matrix-times-second row1 = 2

u form first-slot coordinate0 term := mul(-1, 12)

u form first-slot coordinate0 term = -12

u form first-slot coordinate1 term := mul(-3, 2)

u form first-slot coordinate1 term = -6

u form final first-slot dot sum := sum(-12, -6)

u form final first-slot dot sum = -18

v form:matrix=[[9, -3], [1, -1]];first slot=[-1, -3];second slot=[-3, 1]

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

v form matrix row0 column0 term := mul(9, -3)

v form matrix row0 column0 term = -27

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

v form matrix row0 column1 term = -3

v form matrix-times-second row0 := sum(-27, -3)

v form matrix-times-second row0 = -30

v form matrix row1 column0 term := mul(1, -3)

v form matrix row1 column0 term = -3

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

v form matrix row1 column1 term = -1

v form matrix-times-second row1 := sum(-3, -1)

v form matrix-times-second row1 = -4

v form first-slot coordinate0 term := mul(-1, -30)

v form first-slot coordinate0 term = 30

v form first-slot coordinate1 term := mul(-3, -4)

v form first-slot coordinate1 term = 12

v form final first-slot dot sum := sum(30, 12)

v form final first-slot dot sum = 42

rhs alpha form value := mul(-1, -18)

rhs alpha form value = 18

rhs beta form value := mul(-3, 42)

rhs beta form value = -126

combination rhs sum := sum(18, -126)

combination rhs sum = -108

fixed matrix formula;second slot expanded;lhs=-108;rhs=-108

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, -3, [8, -2], -108, -108, 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.