Question 12775: 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=[[12, 0], [5, 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, 0], u=[1, 2], v=[0, 1], alpha=2, beta=0. 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=[[12, 0], [5, 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, 0], u=[1, 2], v=[0, 1], alpha=2, beta=0. 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=[[12, 0], [5, 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(0, 0)

combined coordinate0 beta term = 0

combined coordinate0 sum := sum(2, 0)

combined coordinate0 sum = 2

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

combined coordinate1 alpha term = 4

combined coordinate1 beta term := mul(0, 1)

combined coordinate1 beta term = 0

combined coordinate1 sum := sum(4, 0)

combined coordinate1 sum = 4

expanding second slot;alpha=2;beta=0;u=[1, 2];v=[0, 1];fixed other input=[2, 0]

combined=[2, 4];other slot held fixed

combination lhs:matrix=[[12, 0], [5, 2]];first slot=[2, 0];second slot=[2, 4]

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

combination lhs matrix row0 column0 term := mul(12, 2)

combination lhs matrix row0 column0 term = 24

combination lhs matrix row0 column1 term := mul(0, 4)

combination lhs matrix row0 column1 term = 0

combination lhs matrix-times-second row0 := sum(24, 0)

combination lhs matrix-times-second row0 = 24

combination lhs matrix row1 column0 term := mul(5, 2)

combination lhs matrix row1 column0 term = 10

combination lhs matrix row1 column1 term := mul(2, 4)

combination lhs matrix row1 column1 term = 8

combination lhs matrix-times-second row1 := sum(10, 8)

combination lhs matrix-times-second row1 = 18

combination lhs first-slot coordinate0 term := mul(2, 24)

combination lhs first-slot coordinate0 term = 48

combination lhs first-slot coordinate1 term := mul(0, 18)

combination lhs first-slot coordinate1 term = 0

combination lhs final first-slot dot sum := sum(48, 0)

combination lhs final first-slot dot sum = 48

u form:matrix=[[12, 0], [5, 2]];first slot=[2, 0];second slot=[1, 2]

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

u form matrix row0 column0 term := mul(12, 1)

u form matrix row0 column0 term = 12

u form matrix row0 column1 term := mul(0, 2)

u form matrix row0 column1 term = 0

u form matrix-times-second row0 := sum(12, 0)

u form matrix-times-second row0 = 12

u form matrix row1 column0 term := mul(5, 1)

u form matrix row1 column0 term = 5

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

u form matrix row1 column1 term = 4

u form matrix-times-second row1 := sum(5, 4)

u form matrix-times-second row1 = 9

u form first-slot coordinate0 term := mul(2, 12)

u form first-slot coordinate0 term = 24

u form first-slot coordinate1 term := mul(0, 9)

u form first-slot coordinate1 term = 0

u form final first-slot dot sum := sum(24, 0)

u form final first-slot dot sum = 24

v form:matrix=[[12, 0], [5, 2]];first slot=[2, 0];second slot=[0, 1]

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

v form matrix row0 column0 term := mul(12, 0)

v form matrix row0 column0 term = 0

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

v form matrix row0 column1 term = 0

v form matrix-times-second row0 := sum(0, 0)

v form matrix-times-second row0 = 0

v form matrix row1 column0 term := mul(5, 0)

v form matrix row1 column0 term = 0

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

v form matrix row1 column1 term = 2

v form matrix-times-second row1 := sum(0, 2)

v form matrix-times-second row1 = 2

v form first-slot coordinate0 term := mul(2, 0)

v form first-slot coordinate0 term = 0

v form first-slot coordinate1 term := mul(0, 2)

v form first-slot coordinate1 term = 0

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

v form final first-slot dot sum = 0

rhs alpha form value := mul(2, 24)

rhs alpha form value = 48

rhs beta form value := mul(0, 0)

rhs beta form value = 0

combination rhs sum := sum(48, 0)

combination rhs sum = 48

fixed matrix formula;second slot expanded;lhs=48;rhs=48

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, 0, [2, 4], 48, 48, 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.