Question 13809: Evaluate a Candidate Bilinear Form without Presuming an Inner Product

Original question and givens

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

• G001 — Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.

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-inner-product-space

Step-by-step derivation

candidate real R2 form;matrix=[[10, -1], [-1, 2]];B(x,y)=x^T*A*y

matrix distributivity supplies bilinearity;other inner-product obligations remain separate

candidate evaluation matrix second slot:matrix=[[10, -1], [-1, 2]];input=[3, 4]

input dimension=2;output dimension=2;rows dot input

candidate evaluation matrix second slot row0 column0 product := mul(10, 3)

candidate evaluation matrix second slot row0 column0 product = 30

candidate evaluation matrix second slot row0 column1 product := mul(-1, 4)

candidate evaluation matrix second slot row0 column1 product = -4

candidate evaluation matrix second slot row0 sum := sum(30, -4)

candidate evaluation matrix second slot row0 sum = 26

candidate evaluation matrix second slot row1 column0 product := mul(-1, 3)

candidate evaluation matrix second slot row1 column0 product = -3

candidate evaluation matrix second slot row1 column1 product := mul(2, 4)

candidate evaluation matrix second slot row1 column1 product = 8

candidate evaluation matrix second slot row1 sum := sum(-3, 8)

candidate evaluation matrix second slot row1 sum = 5

candidate evaluation first coordinate0 product := mul(1, 26)

candidate evaluation first coordinate0 product = 26

candidate evaluation first coordinate1 product := mul(-2, 5)

candidate evaluation first coordinate1 product = -10

candidate evaluation bilinear sum := sum(26, -10)

candidate evaluation bilinear sum = 16

Reverse checks

• S011 · premise_and_result_audit — Recompute candidate evaluation bilinear sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute candidate evaluation first coordinate1 product with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute candidate evaluation first coordinate0 product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute candidate evaluation matrix second slot row1 sum with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute candidate evaluation matrix second slot row1 column1 product with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute candidate evaluation matrix second slot row1 column0 product with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute candidate evaluation matrix second slot row0 sum with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute candidate evaluation matrix second slot row0 column1 product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute candidate evaluation matrix second slot row0 column0 product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[10, -1], [-1, 2]], [1, -2], [3, 4], 16)

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.