# Question 13824: Evaluate a Candidate Bilinear Form without Presuming an Inner Product [Return to catalogue page 70](https://ocme.evemisslab.com/en/questions/page/70/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r04-real-inner-product-space-foundations/#q-20261014-advance-20261028-r04-real-inner-product-space-0024) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0026-13824 · q-20261014-advance-20261028-r04-real-inner-product-space-0024 ## Original question and givens Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. 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, -1], y=[-4, -11]. 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=[[25, 2], [2, 5]]. 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, -1], y=[-4, -11]. 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](https://ocme.evemisslab.com/en/mko/mko-real-inner-product-space/) ## Step-by-step derivation S001 · impliesBefore candidate real R2 form;matrix=[[25, 2], [2, 5]];B(x,y)=x^T*A*y After matrix distributivity supplies bilinearity;other inner-product obligations remain separate Conditions: none The matrix formula is bilinear without presuming symmetry or positive definiteness.This protocol is real R²,not a complex-conjugate or arbitrary-dimensional claim. Step identifiers and dependenciesRule: inner_candidate_bilinear_not_presumed_inner_product · Uses: G001 S002 · impliesBefore candidate evaluation matrix second slot:matrix=[[25, 2], [2, 5]];input=[-4, -11] After input dimension=2;output dimension=2;rows dot input Conditions: none Retain column-input/row-output roles for rectangular matrices.Computed values can exceed raw input bounds and are not clipped. Step identifiers and dependenciesRule: matrix_row_column_roles · Uses: G001, S001 S003 · calculationBefore candidate evaluation matrix second slot row0 column0 product := mul(25, -4) After candidate evaluation matrix second slot row0 column0 product = -100 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore candidate evaluation matrix second slot row0 column1 product := mul(2, -11) After candidate evaluation matrix second slot row0 column1 product = -22 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore candidate evaluation matrix second slot row0 sum := sum(-100, -22) After candidate evaluation matrix second slot row0 sum = -122 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore candidate evaluation matrix second slot row1 column0 product := mul(2, -4) After candidate evaluation matrix second slot row1 column0 product = -8 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore candidate evaluation matrix second slot row1 column1 product := mul(5, -11) After candidate evaluation matrix second slot row1 column1 product = -55 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore candidate evaluation matrix second slot row1 sum := sum(-8, -55) After candidate evaluation matrix second slot row1 sum = -63 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore candidate evaluation first coordinate0 product := mul(1, -122) After candidate evaluation first coordinate0 product = -122 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore candidate evaluation first coordinate1 product := mul(-1, -63) After candidate evaluation first coordinate1 product = 63 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore candidate evaluation bilinear sum := sum(-122, 63) After candidate evaluation bilinear sum = -59 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 ## 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 ([[25, 2], [2, 5]], [1, -1], [-4, -11], -59) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 08a5ad350ef46a0150e3590bc9fd0a545cba252e80c498a3a90575620fb7f1fb - candidate record SHA-256: 9992b3dffce17503133511e7b190711af7b2181012cfeaca96b82ae9fd811b3a - semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798 - question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r04-real-inner-product-space-0024.json)