# Question 12756: Linearity in the Second Slot with the First Slot Fixed [Return to catalogue page 64](https://ocme.evemisslab.com/en/questions/page/64/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261027-r04-real-bilinear-form-foundations/#q-20261014-advance-20261027-r04-real-bilinear-form-0056) 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. R0024-12756 · q-20261014-advance-20261027-r04-real-bilinear-form-0056 ## Original question and givens Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 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=[-2, 2], u=[1, -2], v=[2, 1], alpha=-2, beta=2. 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=[[-7, 2], [-2, 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=[-2, 2], u=[1, -2], v=[2, 1], alpha=-2, beta=2. 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](https://ocme.evemisslab.com/en/mko/mko-real-bilinear-form/) ## Step-by-step derivation S001 · impliesBefore fixed real matrix=[[-7, 2], [-2, 1]];B(x,y)=sum_i sum_j x_i*A_ij*y_j After distributivity proves real linearity in each slot separately for all vectors and scalars Conditions: none Distributivity expands any real linear combination in either slot of the finite double sum,giving separate universal linearity certificates.Integer instances evaluate this fixed formula,not prove a black box by sampling. Step identifiers and dependenciesRule: matrix_formula_bilinearity_universal_not_samples · Uses: G001 S002 · calculationBefore combined coordinate0 alpha term := mul(-2, 1) After combined coordinate0 alpha term = -2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore combined coordinate0 beta term := mul(2, 2) After combined coordinate0 beta term = 4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore combined coordinate0 sum := sum(-2, 4) After combined coordinate0 sum = 2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore combined coordinate1 alpha term := mul(-2, -2) After combined coordinate1 alpha term = 4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore combined coordinate1 beta term := mul(2, 1) After combined coordinate1 beta term = 2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore combined coordinate1 sum := sum(4, 2) After combined coordinate1 sum = 6 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore expanding second slot;alpha=-2;beta=2;u=[1, -2];v=[2, 1];fixed other input=[-2, 2] After combined=[2, 6];other slot held fixed Conditions: none Substitute alpha*u+beta*v only in the selected slot,holding the other input fixed.This does not exchange slots by symmetry. Step identifiers and dependenciesRule: bilinear_selected_slot_not_symmetry · Uses: G001, S007 S009 · impliesBefore combination lhs:matrix=[[-7, 2], [-2, 1]];first slot=[-2, 2];second slot=[2, 6] After B(first,second)=first^T*(matrix*second);slot order fixed Conditions: none First multiply A by the second slot,then dot with the first.A general matrix is not symmetric,so the two slots cannot be exchanged. Step identifiers and dependenciesRule: bilinear_ordered_matrix_slots · Uses: G001, S008 S010 · calculationBefore combination lhs matrix row0 column0 term := mul(-7, 2) After combination lhs matrix row0 column0 term = -14 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore combination lhs matrix row0 column1 term := mul(2, 6) After combination lhs matrix row0 column1 term = 12 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore combination lhs matrix-times-second row0 := sum(-14, 12) After combination lhs matrix-times-second row0 = -2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore combination lhs matrix row1 column0 term := mul(-2, 2) After combination lhs matrix row1 column0 term = -4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore combination lhs matrix row1 column1 term := mul(1, 6) After combination lhs matrix row1 column1 term = 6 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore combination lhs matrix-times-second row1 := sum(-4, 6) After combination lhs matrix-times-second row1 = 2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore combination lhs first-slot coordinate0 term := mul(-2, -2) After combination lhs first-slot coordinate0 term = 4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore combination lhs first-slot coordinate1 term := mul(2, 2) After combination lhs first-slot coordinate1 term = 4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore combination lhs final first-slot dot sum := sum(4, 4) After combination lhs final first-slot dot sum = 8 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore u form:matrix=[[-7, 2], [-2, 1]];first slot=[-2, 2];second slot=[1, -2] After B(first,second)=first^T*(matrix*second);slot order fixed Conditions: none First multiply A by the second slot,then dot with the first.A general matrix is not symmetric,so the two slots cannot be exchanged. Step identifiers and dependenciesRule: bilinear_ordered_matrix_slots · Uses: G001, S018 S020 · calculationBefore u form matrix row0 column0 term := mul(-7, 1) After u form matrix row0 column0 term = -7 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore u form matrix row0 column1 term := mul(2, -2) After u form matrix row0 column1 term = -4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore u form matrix-times-second row0 := sum(-7, -4) After u form matrix-times-second row0 = -11 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore u form matrix row1 column0 term := mul(-2, 1) After u form matrix row1 column0 term = -2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore u form matrix row1 column1 term := mul(1, -2) After u form matrix row1 column1 term = -2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore u form matrix-times-second row1 := sum(-2, -2) After u form matrix-times-second row1 = -4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore u form first-slot coordinate0 term := mul(-2, -11) After u form first-slot coordinate0 term = 22 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore u form first-slot coordinate1 term := mul(2, -4) After u form first-slot coordinate1 term = -8 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore u form final first-slot dot sum := sum(22, -8) After u form final first-slot dot sum = 14 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore v form:matrix=[[-7, 2], [-2, 1]];first slot=[-2, 2];second slot=[2, 1] After B(first,second)=first^T*(matrix*second);slot order fixed Conditions: none First multiply A by the second slot,then dot with the first.A general matrix is not symmetric,so the two slots cannot be exchanged. Step identifiers and dependenciesRule: bilinear_ordered_matrix_slots · Uses: G001, S028 S030 · calculationBefore v form matrix row0 column0 term := mul(-7, 2) After v form matrix row0 column0 term = -14 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore v form matrix row0 column1 term := mul(2, 1) After v form matrix row0 column1 term = 2 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore v form matrix-times-second row0 := sum(-14, 2) After v form matrix-times-second row0 = -12 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore v form matrix row1 column0 term := mul(-2, 2) After v form matrix row1 column0 term = -4 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore v form matrix row1 column1 term := mul(1, 1) After v form matrix row1 column1 term = 1 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore v form matrix-times-second row1 := sum(-4, 1) After v form matrix-times-second row1 = -3 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore v form first-slot coordinate0 term := mul(-2, -12) After v form first-slot coordinate0 term = 24 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore v form first-slot coordinate1 term := mul(2, -3) After v form first-slot coordinate1 term = -6 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore v form final first-slot dot sum := sum(24, -6) After v form final first-slot dot sum = 18 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore rhs alpha form value := mul(-2, 14) After rhs alpha form value = -28 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore rhs beta form value := mul(2, 18) After rhs beta form value = 36 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore combination rhs sum := sum(-28, 36) After combination rhs sum = 8 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · impliesBefore fixed matrix formula;second slot expanded;lhs=8;rhs=8 After equal=Yes;universal justification is matrix distributivity,not one passing sample Conditions: none The equal numerical sides check this substitution.The all-real-vector-and-scalar result follows from matrix distributivity above,not extrapolation from one passing instance. Step identifiers and dependenciesRule: bilinear_slot_identity_and_formula_certificate · Uses: G001, S041 ## 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, 2, [2, 6], 8, 8, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 9798a009a8d8f5b026498944893883faa5fe2404274ae61297e69d5b4bdd24f5 - candidate record SHA-256: 2997de12877fb9e756673542a51db56f81bc9fac724ee4fce9e4d663fc529a99 - semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35 - question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261027-r04-real-bilinear-form-0056.json)