第 12751 題：實雙線性形式

原題與已知條件

雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={"matrix":[[-12,-3],[-5,-4]],"x":[-2,-3],"u":[1,-2],"v":[-3,1],"alpha":-2,"beta":-3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。

• G001 — 雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={"matrix":[[-12,-3],[-5,-4]],"x":[-2,-3],"u":[1,-2],"v":[-3,1],"alpha":-2,"beta":-3}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-real-bilinear-form

逐步推導

fixed real matrix=[[-12, -3], [-5, -4]];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(-3, -3)

combined coordinate0 beta term = 9

combined coordinate0 sum := sum(-2, 9)

combined coordinate0 sum = 7

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

combined coordinate1 alpha term = 4

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

combined coordinate1 beta term = -3

combined coordinate1 sum := sum(4, -3)

combined coordinate1 sum = 1

expanding second slot;alpha=-2;beta=-3;u=[1, -2];v=[-3, 1];fixed other input=[-2, -3]

combined=[7, 1];other slot held fixed

combination lhs:matrix=[[-12, -3], [-5, -4]];first slot=[-2, -3];second slot=[7, 1]

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

combination lhs matrix row0 column0 term := mul(-12, 7)

combination lhs matrix row0 column0 term = -84

combination lhs matrix row0 column1 term := mul(-3, 1)

combination lhs matrix row0 column1 term = -3

combination lhs matrix-times-second row0 := sum(-84, -3)

combination lhs matrix-times-second row0 = -87

combination lhs matrix row1 column0 term := mul(-5, 7)

combination lhs matrix row1 column0 term = -35

combination lhs matrix row1 column1 term := mul(-4, 1)

combination lhs matrix row1 column1 term = -4

combination lhs matrix-times-second row1 := sum(-35, -4)

combination lhs matrix-times-second row1 = -39

combination lhs first-slot coordinate0 term := mul(-2, -87)

combination lhs first-slot coordinate0 term = 174

combination lhs first-slot coordinate1 term := mul(-3, -39)

combination lhs first-slot coordinate1 term = 117

combination lhs final first-slot dot sum := sum(174, 117)

combination lhs final first-slot dot sum = 291

u form:matrix=[[-12, -3], [-5, -4]];first slot=[-2, -3];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(-3, -2)

u form matrix row0 column1 term = 6

u form matrix-times-second row0 := sum(-12, 6)

u form matrix-times-second row0 = -6

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

u form matrix row1 column0 term = -5

u form matrix row1 column1 term := mul(-4, -2)

u form matrix row1 column1 term = 8

u form matrix-times-second row1 := sum(-5, 8)

u form matrix-times-second row1 = 3

u form first-slot coordinate0 term := mul(-2, -6)

u form first-slot coordinate0 term = 12

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

u form first-slot coordinate1 term = -9

u form final first-slot dot sum := sum(12, -9)

u form final first-slot dot sum = 3

v form:matrix=[[-12, -3], [-5, -4]];first slot=[-2, -3];second slot=[-3, 1]

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

v form matrix row0 column0 term := mul(-12, -3)

v form matrix row0 column0 term = 36

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

v form matrix row0 column1 term = -3

v form matrix-times-second row0 := sum(36, -3)

v form matrix-times-second row0 = 33

v form matrix row1 column0 term := mul(-5, -3)

v form matrix row1 column0 term = 15

v form matrix row1 column1 term := mul(-4, 1)

v form matrix row1 column1 term = -4

v form matrix-times-second row1 := sum(15, -4)

v form matrix-times-second row1 = 11

v form first-slot coordinate0 term := mul(-2, 33)

v form first-slot coordinate0 term = -66

v form first-slot coordinate1 term := mul(-3, 11)

v form first-slot coordinate1 term = -33

v form final first-slot dot sum := sum(-66, -33)

v form final first-slot dot sum = -99

rhs alpha form value := mul(-2, 3)

rhs alpha form value = -6

rhs beta form value := mul(-3, -99)

rhs beta form value = 297

combination rhs sum := sum(-6, 297)

combination rhs sum = 291

fixed matrix formula;second slot expanded;lhs=291;rhs=291

equal=是;universal justification is matrix distributivity,not one passing sample

逐步反向核對

• S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 以精確有理數重算 combination rhs sum，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 rhs beta form value，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 rhs alpha form value，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 v form final first-slot dot sum，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 v form first-slot coordinate1 term，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 v form first-slot coordinate0 term，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 v form matrix-times-second row1，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 v form matrix row1 column1 term，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 v form matrix row1 column0 term，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 v form matrix-times-second row0，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 v form matrix row0 column1 term，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 v form matrix row0 column0 term，不以小數近似替代。
• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 以精確有理數重算 u form final first-slot dot sum，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 u form first-slot coordinate1 term，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 u form first-slot coordinate0 term，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 u form matrix-times-second row1，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 u form matrix row1 column1 term，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 u form matrix row1 column0 term，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 u form matrix-times-second row0，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 u form matrix row0 column1 term，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 u form matrix row0 column0 term，不以小數近似替代。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 以精確有理數重算 combination lhs final first-slot dot sum，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 combination lhs first-slot coordinate1 term，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 combination lhs first-slot coordinate0 term，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 combination lhs matrix-times-second row1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 combination lhs matrix row1 column1 term，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 combination lhs matrix row1 column0 term，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 combination lhs matrix-times-second row0，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 combination lhs matrix row0 column1 term，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 combination lhs matrix row0 column0 term，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 combined coordinate1 sum，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 combined coordinate1 beta term，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 combined coordinate1 alpha term，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 combined coordinate0 sum，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 combined coordinate0 beta term，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 combined coordinate0 alpha term，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(-2, -3, [7, 1], 291, 291, 是)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。