第 12701 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-first-slot-identity；資料={"matrix":[[-12,-3],[-5,-4]],"u":[1,-2],"v":[-3,1],"alpha":-2,"beta":-3,"y":[-5,12]}。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-first-slot-identity；資料={"matrix":[[-12,-3],[-5,-4]],"u":[1,-2],"v":[-3,1],"alpha":-2,"beta":-3,"y":[-5,12]}。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 first slot;alpha=-2;beta=-3;u=[1, -2];v=[-3, 1];fixed other input=[-5, 12]

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

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

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

combination lhs matrix row0 column0 term := mul(-12, -5)

combination lhs matrix row0 column0 term = 60

combination lhs matrix row0 column1 term := mul(-3, 12)

combination lhs matrix row0 column1 term = -36

combination lhs matrix-times-second row0 := sum(60, -36)

combination lhs matrix-times-second row0 = 24

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

combination lhs matrix row1 column0 term = 25

combination lhs matrix row1 column1 term := mul(-4, 12)

combination lhs matrix row1 column1 term = -48

combination lhs matrix-times-second row1 := sum(25, -48)

combination lhs matrix-times-second row1 = -23

combination lhs first-slot coordinate0 term := mul(7, 24)

combination lhs first-slot coordinate0 term = 168

combination lhs first-slot coordinate1 term := mul(1, -23)

combination lhs first-slot coordinate1 term = -23

combination lhs final first-slot dot sum := sum(168, -23)

combination lhs final first-slot dot sum = 145

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

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

u form matrix row0 column0 term := mul(-12, -5)

u form matrix row0 column0 term = 60

u form matrix row0 column1 term := mul(-3, 12)

u form matrix row0 column1 term = -36

u form matrix-times-second row0 := sum(60, -36)

u form matrix-times-second row0 = 24

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

u form matrix row1 column0 term = 25

u form matrix row1 column1 term := mul(-4, 12)

u form matrix row1 column1 term = -48

u form matrix-times-second row1 := sum(25, -48)

u form matrix-times-second row1 = -23

u form first-slot coordinate0 term := mul(1, 24)

u form first-slot coordinate0 term = 24

u form first-slot coordinate1 term := mul(-2, -23)

u form first-slot coordinate1 term = 46

u form final first-slot dot sum := sum(24, 46)

u form final first-slot dot sum = 70

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

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

v form matrix row0 column0 term := mul(-12, -5)

v form matrix row0 column0 term = 60

v form matrix row0 column1 term := mul(-3, 12)

v form matrix row0 column1 term = -36

v form matrix-times-second row0 := sum(60, -36)

v form matrix-times-second row0 = 24

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

v form matrix row1 column0 term = 25

v form matrix row1 column1 term := mul(-4, 12)

v form matrix row1 column1 term = -48

v form matrix-times-second row1 := sum(25, -48)

v form matrix-times-second row1 = -23

v form first-slot coordinate0 term := mul(-3, 24)

v form first-slot coordinate0 term = -72

v form first-slot coordinate1 term := mul(1, -23)

v form first-slot coordinate1 term = -23

v form final first-slot dot sum := sum(-72, -23)

v form final first-slot dot sum = -95

rhs alpha form value := mul(-2, 70)

rhs alpha form value = -140

rhs beta form value := mul(-3, -95)

rhs beta form value = 285

combination rhs sum := sum(-140, 285)

combination rhs sum = 145

fixed matrix formula;first slot expanded;lhs=145;rhs=145

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], 145, 145, 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。