第 12741 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-matrix-evaluate；資料={"matrix":[[3,-2],[4,2]],"x":[-2,-2],"y":[-1,-3]}。請依 [matrix,x,y,value] 欄序回答。

• G001 — 雙線性必須分別核第一槽與第二槽。操作=bilinear-matrix-evaluate；資料={"matrix":[[3,-2],[4,2]],"x":[-2,-2],"y":[-1,-3]}。請依 [matrix,x,y,value] 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=[[3, -2], [4, 2]];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

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

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

requested form matrix row0 column0 term := mul(3, -1)

requested form matrix row0 column0 term = -3

requested form matrix row0 column1 term := mul(-2, -3)

requested form matrix row0 column1 term = 6

requested form matrix-times-second row0 := sum(-3, 6)

requested form matrix-times-second row0 = 3

requested form matrix row1 column0 term := mul(4, -1)

requested form matrix row1 column0 term = -4

requested form matrix row1 column1 term := mul(2, -3)

requested form matrix row1 column1 term = -6

requested form matrix-times-second row1 := sum(-4, -6)

requested form matrix-times-second row1 = -10

requested form first-slot coordinate0 term := mul(-2, 3)

requested form first-slot coordinate0 term = -6

requested form first-slot coordinate1 term := mul(-2, -10)

requested form first-slot coordinate1 term = 20

requested form final first-slot dot sum := sum(-6, 20)

requested form final first-slot dot sum = 14

逐步反向核對

• S011 · premise_and_result_audit — 以精確有理數重算 requested form final first-slot dot sum，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 requested form first-slot coordinate1 term，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 requested form first-slot coordinate0 term，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 requested form matrix-times-second row1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 requested form matrix row1 column1 term，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 requested form matrix row1 column0 term，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 requested form matrix-times-second row0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 requested form matrix row0 column1 term，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 requested form matrix row0 column0 term，不以小數近似替代。
• S002 · 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。
• 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

([[3, -2], [4, 2]], [-2, -2], [-1, -3], 14)

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。