第 12748 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":[[10,-2],[-5,0]],"x":[0,-2],"y":[-5,-10]}。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":[[10,-2],[-5,0]],"x":[0,-2],"y":[-5,-10]}。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=[[10, -2], [-5, 0]];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=[[10, -2], [-5, 0]];first slot=[0, -2];second slot=[-5, -10]

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

requested form matrix row0 column0 term := mul(10, -5)

requested form matrix row0 column0 term = -50

requested form matrix row0 column1 term := mul(-2, -10)

requested form matrix row0 column1 term = 20

requested form matrix-times-second row0 := sum(-50, 20)

requested form matrix-times-second row0 = -30

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

requested form matrix row1 column0 term = 25

requested form matrix row1 column1 term := mul(0, -10)

requested form matrix row1 column1 term = 0

requested form matrix-times-second row1 := sum(25, 0)

requested form matrix-times-second row1 = 25

requested form first-slot coordinate0 term := mul(0, -30)

requested form first-slot coordinate0 term = 0

requested form first-slot coordinate1 term := mul(-2, 25)

requested form first-slot coordinate1 term = -50

requested form final first-slot dot sum := sum(0, -50)

requested form final first-slot dot sum = -50

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

([[10, -2], [-5, 0]], [0, -2], [-5, -10], -50)

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。