第 13820 題：實內積空間

原題與已知條件

內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={"matrix":[[21,2],[2,5]],"x":[2,2],"y":[3,-7]}。請依 [matrix,x,y,value] 欄序回答。

• G001 — 內積須同時雙線性對稱且正定。操作=inner-evaluate；資料={"matrix":[[21,2],[2,5]],"x":[2,2],"y":[3,-7]}。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-inner-product-space

逐步推導

candidate real R2 form;matrix=[[21, 2], [2, 5]];B(x,y)=x^T*A*y

matrix distributivity supplies bilinearity;other inner-product obligations remain separate

candidate evaluation matrix second slot:matrix=[[21, 2], [2, 5]];input=[3, -7]

input dimension=2;output dimension=2;rows dot input

candidate evaluation matrix second slot row0 column0 product := mul(21, 3)

candidate evaluation matrix second slot row0 column0 product = 63

candidate evaluation matrix second slot row0 column1 product := mul(2, -7)

candidate evaluation matrix second slot row0 column1 product = -14

candidate evaluation matrix second slot row0 sum := sum(63, -14)

candidate evaluation matrix second slot row0 sum = 49

candidate evaluation matrix second slot row1 column0 product := mul(2, 3)

candidate evaluation matrix second slot row1 column0 product = 6

candidate evaluation matrix second slot row1 column1 product := mul(5, -7)

candidate evaluation matrix second slot row1 column1 product = -35

candidate evaluation matrix second slot row1 sum := sum(6, -35)

candidate evaluation matrix second slot row1 sum = -29

candidate evaluation first coordinate0 product := mul(2, 49)

candidate evaluation first coordinate0 product = 98

candidate evaluation first coordinate1 product := mul(2, -29)

candidate evaluation first coordinate1 product = -58

candidate evaluation bilinear sum := sum(98, -58)

candidate evaluation bilinear sum = 40

逐步反向核對

• S011 · premise_and_result_audit — 以精確有理數重算 candidate evaluation bilinear sum，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 candidate evaluation first coordinate1 product，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 candidate evaluation first coordinate0 product，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 candidate evaluation matrix second slot row1 sum，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 candidate evaluation matrix second slot row1 column1 product，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 candidate evaluation matrix second slot row1 column0 product，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 candidate evaluation matrix second slot row0 sum，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 candidate evaluation matrix second slot row0 column1 product，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 candidate evaluation matrix second slot row0 column0 product，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

([[21, 2], [2, 5]], [2, 2], [3, -7], 40)

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。