第 13822 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":[[23,0],[0,1]],"x":[-1,-3],"y":[5,-9]}。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":[[23,0],[0,1]],"x":[-1,-3],"y":[5,-9]}。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=[[23, 0], [0, 1]];B(x,y)=x^T*A*y

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

candidate evaluation matrix second slot:matrix=[[23, 0], [0, 1]];input=[5, -9]

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

candidate evaluation matrix second slot row0 column0 product := mul(23, 5)

candidate evaluation matrix second slot row0 column0 product = 115

candidate evaluation matrix second slot row0 column1 product := mul(0, -9)

candidate evaluation matrix second slot row0 column1 product = 0

candidate evaluation matrix second slot row0 sum := sum(115, 0)

candidate evaluation matrix second slot row0 sum = 115

candidate evaluation matrix second slot row1 column0 product := mul(0, 5)

candidate evaluation matrix second slot row1 column0 product = 0

candidate evaluation matrix second slot row1 column1 product := mul(1, -9)

candidate evaluation matrix second slot row1 column1 product = -9

candidate evaluation matrix second slot row1 sum := sum(0, -9)

candidate evaluation matrix second slot row1 sum = -9

candidate evaluation first coordinate0 product := mul(-1, 115)

candidate evaluation first coordinate0 product = -115

candidate evaluation first coordinate1 product := mul(-3, -9)

candidate evaluation first coordinate1 product = 27

candidate evaluation bilinear sum := sum(-115, 27)

candidate evaluation bilinear sum = -88

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

([[23, 0], [0, 1]], [-1, -3], [5, -9], -88)

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。