第 14537 題：內積誘導範數

原題與已知條件

向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={"matrix":[[13,2],[2,5]],"x":[1,-1],"y":[-5,1]}。請依 [x,y,difference,distance_square,exact_distance] 欄序回答。

• G001 — 向量與已認證內積綁定同一實空間。操作=innernorm-distance-square；資料={"matrix":[[13,2],[2,5]],"x":[1,-1],"y":[-5,1]}。請依 [x,y,difference,distance_square,exact_distance] 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-inner-product-induced-norm

逐步推導

inner parent diagonal product := mul(13, 5)

inner parent diagonal product = 65

inner parent off-diagonal product := mul(2, 2)

inner parent off-diagonal product = 4

inner parent determinant := sub(65, 4)

inner parent determinant = 61

same real R2 inner-product candidate matrix=[[13, 2], [2, 5]]

symmetric=是;leading=13;determinant=61

inner parent completed-square slope := div(2, 13)

inner parent completed-square slope = 2/13

inner parent completed-square remainder := div(61, 13)

inner parent completed-square remainder = 61/13

positive leading=13;slope=2/13;positive remainder=61/13

q(s,t)=(13)*(s+(2/13)*t)^2+(61/13)*t^2;positive iff(s,t)!=0

same certified real inner product;arbitrary real x,y

if y=0 then inner(x,y)=0;otherwise yy>0 and q(x-a*y)=xx-2*a*xy+a^2*yy>=0

norm(x)=nonnegative sqrt(inner(x,x));inner product positive definite and bilinear

norm(x)>=0;norm(x)=0 iff x=0;norm(a*x)=abs(a)*norm(x)

Cauchy-Schwarz holds for arbitrary x,y in this same inner product

norm(x+y)^2=xx+2*xy+yy<=(norm(x)+norm(y))^2;nonnegative roots give triangle inequality

distance signed coordinate0 := sub(1, -5)

distance signed coordinate0 = 6

distance signed coordinate1 := sub(-1, 1)

distance signed coordinate1 = -2

distance difference square:same matrix=[[13, 2], [2, 5]];first vector=[6, -2];second vector=[6, -2]

inner product=first^T*(matrix*second)

distance difference square matrix term0,0 := mul(13, 6)

distance difference square matrix term0,0 = 78

distance difference square matrix term0,1 := mul(2, -2)

distance difference square matrix term0,1 = -4

distance difference square image coordinate0 := sum(78, -4)

distance difference square image coordinate0 = 74

distance difference square matrix term1,0 := mul(2, 6)

distance difference square matrix term1,0 = 12

distance difference square matrix term1,1 := mul(5, -2)

distance difference square matrix term1,1 = -10

distance difference square image coordinate1 := sum(12, -10)

distance difference square image coordinate1 = 2

distance difference square first-slot term0 := mul(6, 74)

distance difference square first-slot term0 = 444

distance difference square first-slot term1 := mul(-2, 2)

distance difference square first-slot term1 = -4

distance difference square inner sum := sum(444, -4)

distance difference square inner sum = 440

induced distance:same-form self-inner-product=440>=0

nonnegative exact root=sqrt(440);source expression kept unsimplified

逐步反向核對

• S023 · 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。
• S022 · premise_and_result_audit — 以精確有理數重算 distance difference square inner sum，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 distance difference square first-slot term1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 distance difference square first-slot term0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 distance difference square image coordinate1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 distance difference square matrix term1,1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 distance difference square matrix term1,0，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 distance difference square image coordinate0，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 distance difference square matrix term0,1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 distance difference square matrix term0,0，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 distance signed coordinate1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 distance signed coordinate0，不以小數近似替代。
• S010 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 inner parent completed-square remainder，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 inner parent completed-square slope，不以小數近似替代。
• S004 · 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。
• S003 · premise_and_result_audit — 以精確有理數重算 inner parent determinant，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 inner parent off-diagonal product，不以小數近似替代。
• S001 · premise_and_result_audit — 以精確有理數重算 inner parent diagonal product，不以小數近似替代。

結論

([1, -1], [-5, 1], [6, -2], 440, sqrt(440))

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。