第 14571 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-parallelogram；資料={"matrix":[[22,2],[2,5]],"x":[3,-2],"y":[4,-8]}。請依 [x,y,lhs,rhs,equal] 欄序回答。

• G001 — 向量與已認證內積綁定同一實空間。操作=innernorm-parallelogram；資料={"matrix":[[22,2],[2,5]],"x":[3,-2],"y":[4,-8]}。請依 [x,y,lhs,rhs,equal] 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(22, 5)

inner parent diagonal product = 110

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

inner parent off-diagonal product = 4

inner parent determinant := sub(110, 4)

inner parent determinant = 106

same real R2 inner-product candidate matrix=[[22, 2], [2, 5]]

symmetric=是;leading=22;determinant=106

inner parent completed-square slope := div(2, 22)

inner parent completed-square slope = 1/11

inner parent completed-square remainder := div(106, 22)

inner parent completed-square remainder = 53/11

positive leading=22;slope=1/11;positive remainder=53/11

q(s,t)=(22)*(s+(1/11)*t)^2+(53/11)*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

x self-inner-product:same matrix=[[22, 2], [2, 5]];first vector=[3, -2];second vector=[3, -2]

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

x self-inner-product matrix term0,0 := mul(22, 3)

x self-inner-product matrix term0,0 = 66

x self-inner-product matrix term0,1 := mul(2, -2)

x self-inner-product matrix term0,1 = -4

x self-inner-product image coordinate0 := sum(66, -4)

x self-inner-product image coordinate0 = 62

x self-inner-product matrix term1,0 := mul(2, 3)

x self-inner-product matrix term1,0 = 6

x self-inner-product matrix term1,1 := mul(5, -2)

x self-inner-product matrix term1,1 = -10

x self-inner-product image coordinate1 := sum(6, -10)

x self-inner-product image coordinate1 = -4

x self-inner-product first-slot term0 := mul(3, 62)

x self-inner-product first-slot term0 = 186

x self-inner-product first-slot term1 := mul(-2, -4)

x self-inner-product first-slot term1 = 8

x self-inner-product inner sum := sum(186, 8)

x self-inner-product inner sum = 194

y self-inner-product:same matrix=[[22, 2], [2, 5]];first vector=[4, -8];second vector=[4, -8]

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

y self-inner-product matrix term0,0 := mul(22, 4)

y self-inner-product matrix term0,0 = 88

y self-inner-product matrix term0,1 := mul(2, -8)

y self-inner-product matrix term0,1 = -16

y self-inner-product image coordinate0 := sum(88, -16)

y self-inner-product image coordinate0 = 72

y self-inner-product matrix term1,0 := mul(2, 4)

y self-inner-product matrix term1,0 = 8

y self-inner-product matrix term1,1 := mul(5, -8)

y self-inner-product matrix term1,1 = -40

y self-inner-product image coordinate1 := sum(8, -40)

y self-inner-product image coordinate1 = -32

y self-inner-product first-slot term0 := mul(4, 72)

y self-inner-product first-slot term0 = 288

y self-inner-product first-slot term1 := mul(-8, -32)

y self-inner-product first-slot term1 = 256

y self-inner-product inner sum := sum(288, 256)

y self-inner-product inner sum = 544

parallelogram sum coordinate0 := sum(3, 4)

parallelogram sum coordinate0 = 7

parallelogram sum coordinate1 := sum(-2, -8)

parallelogram sum coordinate1 = -10

parallelogram difference coordinate0 := sub(3, 4)

parallelogram difference coordinate0 = -1

parallelogram difference coordinate1 := sub(-2, -8)

parallelogram difference coordinate1 = 6

sum self-inner-product:same matrix=[[22, 2], [2, 5]];first vector=[7, -10];second vector=[7, -10]

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

sum self-inner-product matrix term0,0 := mul(22, 7)

sum self-inner-product matrix term0,0 = 154

sum self-inner-product matrix term0,1 := mul(2, -10)

sum self-inner-product matrix term0,1 = -20

sum self-inner-product image coordinate0 := sum(154, -20)

sum self-inner-product image coordinate0 = 134

sum self-inner-product matrix term1,0 := mul(2, 7)

sum self-inner-product matrix term1,0 = 14

sum self-inner-product matrix term1,1 := mul(5, -10)

sum self-inner-product matrix term1,1 = -50

sum self-inner-product image coordinate1 := sum(14, -50)

sum self-inner-product image coordinate1 = -36

sum self-inner-product first-slot term0 := mul(7, 134)

sum self-inner-product first-slot term0 = 938

sum self-inner-product first-slot term1 := mul(-10, -36)

sum self-inner-product first-slot term1 = 360

sum self-inner-product inner sum := sum(938, 360)

sum self-inner-product inner sum = 1298

difference self-inner-product:same matrix=[[22, 2], [2, 5]];first vector=[-1, 6];second vector=[-1, 6]

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

difference self-inner-product matrix term0,0 := mul(22, -1)

difference self-inner-product matrix term0,0 = -22

difference self-inner-product matrix term0,1 := mul(2, 6)

difference self-inner-product matrix term0,1 = 12

difference self-inner-product image coordinate0 := sum(-22, 12)

difference self-inner-product image coordinate0 = -10

difference self-inner-product matrix term1,0 := mul(2, -1)

difference self-inner-product matrix term1,0 = -2

difference self-inner-product matrix term1,1 := mul(5, 6)

difference self-inner-product matrix term1,1 = 30

difference self-inner-product image coordinate1 := sum(-2, 30)

difference self-inner-product image coordinate1 = 28

difference self-inner-product first-slot term0 := mul(-1, -10)

difference self-inner-product first-slot term0 = 10

difference self-inner-product first-slot term1 := mul(6, 28)

difference self-inner-product first-slot term1 = 168

difference self-inner-product inner sum := sum(10, 168)

difference self-inner-product inner sum = 178

parallelogram left sum := sum(1298, 178)

parallelogram left sum = 1476

twice x self-inner-product := mul(2, 194)

twice x self-inner-product = 388

twice y self-inner-product := mul(2, 544)

twice y self-inner-product = 1088

parallelogram right sum := sum(388, 1088)

parallelogram right sum = 1476

same inner product;left=1476;right=1476

bilinear expansions have opposite cross terms which cancel;identity=是

逐步反向核對

• S059 · 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。
• S058 · premise_and_result_audit — 以精確有理數重算 parallelogram right sum，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 twice y self-inner-product，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 twice x self-inner-product，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 parallelogram left sum，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product inner sum，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product first-slot term1，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product first-slot term0，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product image coordinate1，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product matrix term1,1，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product matrix term1,0，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product image coordinate0，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product matrix term0,1，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 difference self-inner-product matrix term0,0，不以小數近似替代。
• S045 · 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。
• S044 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product inner sum，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product first-slot term1，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product first-slot term0，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product image coordinate1，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product matrix term1,1，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product matrix term1,0，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product image coordinate0，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product matrix term0,1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 sum self-inner-product matrix term0,0，不以小數近似替代。
• S035 · 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。
• S034 · premise_and_result_audit — 以精確有理數重算 parallelogram difference coordinate1，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 parallelogram difference coordinate0，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 parallelogram sum coordinate1，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 parallelogram sum coordinate0，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 y self-inner-product inner sum，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 y self-inner-product first-slot term1，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 y self-inner-product first-slot term0，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 y self-inner-product image coordinate1，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 y self-inner-product matrix term1,1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 y self-inner-product matrix term1,0，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 y self-inner-product image coordinate0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 y self-inner-product matrix term0,1，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 y self-inner-product matrix term0,0，不以小數近似替代。
• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 x self-inner-product inner sum，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 x self-inner-product first-slot term1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 x self-inner-product first-slot term0，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 x self-inner-product image coordinate1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 x self-inner-product matrix term1,1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 x self-inner-product matrix term1,0，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 x self-inner-product image coordinate0，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 x self-inner-product matrix term0,1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 x self-inner-product matrix term0,0，不以小數近似替代。
• S011 · 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。
• 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

([3, -2], [4, -8], 1476, 1476, 是)

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。