第 14570 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":[[21,1],[1,2]],"x":[2,2],"y":[3,-7]}。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":[[21,1],[1,2]],"x":[2,2],"y":[3,-7]}。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(21, 2)

inner parent diagonal product = 42

inner parent off-diagonal product := mul(1, 1)

inner parent off-diagonal product = 1

inner parent determinant := sub(42, 1)

inner parent determinant = 41

same real R2 inner-product candidate matrix=[[21, 1], [1, 2]]

symmetric=是;leading=21;determinant=41

inner parent completed-square slope := div(1, 21)

inner parent completed-square slope = 1/21

inner parent completed-square remainder := div(41, 21)

inner parent completed-square remainder = 41/21

positive leading=21;slope=1/21;positive remainder=41/21

q(s,t)=(21)*(s+(1/21)*t)^2+(41/21)*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=[[21, 1], [1, 2]];first vector=[2, 2];second vector=[2, 2]

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

x self-inner-product matrix term0,0 := mul(21, 2)

x self-inner-product matrix term0,0 = 42

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

x self-inner-product matrix term0,1 = 2

x self-inner-product image coordinate0 := sum(42, 2)

x self-inner-product image coordinate0 = 44

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

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

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

x self-inner-product matrix term1,1 = 4

x self-inner-product image coordinate1 := sum(2, 4)

x self-inner-product image coordinate1 = 6

x self-inner-product first-slot term0 := mul(2, 44)

x self-inner-product first-slot term0 = 88

x self-inner-product first-slot term1 := mul(2, 6)

x self-inner-product first-slot term1 = 12

x self-inner-product inner sum := sum(88, 12)

x self-inner-product inner sum = 100

y self-inner-product:same matrix=[[21, 1], [1, 2]];first vector=[3, -7];second vector=[3, -7]

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

y self-inner-product matrix term0,0 := mul(21, 3)

y self-inner-product matrix term0,0 = 63

y self-inner-product matrix term0,1 := mul(1, -7)

y self-inner-product matrix term0,1 = -7

y self-inner-product image coordinate0 := sum(63, -7)

y self-inner-product image coordinate0 = 56

y self-inner-product matrix term1,0 := mul(1, 3)

y self-inner-product matrix term1,0 = 3

y self-inner-product matrix term1,1 := mul(2, -7)

y self-inner-product matrix term1,1 = -14

y self-inner-product image coordinate1 := sum(3, -14)

y self-inner-product image coordinate1 = -11

y self-inner-product first-slot term0 := mul(3, 56)

y self-inner-product first-slot term0 = 168

y self-inner-product first-slot term1 := mul(-7, -11)

y self-inner-product first-slot term1 = 77

y self-inner-product inner sum := sum(168, 77)

y self-inner-product inner sum = 245

parallelogram sum coordinate0 := sum(2, 3)

parallelogram sum coordinate0 = 5

parallelogram sum coordinate1 := sum(2, -7)

parallelogram sum coordinate1 = -5

parallelogram difference coordinate0 := sub(2, 3)

parallelogram difference coordinate0 = -1

parallelogram difference coordinate1 := sub(2, -7)

parallelogram difference coordinate1 = 9

sum self-inner-product:same matrix=[[21, 1], [1, 2]];first vector=[5, -5];second vector=[5, -5]

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

sum self-inner-product matrix term0,0 := mul(21, 5)

sum self-inner-product matrix term0,0 = 105

sum self-inner-product matrix term0,1 := mul(1, -5)

sum self-inner-product matrix term0,1 = -5

sum self-inner-product image coordinate0 := sum(105, -5)

sum self-inner-product image coordinate0 = 100

sum self-inner-product matrix term1,0 := mul(1, 5)

sum self-inner-product matrix term1,0 = 5

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

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

sum self-inner-product image coordinate1 := sum(5, -10)

sum self-inner-product image coordinate1 = -5

sum self-inner-product first-slot term0 := mul(5, 100)

sum self-inner-product first-slot term0 = 500

sum self-inner-product first-slot term1 := mul(-5, -5)

sum self-inner-product first-slot term1 = 25

sum self-inner-product inner sum := sum(500, 25)

sum self-inner-product inner sum = 525

difference self-inner-product:same matrix=[[21, 1], [1, 2]];first vector=[-1, 9];second vector=[-1, 9]

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

difference self-inner-product matrix term0,0 := mul(21, -1)

difference self-inner-product matrix term0,0 = -21

difference self-inner-product matrix term0,1 := mul(1, 9)

difference self-inner-product matrix term0,1 = 9

difference self-inner-product image coordinate0 := sum(-21, 9)

difference self-inner-product image coordinate0 = -12

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

difference self-inner-product matrix term1,0 = -1

difference self-inner-product matrix term1,1 := mul(2, 9)

difference self-inner-product matrix term1,1 = 18

difference self-inner-product image coordinate1 := sum(-1, 18)

difference self-inner-product image coordinate1 = 17

difference self-inner-product first-slot term0 := mul(-1, -12)

difference self-inner-product first-slot term0 = 12

difference self-inner-product first-slot term1 := mul(9, 17)

difference self-inner-product first-slot term1 = 153

difference self-inner-product inner sum := sum(12, 153)

difference self-inner-product inner sum = 165

parallelogram left sum := sum(525, 165)

parallelogram left sum = 690

twice x self-inner-product := mul(2, 100)

twice x self-inner-product = 200

twice y self-inner-product := mul(2, 245)

twice y self-inner-product = 490

parallelogram right sum := sum(200, 490)

parallelogram right sum = 690

same inner product;left=690;right=690

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

([2, 2], [3, -7], 690, 690, 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。