第 14572 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":[[23,0],[0,1]],"x":[-3,-1],"y":[5,-9]}。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":[[23,0],[0,1]],"x":[-3,-1],"y":[5,-9]}。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(23, 1)

inner parent diagonal product = 23

inner parent off-diagonal product := mul(0, 0)

inner parent off-diagonal product = 0

inner parent determinant := sub(23, 0)

inner parent determinant = 23

same real R2 inner-product candidate matrix=[[23, 0], [0, 1]]

symmetric=是;leading=23;determinant=23

inner parent completed-square slope := div(0, 23)

inner parent completed-square slope = 0

inner parent completed-square remainder := div(23, 23)

inner parent completed-square remainder = 1

positive leading=23;slope=0;positive remainder=1

q(s,t)=(23)*(s+(0)*t)^2+(1)*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=[[23, 0], [0, 1]];first vector=[-3, -1];second vector=[-3, -1]

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

x self-inner-product matrix term0,0 := mul(23, -3)

x self-inner-product matrix term0,0 = -69

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

x self-inner-product matrix term0,1 = 0

x self-inner-product image coordinate0 := sum(-69, 0)

x self-inner-product image coordinate0 = -69

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

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

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

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

x self-inner-product image coordinate1 := sum(0, -1)

x self-inner-product image coordinate1 = -1

x self-inner-product first-slot term0 := mul(-3, -69)

x self-inner-product first-slot term0 = 207

x self-inner-product first-slot term1 := mul(-1, -1)

x self-inner-product first-slot term1 = 1

x self-inner-product inner sum := sum(207, 1)

x self-inner-product inner sum = 208

y self-inner-product:same matrix=[[23, 0], [0, 1]];first vector=[5, -9];second vector=[5, -9]

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

y self-inner-product matrix term0,0 := mul(23, 5)

y self-inner-product matrix term0,0 = 115

y self-inner-product matrix term0,1 := mul(0, -9)

y self-inner-product matrix term0,1 = 0

y self-inner-product image coordinate0 := sum(115, 0)

y self-inner-product image coordinate0 = 115

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

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

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

y self-inner-product matrix term1,1 = -9

y self-inner-product image coordinate1 := sum(0, -9)

y self-inner-product image coordinate1 = -9

y self-inner-product first-slot term0 := mul(5, 115)

y self-inner-product first-slot term0 = 575

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

y self-inner-product first-slot term1 = 81

y self-inner-product inner sum := sum(575, 81)

y self-inner-product inner sum = 656

parallelogram sum coordinate0 := sum(-3, 5)

parallelogram sum coordinate0 = 2

parallelogram sum coordinate1 := sum(-1, -9)

parallelogram sum coordinate1 = -10

parallelogram difference coordinate0 := sub(-3, 5)

parallelogram difference coordinate0 = -8

parallelogram difference coordinate1 := sub(-1, -9)

parallelogram difference coordinate1 = 8

sum self-inner-product:same matrix=[[23, 0], [0, 1]];first vector=[2, -10];second vector=[2, -10]

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

sum self-inner-product matrix term0,0 := mul(23, 2)

sum self-inner-product matrix term0,0 = 46

sum self-inner-product matrix term0,1 := mul(0, -10)

sum self-inner-product matrix term0,1 = 0

sum self-inner-product image coordinate0 := sum(46, 0)

sum self-inner-product image coordinate0 = 46

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

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

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

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

sum self-inner-product image coordinate1 := sum(0, -10)

sum self-inner-product image coordinate1 = -10

sum self-inner-product first-slot term0 := mul(2, 46)

sum self-inner-product first-slot term0 = 92

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

sum self-inner-product first-slot term1 = 100

sum self-inner-product inner sum := sum(92, 100)

sum self-inner-product inner sum = 192

difference self-inner-product:same matrix=[[23, 0], [0, 1]];first vector=[-8, 8];second vector=[-8, 8]

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

difference self-inner-product matrix term0,0 := mul(23, -8)

difference self-inner-product matrix term0,0 = -184

difference self-inner-product matrix term0,1 := mul(0, 8)

difference self-inner-product matrix term0,1 = 0

difference self-inner-product image coordinate0 := sum(-184, 0)

difference self-inner-product image coordinate0 = -184

difference self-inner-product matrix term1,0 := mul(0, -8)

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

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

difference self-inner-product matrix term1,1 = 8

difference self-inner-product image coordinate1 := sum(0, 8)

difference self-inner-product image coordinate1 = 8

difference self-inner-product first-slot term0 := mul(-8, -184)

difference self-inner-product first-slot term0 = 1472

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

difference self-inner-product first-slot term1 = 64

difference self-inner-product inner sum := sum(1472, 64)

difference self-inner-product inner sum = 1536

parallelogram left sum := sum(192, 1536)

parallelogram left sum = 1728

twice x self-inner-product := mul(2, 208)

twice x self-inner-product = 416

twice y self-inner-product := mul(2, 656)

twice y self-inner-product = 1312

parallelogram right sum := sum(416, 1312)

parallelogram right sum = 1728

same inner product;left=1728;right=1728

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, -1], [5, -9], 1728, 1728, 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。