第 14515 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-cauchy-schwarz；資料={"matrix":[[16,2],[2,5]],"x":[-3,2],"y":[-2,-2]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 欄序回答。

• G001 — 向量與已認證內積綁定同一實空間。操作=innernorm-cauchy-schwarz；資料={"matrix":[[16,2],[2,5]],"x":[-3,2],"y":[-2,-2]}。請依 [inner_xy,norm_x_square,norm_y_square,lhs_square,rhs,pass,certificate] 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(16, 5)

inner parent diagonal product = 80

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

inner parent off-diagonal product = 4

inner parent determinant := sub(80, 4)

inner parent determinant = 76

same real R2 inner-product candidate matrix=[[16, 2], [2, 5]]

symmetric=是;leading=16;determinant=76

inner parent completed-square slope := div(2, 16)

inner parent completed-square slope = 1/8

inner parent completed-square remainder := div(76, 16)

inner parent completed-square remainder = 19/4

positive leading=16;slope=1/8;positive remainder=19/4

q(s,t)=(16)*(s+(1/8)*t)^2+(19/4)*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=[[16, 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(16, -3)

x self-inner-product matrix term0,0 = -48

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(-48, 4)

x self-inner-product image coordinate0 = -44

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, -44)

x self-inner-product first-slot term0 = 132

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(132, 8)

x self-inner-product inner sum = 140

y self-inner-product:same matrix=[[16, 2], [2, 5]];first vector=[-2, -2];second vector=[-2, -2]

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

y self-inner-product matrix term0,0 := mul(16, -2)

y self-inner-product matrix term0,0 = -32

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

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

y self-inner-product image coordinate0 := sum(-32, -4)

y self-inner-product image coordinate0 = -36

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

y self-inner-product matrix term1,0 = -4

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

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

y self-inner-product image coordinate1 := sum(-4, -10)

y self-inner-product image coordinate1 = -14

y self-inner-product first-slot term0 := mul(-2, -36)

y self-inner-product first-slot term0 = 72

y self-inner-product first-slot term1 := mul(-2, -14)

y self-inner-product first-slot term1 = 28

y self-inner-product inner sum := sum(72, 28)

y self-inner-product inner sum = 100

ordered cross inner-product:same matrix=[[16, 2], [2, 5]];first vector=[-3, 2];second vector=[-2, -2]

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

ordered cross inner-product matrix term0,0 := mul(16, -2)

ordered cross inner-product matrix term0,0 = -32

ordered cross inner-product matrix term0,1 := mul(2, -2)

ordered cross inner-product matrix term0,1 = -4

ordered cross inner-product image coordinate0 := sum(-32, -4)

ordered cross inner-product image coordinate0 = -36

ordered cross inner-product matrix term1,0 := mul(2, -2)

ordered cross inner-product matrix term1,0 = -4

ordered cross inner-product matrix term1,1 := mul(5, -2)

ordered cross inner-product matrix term1,1 = -10

ordered cross inner-product image coordinate1 := sum(-4, -10)

ordered cross inner-product image coordinate1 = -14

ordered cross inner-product first-slot term0 := mul(-3, -36)

ordered cross inner-product first-slot term0 = 108

ordered cross inner-product first-slot term1 := mul(2, -14)

ordered cross inner-product first-slot term1 = -28

ordered cross inner-product inner sum := sum(108, -28)

ordered cross inner-product inner sum = 80

Cauchy-Schwarz signed inner squared := mul(80, 80)

Cauchy-Schwarz signed inner squared = 6400

Cauchy-Schwarz product of self-inner-products := mul(140, 100)

Cauchy-Schwarz product of self-inner-products = 14000

Cauchy-Schwarz projection scalar := div(80, 100)

Cauchy-Schwarz projection scalar = 4/5

projection scaled y coordinate0 := mul(4/5, -2)

projection scaled y coordinate0 = -8/5

projection residual coordinate0 := sub(-3, -8/5)

projection residual coordinate0 = -7/5

projection scaled y coordinate1 := mul(4/5, -2)

projection scaled y coordinate1 = -8/5

projection residual coordinate1 := sub(2, -8/5)

projection residual coordinate1 = 18/5

projection residual self-inner-product:same matrix=[[16, 2], [2, 5]];first vector=[-7/5, 18/5];second vector=[-7/5, 18/5]

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

projection residual self-inner-product matrix term0,0 := mul(16, -7/5)

projection residual self-inner-product matrix term0,0 = -112/5

projection residual self-inner-product matrix term0,1 := mul(2, 18/5)

projection residual self-inner-product matrix term0,1 = 36/5

projection residual self-inner-product image coordinate0 := sum(-112/5, 36/5)

projection residual self-inner-product image coordinate0 = -76/5

projection residual self-inner-product matrix term1,0 := mul(2, -7/5)

projection residual self-inner-product matrix term1,0 = -14/5

projection residual self-inner-product matrix term1,1 := mul(5, 18/5)

projection residual self-inner-product matrix term1,1 = 18

projection residual self-inner-product image coordinate1 := sum(-14/5, 18)

projection residual self-inner-product image coordinate1 = 76/5

projection residual self-inner-product first-slot term0 := mul(-7/5, -76/5)

projection residual self-inner-product first-slot term0 = 532/25

projection residual self-inner-product first-slot term1 := mul(18/5, 76/5)

projection residual self-inner-product first-slot term1 = 1368/25

projection residual self-inner-product inner sum := sum(532/25, 1368/25)

projection residual self-inner-product inner sum = 76

yy=100>0;projection scalar=4/5;residual=[-7/5, 18/5];residual square=76

q=xx-xy^2/yy>=0;positive multiplication gives xy^2<=xx*yy

逐步反向核對

• S058 · 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。
• S057 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product inner sum，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product first-slot term1，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product first-slot term0，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product image coordinate1，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product matrix term1,1，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product matrix term1,0，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product image coordinate0，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product matrix term0,1，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 projection residual self-inner-product matrix term0,0，不以小數近似替代。
• S048 · 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。
• S047 · premise_and_result_audit — 以精確有理數重算 projection residual coordinate1，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 projection scaled y coordinate1，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 projection residual coordinate0，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 projection scaled y coordinate0，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Cauchy-Schwarz projection scalar，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Cauchy-Schwarz product of self-inner-products，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Cauchy-Schwarz signed inner squared，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product inner sum，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product first-slot term1，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product first-slot term0，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product image coordinate1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product matrix term1,1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product matrix term1,0，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product image coordinate0，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product matrix term0,1，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 ordered cross inner-product matrix term0,0，不以小數近似替代。
• S031 · 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。
• 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

(80, 140, 100, 6400, 14000, 是, Cauchy_Schwarz_universal_certificate)

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。