第 18778 題：對偶向量空間

原題與已知條件

有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={"coefficients":[3,-2],"u":[1,4],"v":[5,-1],"alpha":3,"beta":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 有限維R^n中的線性泛函與明示基底座標。操作=dual-linearity-certificate；資料={"coefficients":[3,-2],"u":[1,4],"v":[5,-1],"alpha":3,"beta":-2}。請依 exact_ordered_tuple_declared_by_interface 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-dual-vector-space

逐步推導

source operation=dual-linearity-certificate;original snapshot and mathematics retained

explicit ordered source output fields=[left, right, equal, certificate]

fixed original covector ROW=[3, -2];coordinate dimension=2

define phi(x)=sum_j row[j]*x[j] for EVERY real coordinate vector x;finite distributivity gives phi(alpha*u+beta*v)=alpha*phi(u)+beta*phi(v)

dual linear combination coordinate0 alpha u := mul(3, 1)

dual linear combination coordinate0 alpha u = 3

dual linear combination coordinate0 beta v := mul(-2, 5)

dual linear combination coordinate0 beta v = -10

dual linear combination coordinate0 sum := sum(3, -10)

dual linear combination coordinate0 sum = -7

dual linear combination coordinate1 alpha u := mul(3, 4)

dual linear combination coordinate1 alpha u = 12

dual linear combination coordinate1 beta v := mul(-2, -1)

dual linear combination coordinate1 beta v = 2

dual linear combination coordinate1 sum := sum(12, 2)

dual linear combination coordinate1 sum = 14

dual linearity left evaluation coordinate0 product := mul(3, -7)

dual linearity left evaluation coordinate0 product = -21

dual linearity left evaluation coordinate1 product := mul(-2, 14)

dual linearity left evaluation coordinate1 product = -28

dual linearity left evaluation complete sum := sum(-21, -28)

dual linearity left evaluation complete sum = -49

dual linearity phi u coordinate0 product := mul(3, 1)

dual linearity phi u coordinate0 product = 3

dual linearity phi u coordinate1 product := mul(-2, 4)

dual linearity phi u coordinate1 product = -8

dual linearity phi u complete sum := sum(3, -8)

dual linearity phi u complete sum = -5

dual linearity phi v coordinate0 product := mul(3, 5)

dual linearity phi v coordinate0 product = 15

dual linearity phi v coordinate1 product := mul(-2, -1)

dual linearity phi v coordinate1 product = 2

dual linearity phi v complete sum := sum(15, 2)

dual linearity phi v complete sum = 17

dual linearity alpha times phi u := mul(3, -5)

dual linearity alpha times phi u = -15

dual linearity beta times phi v := mul(-2, 17)

dual linearity beta times phi v = -34

dual linearity right complete sum := sum(-15, -34)

dual linearity right complete sum = -49

逐步反向核對

• S020 · premise_and_result_audit — 以精確有理數重算 dual linearity right complete sum，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 dual linearity beta times phi v，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 dual linearity alpha times phi u，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 dual linearity phi v complete sum，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 dual linearity phi v coordinate1 product，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 dual linearity phi v coordinate0 product，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 dual linearity phi u complete sum，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 dual linearity phi u coordinate1 product，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 dual linearity phi u coordinate0 product，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 dual linearity left evaluation complete sum，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 dual linearity left evaluation coordinate1 product，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 dual linearity left evaluation coordinate0 product，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 dual linear combination coordinate1 sum，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 dual linear combination coordinate1 beta v，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 dual linear combination coordinate1 alpha u，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 dual linear combination coordinate0 sum，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 dual linear combination coordinate0 beta v，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 dual linear combination coordinate0 alpha u，不以小數近似替代。
• S002 · 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。
• S001 · 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。

結論

(-49, -49, 是, row_covector_gives_universal_linearity_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。