第 18721 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-coordinate-change-boundary；資料={"change_of_basis":[[1,21],[0,1]],"primal_coordinates":[22,3],"covector_coordinates":[2,21]}。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-coordinate-change-boundary；資料={"change_of_basis":[[1,21],[0,1]],"primal_coordinates":[22,3],"covector_coordinates":[2,21]}。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-coordinate-change-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary]

dual change augmented P with identity:complete ordered matrix=[[1, 21, 1, 0], [0, 1, 0, 1]];rows=2;columns=4

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

dual change augmented P with identity pivot0 column0 normalized entry0 := div(1, 1)

dual change augmented P with identity pivot0 column0 normalized entry0 = 1

dual change augmented P with identity pivot0 column0 normalized entry1 := div(21, 1)

dual change augmented P with identity pivot0 column0 normalized entry1 = 21

dual change augmented P with identity pivot0 column0 normalized entry2 := div(1, 1)

dual change augmented P with identity pivot0 column0 normalized entry2 = 1

dual change augmented P with identity pivot0 column0 normalized entry3 := div(0, 1)

dual change augmented P with identity pivot0 column0 normalized entry3 = 0

dual change augmented P with identity:pivot row=0;column=0;nonzero pivot=1;old row=[1, 21, 1, 0]

normalized full row=[1, 21, 1, 0]

dual change augmented P with identity pivot1 column1 normalized entry0 := div(0, 1)

dual change augmented P with identity pivot1 column1 normalized entry0 = 0

dual change augmented P with identity pivot1 column1 normalized entry1 := div(1, 1)

dual change augmented P with identity pivot1 column1 normalized entry1 = 1

dual change augmented P with identity pivot1 column1 normalized entry2 := div(0, 1)

dual change augmented P with identity pivot1 column1 normalized entry2 = 0

dual change augmented P with identity pivot1 column1 normalized entry3 := div(1, 1)

dual change augmented P with identity pivot1 column1 normalized entry3 = 1

dual change augmented P with identity:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1, 0, 1]

normalized full row=[0, 1, 0, 1]

dual change augmented P with identity eliminate row0 pivot1 entry0 product := mul(21, 0)

dual change augmented P with identity eliminate row0 pivot1 entry0 product = 0

dual change augmented P with identity eliminate row0 pivot1 entry0 difference := sub(1, 0)

dual change augmented P with identity eliminate row0 pivot1 entry0 difference = 1

dual change augmented P with identity eliminate row0 pivot1 entry1 product := mul(21, 1)

dual change augmented P with identity eliminate row0 pivot1 entry1 product = 21

dual change augmented P with identity eliminate row0 pivot1 entry1 difference := sub(21, 21)

dual change augmented P with identity eliminate row0 pivot1 entry1 difference = 0

dual change augmented P with identity eliminate row0 pivot1 entry2 product := mul(21, 0)

dual change augmented P with identity eliminate row0 pivot1 entry2 product = 0

dual change augmented P with identity eliminate row0 pivot1 entry2 difference := sub(1, 0)

dual change augmented P with identity eliminate row0 pivot1 entry2 difference = 1

dual change augmented P with identity eliminate row0 pivot1 entry3 product := mul(21, 1)

dual change augmented P with identity eliminate row0 pivot1 entry3 product = 21

dual change augmented P with identity eliminate row0 pivot1 entry3 difference := sub(0, 21)

dual change augmented P with identity eliminate row0 pivot1 entry3 difference = -21

dual change augmented P with identity:target row=0;old=[1, 21, 1, 0];factor=21;pivot row=[0, 1, 0, 1]

new full row=[1, 0, 1, -21]

dual change augmented P with identity:complete exact reduced matrix=[[1, 0, 1, -21], [0, 1, 0, 1]];pivot columns=[0, 1]

rank=2;pivot columns independent and all original column relations preserved;zero rows retained

dual P times inverse row0 column0 coordinate0 product := mul(1, 1)

dual P times inverse row0 column0 coordinate0 product = 1

dual P times inverse row0 column0 coordinate1 product := mul(21, 0)

dual P times inverse row0 column0 coordinate1 product = 0

dual P times inverse row0 column0 complete sum := sum(1, 0)

dual P times inverse row0 column0 complete sum = 1

dual P times inverse row0 column1 coordinate0 product := mul(1, -21)

dual P times inverse row0 column1 coordinate0 product = -21

dual P times inverse row0 column1 coordinate1 product := mul(21, 1)

dual P times inverse row0 column1 coordinate1 product = 21

dual P times inverse row0 column1 complete sum := sum(-21, 21)

dual P times inverse row0 column1 complete sum = 0

dual P times inverse row1 column0 coordinate0 product := mul(0, 1)

dual P times inverse row1 column0 coordinate0 product = 0

dual P times inverse row1 column0 coordinate1 product := mul(1, 0)

dual P times inverse row1 column0 coordinate1 product = 0

dual P times inverse row1 column0 complete sum := sum(0, 0)

dual P times inverse row1 column0 complete sum = 0

dual P times inverse row1 column1 coordinate0 product := mul(0, -21)

dual P times inverse row1 column1 coordinate0 product = 0

dual P times inverse row1 column1 coordinate1 product := mul(1, 1)

dual P times inverse row1 column1 coordinate1 product = 1

dual P times inverse row1 column1 complete sum := sum(0, 1)

dual P times inverse row1 column1 complete sum = 1

original P=[[1, 21], [0, 1]];exact row reduction [P|I] to [I|inverse];inverse=[[1, -21], [0, 1]]

complete P*inverse=identity;given old_primal=P*new_primal therefore new_primal=inverse*old_primal

dual new primal coordinate0 coordinate0 product := mul(1, 22)

dual new primal coordinate0 coordinate0 product = 22

dual new primal coordinate0 coordinate1 product := mul(-21, 3)

dual new primal coordinate0 coordinate1 product = -63

dual new primal coordinate0 complete sum := sum(22, -63)

dual new primal coordinate0 complete sum = -41

dual new primal coordinate1 coordinate0 product := mul(0, 22)

dual new primal coordinate1 coordinate0 product = 0

dual new primal coordinate1 coordinate1 product := mul(1, 3)

dual new primal coordinate1 coordinate1 product = 3

dual new primal coordinate1 complete sum := sum(0, 3)

dual new primal coordinate1 complete sum = 3

dual new covector coordinate0 coordinate0 product := mul(2, 1)

dual new covector coordinate0 coordinate0 product = 2

dual new covector coordinate0 coordinate1 product := mul(21, 0)

dual new covector coordinate0 coordinate1 product = 0

dual new covector coordinate0 complete sum := sum(2, 0)

dual new covector coordinate0 complete sum = 2

dual new covector coordinate1 coordinate0 product := mul(2, 21)

dual new covector coordinate1 coordinate0 product = 42

dual new covector coordinate1 coordinate1 product := mul(21, 1)

dual new covector coordinate1 coordinate1 product = 21

dual new covector coordinate1 complete sum := sum(42, 21)

dual new covector coordinate1 complete sum = 63

dual recover old primal coordinate0 coordinate0 product := mul(1, -41)

dual recover old primal coordinate0 coordinate0 product = -41

dual recover old primal coordinate0 coordinate1 product := mul(21, 3)

dual recover old primal coordinate0 coordinate1 product = 63

dual recover old primal coordinate0 complete sum := sum(-41, 63)

dual recover old primal coordinate0 complete sum = 22

dual recover old primal coordinate1 coordinate0 product := mul(0, -41)

dual recover old primal coordinate1 coordinate0 product = 0

dual recover old primal coordinate1 coordinate1 product := mul(1, 3)

dual recover old primal coordinate1 coordinate1 product = 3

dual recover old primal coordinate1 complete sum := sum(0, 3)

dual recover old primal coordinate1 complete sum = 3

dual old covector primal pairing coordinate0 product := mul(2, 22)

dual old covector primal pairing coordinate0 product = 44

dual old covector primal pairing coordinate1 product := mul(21, 3)

dual old covector primal pairing coordinate1 product = 63

dual old covector primal pairing complete sum := sum(44, 63)

dual old covector primal pairing complete sum = 107

dual new covector primal pairing coordinate0 product := mul(2, -41)

dual new covector primal pairing coordinate0 product = -82

dual new covector primal pairing coordinate1 product := mul(63, 3)

dual new covector primal pairing coordinate1 product = 189

dual new covector primal pairing complete sum := sum(-82, 189)

dual new covector primal pairing complete sum = 107

original old primal=[22, 3];old covector ROW=[2, 21];given old_primal=P*new_primal

new primal=[-41, 3];new covector=old_covector*P=[2, 63];pairing preserved=107=107;canonical identification CLAIM=否

逐步反向核對

• S060 · 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。
• S059 · premise_and_result_audit — 以精確有理數重算 dual new covector primal pairing complete sum，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 dual new covector primal pairing coordinate1 product，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 dual new covector primal pairing coordinate0 product，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 dual old covector primal pairing complete sum，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 dual old covector primal pairing coordinate1 product，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 dual old covector primal pairing coordinate0 product，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 dual recover old primal coordinate1 complete sum，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 dual recover old primal coordinate1 coordinate1 product，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 dual recover old primal coordinate1 coordinate0 product，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 dual recover old primal coordinate0 complete sum，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 dual recover old primal coordinate0 coordinate1 product，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 dual recover old primal coordinate0 coordinate0 product，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 dual new covector coordinate1 complete sum，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 dual new covector coordinate1 coordinate1 product，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 dual new covector coordinate1 coordinate0 product，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 dual new covector coordinate0 complete sum，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 dual new covector coordinate0 coordinate1 product，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 dual new covector coordinate0 coordinate0 product，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 dual new primal coordinate1 complete sum，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 dual new primal coordinate1 coordinate1 product，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 dual new primal coordinate1 coordinate0 product，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 dual new primal coordinate0 complete sum，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 dual new primal coordinate0 coordinate1 product，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 dual new primal coordinate0 coordinate0 product，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 dual P times inverse row1 column1 complete sum，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row1 column1 coordinate1 product，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row1 column1 coordinate0 product，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row1 column0 complete sum，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row1 column0 coordinate1 product，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row1 column0 coordinate0 product，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row0 column1 complete sum，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row0 column1 coordinate1 product，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row0 column1 coordinate0 product，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row0 column0 complete sum，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row0 column0 coordinate1 product，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 dual P times inverse row0 column0 coordinate0 product，不以小數近似替代。
• S022 · 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。
• 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 dual change augmented P with identity eliminate row0 pivot1 entry3 difference，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry3 product，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry2 difference，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry2 product，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry1 difference，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry1 product，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry0 difference，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry0 product，不以小數近似替代。
• S012 · 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。
• S011 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity pivot1 column1 normalized entry3，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity pivot1 column1 normalized entry2，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity pivot1 column1 normalized entry1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity pivot1 column1 normalized entry0，不以小數近似替代。
• 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 dual change augmented P with identity pivot0 column0 normalized entry3，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity pivot0 column0 normalized entry2，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity pivot0 column0 normalized entry1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 dual change augmented P with identity pivot0 column0 normalized entry0，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

([-41, 3], [2, 63], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)

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。