第 19922 題：商向量空間

原題與已知條件

有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={"ambient_dimension":3,"basis_w":[[1,22,0]],"v":[23,2,3],"u":[25,46,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 有限維實向量空間、明示子空間基底與exact membership。操作=quotient-coset-equivalence；資料={"ambient_dimension":3,"basis_w":[[1,22,0]],"v":[23,2,3],"u":[25,46,3]}。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-quotient-vector-space

逐步推導

source operation=quotient-coset-equivalence;original snapshot and mathematics retained

explicit ordered source output fields=[difference, equivalent, criterion, boundary]

quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [22], [0]];rows=3;columns=1

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 := div(1, 1)

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 = 1

quotient ORIGINAL ordered W basis:row=0;column=0;nonzero exact pivot=1;old FULL row=[1]

normalized FULL row=[1]

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(22, 1)

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 22

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(22, 22)

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference = 0

quotient ORIGINAL ordered W basis:target row=1;old row=[22];subtract factor=22 times FULL pivot row=[1]

new exact FULL row=[0]

quotient ORIGINAL ordered W basis:complete exact reduced matrix=[[1], [0], [0]];ORIGINAL pivot-column indices=[0]

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 22, 0]]

W is the FULL real span of these columns;dim W=1;zero coset is the WHOLE subspace W,not the vector0 alone

quotient ORIGINAL v minus u coordinate0 := sub(23, 25)

quotient ORIGINAL v minus u coordinate0 = -2

quotient ORIGINAL v minus u coordinate1 := sub(2, 46)

quotient ORIGINAL v minus u coordinate1 = -44

quotient ORIGINAL v minus u coordinate2 := sub(3, 3)

quotient ORIGINAL v minus u coordinate2 = 0

quotient coset original difference membership:COMPLETE exact original ordered matrix=[[1, -2], [22, -44], [0, 0]];rows=3;columns=2

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

quotient coset original difference membership pivot0 column0 normalized entry0 := div(1, 1)

quotient coset original difference membership pivot0 column0 normalized entry0 = 1

quotient coset original difference membership pivot0 column0 normalized entry1 := div(-2, 1)

quotient coset original difference membership pivot0 column0 normalized entry1 = -2

quotient coset original difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, -2]

normalized FULL row=[1, -2]

quotient coset original difference membership eliminate row1 pivot0 entry0 product := mul(22, 1)

quotient coset original difference membership eliminate row1 pivot0 entry0 product = 22

quotient coset original difference membership eliminate row1 pivot0 entry0 difference := sub(22, 22)

quotient coset original difference membership eliminate row1 pivot0 entry0 difference = 0

quotient coset original difference membership eliminate row1 pivot0 entry1 product := mul(22, -2)

quotient coset original difference membership eliminate row1 pivot0 entry1 product = -44

quotient coset original difference membership eliminate row1 pivot0 entry1 difference := sub(-44, -44)

quotient coset original difference membership eliminate row1 pivot0 entry1 difference = 0

quotient coset original difference membership:target row=1;old row=[22, -44];subtract factor=22 times FULL pivot row=[1, -2]

new exact FULL row=[0, 0]

quotient coset original difference membership:column=1;remaining exact entries=[0, 0]

no pivot added;current rank=1

quotient coset original difference membership:complete exact reduced matrix=[[1, -2], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

quotient coset original difference membership original coordinate0 reconstruction coefficient0 product := mul(1, -2)

quotient coset original difference membership original coordinate0 reconstruction coefficient0 product = -2

quotient coset original difference membership original coordinate0 reconstruction complete sum := sum(-2)

quotient coset original difference membership original coordinate0 reconstruction complete sum = -2

quotient coset original difference membership original coordinate1 reconstruction coefficient0 product := mul(22, -2)

quotient coset original difference membership original coordinate1 reconstruction coefficient0 product = -44

quotient coset original difference membership original coordinate1 reconstruction complete sum := sum(-44)

quotient coset original difference membership original coordinate1 reconstruction complete sum = -44

quotient coset original difference membership original coordinate2 reconstruction coefficient0 product := mul(0, -2)

quotient coset original difference membership original coordinate2 reconstruction coefficient0 product = 0

quotient coset original difference membership original coordinate2 reconstruction complete sum := sum(0)

quotient coset original difference membership original coordinate2 reconstruction complete sum = 0

quotient coset original difference membership:original proposed vector=[-2, -44, 0];original ordered independent columns=[[1, 22, 0]];complete reduced augmented system=[[1, -2], [0, 0], [0, 0]]

exact real-span coefficient witness=[-2];FULL original reconstruction=[-2, -44, 0];membership=是

ORIGINAL v=[23, 2, 3],u=[25, 46, 3];ORDERED difference v-u=[-2, -44, 0];full-real-span membership=是

v+W=u+W iff v-u in W;coset equality=是;not a test of representative vector equality

逐步反向核對

• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership original coordinate2 reconstruction complete sum，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership original coordinate2 reconstruction coefficient0 product，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership original coordinate1 reconstruction complete sum，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership original coordinate1 reconstruction coefficient0 product，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership original coordinate0 reconstruction complete sum，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership original coordinate0 reconstruction coefficient0 product，不以小數近似替代。
• S023 · 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。
• 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 quotient coset original difference membership eliminate row1 pivot0 entry1 difference，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership eliminate row1 pivot0 entry1 product，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership eliminate row1 pivot0 entry0 difference，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership eliminate row1 pivot0 entry0 product，不以小數近似替代。
• S016 · 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。
• S015 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership pivot0 column0 normalized entry1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 quotient coset original difference membership pivot0 column0 normalized entry0，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 quotient ORIGINAL v minus u coordinate2，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 quotient ORIGINAL v minus u coordinate1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 quotient ORIGINAL v minus u coordinate0，不以小數近似替代。
• 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 quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product，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 quotient ORIGINAL ordered W basis 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

([-2, -44, 0], 是, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

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。