第 19933 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-operations；資料={"ambient_dimension":3,"basis_w":[[1,8,0]],"v":[9,2,3],"u":[3,8,4],"scalar":8}。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-operations；資料={"ambient_dimension":3,"basis_w":[[1,8,0]],"v":[9,2,3],"u":[3,8,4],"scalar":8}。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-operations;original snapshot and mathematics retained

explicit ordered source output fields=[sum_representative, scalar_representative, addition_law, scalar_law, boundary]

quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [8], [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(8, 1)

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 8

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(8, 8)

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

quotient ORIGINAL ordered W basis:target row=1;old row=[8];subtract factor=8 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, 8, 0]]

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

ACTUAL subspace W full real span in R^3;original basis=[[1, 8, 0]]

for ANY representatives v'=v+w,u'=u+z with w,z in W,and ANY real lambda,sum difference=w+z in W and scalar difference=lambda*w in W;thus quotient addition/scaling independent of representatives

quotient sum representative coordinate0 := sum(9, 3)

quotient sum representative coordinate0 = 12

quotient sum representative coordinate1 := sum(2, 8)

quotient sum representative coordinate1 = 10

quotient sum representative coordinate2 := sum(3, 4)

quotient sum representative coordinate2 = 7

quotient scaled representative coordinate0 := mul(8, 9)

quotient scaled representative coordinate0 = 72

quotient scaled representative coordinate1 := mul(8, 2)

quotient scaled representative coordinate1 = 16

quotient scaled representative coordinate2 := mul(8, 3)

quotient scaled representative coordinate2 = 24

ORIGINAL representatives v=[9, 2, 3],u=[3, 8, 4];original real scalar=8

reported representative vectors v+u=[12, 10, 7],lambda*v=[72, 16, 24];resulting objects are their cosets,not a selected generic unique vector name

逐步反向核對

• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 quotient scaled representative coordinate2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 quotient scaled representative coordinate1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 quotient scaled representative coordinate0，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 quotient sum representative coordinate2，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 quotient sum representative coordinate1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 quotient sum representative coordinate0，不以小數近似替代。
• 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 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

([12, 10, 7], [72, 16, 24], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

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。