第 18665 題：子空間直和

原題與已知條件

有限維實向量空間、完整基底與exact rank證書。操作=directsum-internal-audit；資料={"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,15]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 有限維實向量空間、完整基底與exact rank證書。操作=directsum-internal-audit；資料={"ambient_dimension":3,"basis_u":[[1,0,0]],"basis_w":[[0,1,15]]}。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-direct-sum

逐步推導

source operation=directsum-internal-audit;original snapshot and mathematics retained

explicit ordered source output fields=[dimension_u, dimension_w, dimension_sum, dimension_intersection, unique_in_sum, covers_ambient, ambient_direct_sum]

directsum U:complete ordered matrix=[[1], [0], [0]];rows=3;columns=1

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

directsum U pivot0 column0 normalized entry0 := div(1, 1)

directsum U pivot0 column0 normalized entry0 = 1

directsum U:pivot row=0;column=0;nonzero pivot=1;old row=[1]

normalized full row=[1]

directsum U:complete exact reduced matrix=[[1], [0], [0]];pivot columns=[0]

rank=1;pivot columns independent and all original column relations preserved;zero rows retained

directsum U:complete ORIGINAL independent columns=[[1, 0, 0]]

declared subspace dimension=1;basis of its full real span;empty basis means zero subspace

directsum W:complete ordered matrix=[[0], [1], [15]];rows=3;columns=1

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

directsum W:matrix=[[0], [1], [15]];selected row=1

swap rows0,1;matrix=[[1], [0], [15]]

directsum W pivot0 column0 normalized entry0 := div(1, 1)

directsum W pivot0 column0 normalized entry0 = 1

directsum W:pivot row=0;column=0;nonzero pivot=1;old row=[1]

normalized full row=[1]

directsum W eliminate row2 pivot0 entry0 product := mul(15, 1)

directsum W eliminate row2 pivot0 entry0 product = 15

directsum W eliminate row2 pivot0 entry0 difference := sub(15, 15)

directsum W eliminate row2 pivot0 entry0 difference = 0

directsum W:target row=2;old=[15];factor=15;pivot row=[1]

new full row=[0]

directsum W:complete exact reduced matrix=[[1], [0], [0]];pivot columns=[0]

rank=1;pivot columns independent and all original column relations preserved;zero rows retained

directsum W:complete ORIGINAL independent columns=[[0, 1, 15]]

declared subspace dimension=1;basis of its full real span;empty basis means zero subspace

directsum ordered U then W:complete ordered matrix=[[1, 0], [0, 1], [0, 15]];rows=3;columns=2

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

directsum ordered U then W pivot0 column0 normalized entry0 := div(1, 1)

directsum ordered U then W pivot0 column0 normalized entry0 = 1

directsum ordered U then W pivot0 column0 normalized entry1 := div(0, 1)

directsum ordered U then W pivot0 column0 normalized entry1 = 0

directsum ordered U then W:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0]

normalized full row=[1, 0]

directsum ordered U then W pivot1 column1 normalized entry0 := div(0, 1)

directsum ordered U then W pivot1 column1 normalized entry0 = 0

directsum ordered U then W pivot1 column1 normalized entry1 := div(1, 1)

directsum ordered U then W pivot1 column1 normalized entry1 = 1

directsum ordered U then W:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1]

normalized full row=[0, 1]

directsum ordered U then W eliminate row2 pivot1 entry0 product := mul(15, 0)

directsum ordered U then W eliminate row2 pivot1 entry0 product = 0

directsum ordered U then W eliminate row2 pivot1 entry0 difference := sub(0, 0)

directsum ordered U then W eliminate row2 pivot1 entry0 difference = 0

directsum ordered U then W eliminate row2 pivot1 entry1 product := mul(15, 1)

directsum ordered U then W eliminate row2 pivot1 entry1 product = 15

directsum ordered U then W eliminate row2 pivot1 entry1 difference := sub(15, 15)

directsum ordered U then W eliminate row2 pivot1 entry1 difference = 0

directsum ordered U then W:target row=2;old=[0, 15];factor=15;pivot row=[0, 1]

new full row=[0, 0]

directsum ordered U then W:complete exact reduced matrix=[[1, 0], [0, 1], [0, 0]];pivot columns=[0, 1]

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

directsum U plus W:pivot columns=[0, 1];ALL free columns=[];full relation family=[]

EVERY zero combination is uniquely determined by its free coordinates;these free-coordinate unit relations form the full kernel;dimension=0

directsum U and W basis count sum := sum(1, 1)

directsum U and W basis count sum = 2

directsum actual intersection dimension := sub(2, 2)

directsum actual intersection dimension = 0

ORIGINAL U basis=[[1, 0, 0]];W basis=[[0, 1, 15]];combined rank=2;full zero-combination relations=[];common vectors=[]

kernel of sum map consists of(u,-u) for u in U intersection W;intersection dimension=0;unique decomposition INSIDE U+W=是;coverage of R^3=否

逐步反向核對

• S032 · 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。
• S031 · premise_and_result_audit — 以精確有理數重算 directsum actual intersection dimension，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 directsum U and W basis count sum，不以小數近似替代。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · premise_and_result_audit — 以精確有理數重算 directsum ordered U then W eliminate row2 pivot1 entry1 difference，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 directsum ordered U then W eliminate row2 pivot1 entry1 product，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 directsum ordered U then W eliminate row2 pivot1 entry0 difference，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 directsum ordered U then W eliminate row2 pivot1 entry0 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 — 以精確有理數重算 directsum ordered U then W pivot1 column1 normalized entry1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 directsum ordered U then W pivot1 column1 normalized entry0，不以小數近似替代。
• S019 · 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。
• S018 · premise_and_result_audit — 以精確有理數重算 directsum ordered U then W pivot0 column0 normalized entry1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 directsum ordered U then W pivot0 column0 normalized entry0，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · 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。
• 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 directsum W eliminate row2 pivot0 entry0 difference，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 directsum W eliminate row2 pivot0 entry0 product，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 — 以精確有理數重算 directsum W pivot0 column0 normalized entry0，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · 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。
• 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 directsum U 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

(1, 1, 2, 0, 是, 否, 否)

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。