第 10803 題：有限線性組合

原題與已知條件

固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={"v1":[1,1],"v2":[1,-1],"target":[-10,-1]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。

• G001 — 固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={"v1":[1,1],"v2":[1,-1],"target":[-10,-1]}。請依 [v1,v2,target,det,a,b,reconstructed] 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-linear-combination-span

逐步推導

fixed real coordinate space;span uses all finite real linear combinations

given coefficient witness and existence of some coefficients are separate

column determinant first product := mul(1, -1)

column determinant first product = -1

column determinant second product := mul(1, 1)

column determinant second product = 1

signed column determinant := sub(-1, 1)

signed column determinant = -2

first column=[1, 1];second column=[1, -1];target=[-10, -1]

signed determinant=-2;independent generators=是

first coordinate numerator first product := mul(-10, -1)

first coordinate numerator first product = 10

first coordinate numerator second product := mul(1, -1)

first coordinate numerator second product = -1

first coordinate numerator := sub(10, -1)

first coordinate numerator = 11

second coordinate numerator first product := mul(1, -1)

second coordinate numerator first product = -1

second coordinate numerator second product := mul(-10, 1)

second coordinate numerator second product = -10

second coordinate numerator := sub(-1, -10)

second coordinate numerator = 9

coefficient of first generator := div(11, -2)

coefficient of first generator = -11/2

coefficient of second generator := div(9, -2)

coefficient of second generator = -9/2

R2 reconstructed coordinate0 first contribution := mul(-11/2, 1)

R2 reconstructed coordinate0 first contribution = -11/2

R2 reconstructed coordinate0 second contribution := mul(-9/2, 1)

R2 reconstructed coordinate0 second contribution = -9/2

R2 reconstructed coordinate0 sum := sum(-11/2, -9/2)

R2 reconstructed coordinate0 sum = -10

R2 reconstructed coordinate1 first contribution := mul(-11/2, 1)

R2 reconstructed coordinate1 first contribution = -11/2

R2 reconstructed coordinate1 second contribution := mul(-9/2, -1)

R2 reconstructed coordinate1 second contribution = 9/2

R2 reconstructed coordinate1 sum := sum(-11/2, 9/2)

R2 reconstructed coordinate1 sum = -1

first-generator coefficient=-11/2;second-generator coefficient=-9/2

reconstructed target=[-10, -1]

逐步反向核對

• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 R2 reconstructed coordinate1 sum，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 R2 reconstructed coordinate1 second contribution，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 R2 reconstructed coordinate1 first contribution，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 R2 reconstructed coordinate0 sum，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 R2 reconstructed coordinate0 second contribution，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 R2 reconstructed coordinate0 first contribution，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 coefficient of second generator，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 coefficient of first generator，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 second coordinate numerator，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 second coordinate numerator second product，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 second coordinate numerator first product，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 first coordinate numerator，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 first coordinate numerator second product，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 first coordinate numerator first product，不以小數近似替代。
• 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 — 以精確有理數重算 signed column determinant，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 column determinant second product，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 column determinant first product，不以小數近似替代。
• 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], [1, -1], [-10, -1], -2, -11/2, -9/2, [-10, -1])

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。