第 11226 題：線性獨立與基底

原題與已知條件

明示有限座標向量族。操作=linbasis-coordinate-witness；資料={"basis":[[1,-2],[0,1]],"target":[-12,-3]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。

• G001 — 明示有限座標向量族。操作=linbasis-coordinate-witness；資料={"basis":[[1,-2],[0,1]],"target":[-12,-3]}。請依 [basis,target,det,first_coordinate,second_coordinate,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-independence-basis

逐步推導

basis determinant first product := mul(1, 1)

basis determinant first product = 1

basis determinant second product := mul(0, -2)

basis determinant second product = 0

basis determinant signed difference := sub(1, 0)

basis determinant signed difference = 1

ordered basis columns=[[1, -2], [0, 1]];target=[-12, -3];signed determinant=1

nonzero determinant;unique real coordinates;Cramer division permitted

first coordinate numerator first product := mul(-12, 1)

first coordinate numerator first product = -12

first coordinate numerator second product := mul(0, -3)

first coordinate numerator second product = 0

first coordinate numerator signed difference := sub(-12, 0)

first coordinate numerator signed difference = -12

second coordinate numerator first product := mul(1, -3)

second coordinate numerator first product = -3

second coordinate numerator second product := mul(-12, -2)

second coordinate numerator second product = 24

second coordinate numerator signed difference := sub(-3, 24)

second coordinate numerator signed difference = -27

first basis coordinate := div(-12, 1)

first basis coordinate = -12

second basis coordinate := div(-27, 1)

second basis coordinate = -27

coordinate reconstruction:ordered vectors=[[1, -2], [0, 1]];aligned weights=[-12, -27]

dimension=2;preserve duplicates and order

coordinate reconstruction coordinate0 term0 := mul(-12, 1)

coordinate reconstruction coordinate0 term0 = -12

coordinate reconstruction coordinate0 term1 := mul(-27, 0)

coordinate reconstruction coordinate0 term1 = 0

coordinate reconstruction coordinate0 sum := sum(-12, 0)

coordinate reconstruction coordinate0 sum = -12

coordinate reconstruction coordinate1 term0 := mul(-12, -2)

coordinate reconstruction coordinate1 term0 = 24

coordinate reconstruction coordinate1 term1 := mul(-27, 1)

coordinate reconstruction coordinate1 term1 = -27

coordinate reconstruction coordinate1 sum := sum(24, -27)

coordinate reconstruction coordinate1 sum = -3

basis target residual0 := sub(-12, -12)

basis target residual0 = 0

basis target residual1 := sub(-3, -3)

basis target residual1 = 0

coordinates=[-12, -27];reconstructed=[-12, -3];target=[-12, -3];residuals=[0, 0]

both original equations satisfied;uniqueness follows from qualified basis

逐步反向核對

• 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 basis target residual1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 basis target residual0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 coordinate reconstruction coordinate1 sum，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 coordinate reconstruction coordinate1 term1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 coordinate reconstruction coordinate1 term0，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 coordinate reconstruction coordinate0 sum，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 coordinate reconstruction coordinate0 term1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 coordinate reconstruction coordinate0 term0，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 second basis coordinate，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 first basis coordinate，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 second coordinate numerator signed difference，不以小數近似替代。
• S009 · 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。
• S008 · 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。
• S007 · premise_and_result_audit — 以精確有理數重算 first coordinate numerator signed difference，不以小數近似替代。
• S006 · 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。
• S005 · 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。
• 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 basis determinant signed difference，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 basis determinant second product，不以小數近似替代。
• S001 · premise_and_result_audit — 以精確有理數重算 basis determinant first product，不以小數近似替代。

結論

([[1, -2], [0, 1]], [-12, -3], 1, -12, -27, [-12, -3])

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。