第 33287 題：基底密度

原題與已知條件

只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={"vectors":[[1,-1],[-1,1],[0,0]],"zero_index":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 欄序回答。

• G001 — 只深化既有有限R2基底語義。操作=basisdeep-zero-vector-dependence-certificate；資料={"vectors":[[1,-1],[-1,1],[0,0]],"zero_index":2}。請依 [vectors,zero_index,selected_vector,is_zero,coefficients,dependence_certificate] 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

source roles = [qf-mod25-basisdeep-zero-vector-dependence-certificate, basisdeep-zero-vector-dependence-certificate, R2_basis_density_without_dimension_extension, []]

derived = [[vectors, index, selected, is_zero, nontrivial_coefficients, THIS_certificate]]

source roles = [zero certificate ORIGINAL vector0, [1, -1]]

derived = [[1, -1]]

source roles = [zero certificate ORIGINAL vector1, [-1, 1]]

derived = [[-1, 1]]

source roles = [zero certificate ORIGINAL vector2, [0, 0]]

derived = [[0, 0]]

zero proposed nontrivial relation ORIGINAL vector0 coordinate0 weighted term := mul(0, 1)

zero proposed nontrivial relation ORIGINAL vector0 coordinate0 weighted term = 0

zero proposed nontrivial relation ORIGINAL vector1 coordinate0 weighted term := mul(0, -1)

zero proposed nontrivial relation ORIGINAL vector1 coordinate0 weighted term = 0

zero proposed nontrivial relation ORIGINAL vector2 coordinate0 weighted term := mul(1, 0)

zero proposed nontrivial relation ORIGINAL vector2 coordinate0 weighted term = 0

zero proposed nontrivial relation ORIGINAL coordinate0 all terms sum := sum(0, 0, 0)

zero proposed nontrivial relation ORIGINAL coordinate0 all terms sum = 0

source roles = [zero proposed nontrivial relation, 0, [[1, -1], [-1, 1], [0, 0]], [0, 0, 1]]

derived = [[0, 0, 0], 0]

zero proposed nontrivial relation ORIGINAL vector0 coordinate1 weighted term := mul(0, -1)

zero proposed nontrivial relation ORIGINAL vector0 coordinate1 weighted term = 0

zero proposed nontrivial relation ORIGINAL vector1 coordinate1 weighted term := mul(0, 1)

zero proposed nontrivial relation ORIGINAL vector1 coordinate1 weighted term = 0

zero proposed nontrivial relation ORIGINAL vector2 coordinate1 weighted term := mul(1, 0)

zero proposed nontrivial relation ORIGINAL vector2 coordinate1 weighted term = 0

zero proposed nontrivial relation ORIGINAL coordinate1 all terms sum := sum(0, 0, 0)

zero proposed nontrivial relation ORIGINAL coordinate1 all terms sum = 0

source roles = [zero proposed nontrivial relation, 1, [[1, -1], [-1, 1], [0, 0]], [0, 0, 1]]

derived = [[0, 0, 0], 0]

source roles = [[[1, -1], [-1, 1], [0, 0]], 2, [0, 0]]

derived = [[0, 0, 1], [0, 0], 是]

逐步反向核對

• S015 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S014 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S013 · premise_and_result_audit — 以精確有理數重算 zero proposed nontrivial relation ORIGINAL coordinate1 all terms sum，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 zero proposed nontrivial relation ORIGINAL vector2 coordinate1 weighted term，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 zero proposed nontrivial relation ORIGINAL vector1 coordinate1 weighted term，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 zero proposed nontrivial relation ORIGINAL vector0 coordinate1 weighted term，不以小數近似替代。
• S009 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S008 · premise_and_result_audit — 以精確有理數重算 zero proposed nontrivial relation ORIGINAL coordinate0 all terms sum，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 zero proposed nontrivial relation ORIGINAL vector2 coordinate0 weighted term，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 zero proposed nontrivial relation ORIGINAL vector1 coordinate0 weighted term，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 zero proposed nontrivial relation ORIGINAL vector0 coordinate0 weighted term，不以小數近似替代。
• S004 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S003 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S002 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S001 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。

結論

([[1, -1], [-1, 1], [0, 0]], 2, [0, 0], 是, [0, 0, 1], 是)

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。