{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0001","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-12,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-12, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-12, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241001,"parameters":{"vectors":[[1,0],[-12,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-12,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0002","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-11,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-11, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-11, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241002,"parameters":{"vectors":[[1,0],[-11,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-11,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0003","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-10,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-10, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-10, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241003,"parameters":{"vectors":[[1,0],[-10,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-10,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0004","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-9,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-9, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-9, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241004,"parameters":{"vectors":[[1,0],[-9,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-9,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0005","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-8,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-8, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-8, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241005,"parameters":{"vectors":[[1,0],[-8,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-8,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0006","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-7,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-7, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-7, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241006,"parameters":{"vectors":[[1,0],[-7,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-7,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0007","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-6,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-6, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-6, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241007,"parameters":{"vectors":[[1,0],[-6,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0008","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-5,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-5, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-5, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241008,"parameters":{"vectors":[[1,0],[-5,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0009","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-4,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-4, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-4, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241009,"parameters":{"vectors":[[1,0],[-4,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0010","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-3,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-3, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-3, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241010,"parameters":{"vectors":[[1,0],[-3,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0011","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-2,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-2, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-2, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241011,"parameters":{"vectors":[[1,0],[-2,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0012","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[-1,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [-1, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [-1, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241012,"parameters":{"vectors":[[1,0],[-1,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[-1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0013","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[0,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241013,"parameters":{"vectors":[[1,0],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0014","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[1,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [1, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [1, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241014,"parameters":{"vectors":[[1,0],[1,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0015","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[2,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [2, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [2, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241015,"parameters":{"vectors":[[1,0],[2,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0016","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[3,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [3, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [3, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241016,"parameters":{"vectors":[[1,0],[3,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0017","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[4,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [4, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [4, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241017,"parameters":{"vectors":[[1,0],[4,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0018","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[5,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [5, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [5, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241018,"parameters":{"vectors":[[1,0],[5,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0019","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[6,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [6, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [6, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241019,"parameters":{"vectors":[[1,0],[6,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0020","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[7,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [7, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [7, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241020,"parameters":{"vectors":[[1,0],[7,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[7,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0021","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[8,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [8, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [8, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241021,"parameters":{"vectors":[[1,0],[8,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[8,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0022","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[9,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [9, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [9, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241022,"parameters":{"vectors":[[1,0],[9,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[9,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0023","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[10,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [10, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [10, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241023,"parameters":{"vectors":[[1,0],[10,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[10,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0024","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[11,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [11, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [11, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241024,"parameters":{"vectors":[[1,0],[11,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[11,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0025","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-independence-audit；資料={\"vectors\":[[1,0],[12,1]]}。請依 [vectors,rank,independent] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [12, 1]], 2, 是)","explanation_zh":"依賴結構核對=([[1, 0], [12, 1]], 2, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-independence-audit","seed":202610241025,"parameters":{"vectors":[[1,0],[12,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-independence-audit","semantic_template":"qf-mod24-linbasis-independence-audit/{\"vectors\":[[1,0],[12,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0026","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-12],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -12], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -12], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241026,"parameters":{"vectors":[[1,-12],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-12],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0027","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-11],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -11], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -11], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241027,"parameters":{"vectors":[[1,-11],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-11],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0028","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-10],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -10], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -10], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241028,"parameters":{"vectors":[[1,-10],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-10],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0029","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-9],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -9], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -9], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241029,"parameters":{"vectors":[[1,-9],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-9],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0030","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-8],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -8], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -8], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241030,"parameters":{"vectors":[[1,-8],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-8],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0031","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-7],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -7], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -7], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241031,"parameters":{"vectors":[[1,-7],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-7],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0032","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-6],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -6], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -6], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241032,"parameters":{"vectors":[[1,-6],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-6],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0033","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-5],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -5], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -5], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241033,"parameters":{"vectors":[[1,-5],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-5],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0034","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-4],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -4], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -4], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241034,"parameters":{"vectors":[[1,-4],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-4],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0035","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-3],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -3], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -3], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241035,"parameters":{"vectors":[[1,-3],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-3],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0036","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-2],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -2], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241036,"parameters":{"vectors":[[1,-2],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-2],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0037","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,-1],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, -1], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241037,"parameters":{"vectors":[[1,-1],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,-1],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0038","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,0],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241038,"parameters":{"vectors":[[1,0],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,0],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0039","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,1],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 1], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241039,"parameters":{"vectors":[[1,1],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,1],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0040","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,2],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 2], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241040,"parameters":{"vectors":[[1,2],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,2],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0041","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,3],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 3], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241041,"parameters":{"vectors":[[1,3],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,3],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0042","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,4],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 4], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 4], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241042,"parameters":{"vectors":[[1,4],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,4],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0043","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,5],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 5], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241043,"parameters":{"vectors":[[1,5],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,5],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0044","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,6],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 6], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 6], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241044,"parameters":{"vectors":[[1,6],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,6],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0045","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,7],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 7], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 7], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241045,"parameters":{"vectors":[[1,7],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,7],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0046","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,8],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 8], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 8], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241046,"parameters":{"vectors":[[1,8],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,8],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0047","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,9],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 9], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 9], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241047,"parameters":{"vectors":[[1,9],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,9],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0048","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,10],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 10], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 10], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241048,"parameters":{"vectors":[[1,10],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,10],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0049","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,11],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 11], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 11], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241049,"parameters":{"vectors":[[1,11],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,11],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0050","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-basis-r2-audit；資料={\"vectors\":[[1,12],[0,1]]}。請依 [vectors,rank,independent,spans_R2,basis_R2] 欄序回答。","answer":{"kind":"expression","value":"([[1, 12], [0, 1]], 2, 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 12], [0, 1]], 2, 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-basis-r2-audit","seed":202610241050,"parameters":{"vectors":[[1,12],[0,1]],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-basis-r2-audit","semantic_template":"qf-mod24-linbasis-basis-r2-audit/{\"vectors\":[[1,12],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0051","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"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。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-12, -3], 1, -12, -27, [-12, -3])","explanation_zh":"依賴結構核對=([[1, -2], [0, 1]], [-12, -3], 1, -12, -27, [-12, -3])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241051,"parameters":{"basis":[[1,-2],[0,1]],"target":[-12,-3],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[-12,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0052","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-1],[0,1]],\"target\":[-11,-2]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-11, -2], 1, -11, -13, [-11, -2])","explanation_zh":"依賴結構核對=([[1, -1], [0, 1]], [-11, -2], 1, -11, -13, [-11, -2])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241052,"parameters":{"basis":[[1,-1],[0,1]],"target":[-11,-2],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[-11,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0053","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,0],[0,1]],\"target\":[-10,-1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-10, -1], 1, -10, -1, [-10, -1])","explanation_zh":"依賴結構核對=([[1, 0], [0, 1]], [-10, -1], 1, -10, -1, [-10, -1])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241053,"parameters":{"basis":[[1,0],[0,1]],"target":[-10,-1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[-10,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0054","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,1],[0,1]],\"target\":[-9,0]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [-9, 0], 1, -9, 9, [-9, 0])","explanation_zh":"依賴結構核對=([[1, 1], [0, 1]], [-9, 0], 1, -9, 9, [-9, 0])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241054,"parameters":{"basis":[[1,1],[0,1]],"target":[-9,0],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[-9,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0055","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,2],[0,1]],\"target\":[-8,1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [-8, 1], 1, -8, 17, [-8, 1])","explanation_zh":"依賴結構核對=([[1, 2], [0, 1]], [-8, 1], 1, -8, 17, [-8, 1])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241055,"parameters":{"basis":[[1,2],[0,1]],"target":[-8,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0056","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-2],[0,1]],\"target\":[-7,2]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-7, 2], 1, -7, -12, [-7, 2])","explanation_zh":"依賴結構核對=([[1, -2], [0, 1]], [-7, 2], 1, -7, -12, [-7, 2])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241056,"parameters":{"basis":[[1,-2],[0,1]],"target":[-7,2],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[-7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0057","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-1],[0,1]],\"target\":[-6,3]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-6, 3], 1, -6, -3, [-6, 3])","explanation_zh":"依賴結構核對=([[1, -1], [0, 1]], [-6, 3], 1, -6, -3, [-6, 3])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241057,"parameters":{"basis":[[1,-1],[0,1]],"target":[-6,3],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[-6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0058","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,0],[0,1]],\"target\":[-5,-3]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-5, -3], 1, -5, -3, [-5, -3])","explanation_zh":"依賴結構核對=([[1, 0], [0, 1]], [-5, -3], 1, -5, -3, [-5, -3])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241058,"parameters":{"basis":[[1,0],[0,1]],"target":[-5,-3],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[-5,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0059","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,1],[0,1]],\"target\":[-4,-2]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [-4, -2], 1, -4, 2, [-4, -2])","explanation_zh":"依賴結構核對=([[1, 1], [0, 1]], [-4, -2], 1, -4, 2, [-4, -2])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241059,"parameters":{"basis":[[1,1],[0,1]],"target":[-4,-2],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[-4,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0060","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,2],[0,1]],\"target\":[-3,-1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])","explanation_zh":"依賴結構核對=([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241060,"parameters":{"basis":[[1,2],[0,1]],"target":[-3,-1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[-3,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0061","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-2],[0,1]],\"target\":[-2,0]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-2, 0], 1, -2, -4, [-2, 0])","explanation_zh":"依賴結構核對=([[1, -2], [0, 1]], [-2, 0], 1, -2, -4, [-2, 0])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241061,"parameters":{"basis":[[1,-2],[0,1]],"target":[-2,0],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[-2,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0062","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-1],[0,1]],\"target\":[-1,1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-1, 1], 1, -1, 0, [-1, 1])","explanation_zh":"依賴結構核對=([[1, -1], [0, 1]], [-1, 1], 1, -1, 0, [-1, 1])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241062,"parameters":{"basis":[[1,-1],[0,1]],"target":[-1,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0063","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,0],[0,1]],\"target\":[0,2]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [0, 2], 1, 0, 2, [0, 2])","explanation_zh":"依賴結構核對=([[1, 0], [0, 1]], [0, 2], 1, 0, 2, [0, 2])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241063,"parameters":{"basis":[[1,0],[0,1]],"target":[0,2],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0064","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,1],[0,1]],\"target\":[1,3]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [1, 3], 1, 1, 2, [1, 3])","explanation_zh":"依賴結構核對=([[1, 1], [0, 1]], [1, 3], 1, 1, 2, [1, 3])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241064,"parameters":{"basis":[[1,1],[0,1]],"target":[1,3],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0065","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,2],[0,1]],\"target\":[2,-3]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [2, -3], 1, 2, -7, [2, -3])","explanation_zh":"依賴結構核對=([[1, 2], [0, 1]], [2, -3], 1, 2, -7, [2, -3])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241065,"parameters":{"basis":[[1,2],[0,1]],"target":[2,-3],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[2,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0066","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-2],[0,1]],\"target\":[3,-2]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [3, -2], 1, 3, 4, [3, -2])","explanation_zh":"依賴結構核對=([[1, -2], [0, 1]], [3, -2], 1, 3, 4, [3, -2])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241066,"parameters":{"basis":[[1,-2],[0,1]],"target":[3,-2],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[3,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0067","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-1],[0,1]],\"target\":[4,-1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [4, -1], 1, 4, 3, [4, -1])","explanation_zh":"依賴結構核對=([[1, -1], [0, 1]], [4, -1], 1, 4, 3, [4, -1])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241067,"parameters":{"basis":[[1,-1],[0,1]],"target":[4,-1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[4,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0068","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,0],[0,1]],\"target\":[5,0]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [5, 0], 1, 5, 0, [5, 0])","explanation_zh":"依賴結構核對=([[1, 0], [0, 1]], [5, 0], 1, 5, 0, [5, 0])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241068,"parameters":{"basis":[[1,0],[0,1]],"target":[5,0],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[5,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0069","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,1],[0,1]],\"target\":[6,1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [6, 1], 1, 6, -5, [6, 1])","explanation_zh":"依賴結構核對=([[1, 1], [0, 1]], [6, 1], 1, 6, -5, [6, 1])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241069,"parameters":{"basis":[[1,1],[0,1]],"target":[6,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0070","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,2],[0,1]],\"target\":[7,2]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [7, 2], 1, 7, -12, [7, 2])","explanation_zh":"依賴結構核對=([[1, 2], [0, 1]], [7, 2], 1, 7, -12, [7, 2])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241070,"parameters":{"basis":[[1,2],[0,1]],"target":[7,2],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0071","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-2],[0,1]],\"target\":[8,3]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [8, 3], 1, 8, 19, [8, 3])","explanation_zh":"依賴結構核對=([[1, -2], [0, 1]], [8, 3], 1, 8, 19, [8, 3])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241071,"parameters":{"basis":[[1,-2],[0,1]],"target":[8,3],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-2],[0,1]],\"target\":[8,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0072","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,-1],[0,1]],\"target\":[9,-3]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [9, -3], 1, 9, 6, [9, -3])","explanation_zh":"依賴結構核對=([[1, -1], [0, 1]], [9, -3], 1, 9, 6, [9, -3])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241072,"parameters":{"basis":[[1,-1],[0,1]],"target":[9,-3],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,-1],[0,1]],\"target\":[9,-3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0073","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,0],[0,1]],\"target\":[10,-2]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [10, -2], 1, 10, -2, [10, -2])","explanation_zh":"依賴結構核對=([[1, 0], [0, 1]], [10, -2], 1, 10, -2, [10, -2])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241073,"parameters":{"basis":[[1,0],[0,1]],"target":[10,-2],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,0],[0,1]],\"target\":[10,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0074","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,1],[0,1]],\"target\":[11,-1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [11, -1], 1, 11, -12, [11, -1])","explanation_zh":"依賴結構核對=([[1, 1], [0, 1]], [11, -1], 1, 11, -12, [11, -1])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241074,"parameters":{"basis":[[1,1],[0,1]],"target":[11,-1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,1],[0,1]],\"target\":[11,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0075","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,2],[0,1]],\"target\":[12,0]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [12, 0], 1, 12, -24, [12, 0])","explanation_zh":"依賴結構核對=([[1, 2], [0, 1]], [12, 0], 1, 12, -24, [12, 0])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-coordinate-witness","seed":202610241075,"parameters":{"basis":[[1,2],[0,1]],"target":[12,0],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-coordinate-witness","semantic_template":"qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[12,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0076","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-12]],\"coefficients\":[-1,12,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -12]], [-1, 12, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -12]], [-1, 12, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241076,"parameters":{"vectors":[[1,0],[0,1],[1,-12]],"coefficients":[-1,12,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,12,1],\"vectors\":[[1,0],[0,1],[1,-12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0077","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-11]],\"coefficients\":[-1,11,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -11]], [-1, 11, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -11]], [-1, 11, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241077,"parameters":{"vectors":[[1,0],[0,1],[1,-11]],"coefficients":[-1,11,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,11,1],\"vectors\":[[1,0],[0,1],[1,-11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0078","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-10]],\"coefficients\":[-1,10,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -10]], [-1, 10, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -10]], [-1, 10, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241078,"parameters":{"vectors":[[1,0],[0,1],[1,-10]],"coefficients":[-1,10,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,10,1],\"vectors\":[[1,0],[0,1],[1,-10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0079","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-9]],\"coefficients\":[-1,9,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -9]], [-1, 9, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -9]], [-1, 9, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241079,"parameters":{"vectors":[[1,0],[0,1],[1,-9]],"coefficients":[-1,9,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,9,1],\"vectors\":[[1,0],[0,1],[1,-9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0080","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-8]],\"coefficients\":[-1,8,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -8]], [-1, 8, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -8]], [-1, 8, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241080,"parameters":{"vectors":[[1,0],[0,1],[1,-8]],"coefficients":[-1,8,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,8,1],\"vectors\":[[1,0],[0,1],[1,-8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0081","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-7]],\"coefficients\":[-1,7,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -7]], [-1, 7, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -7]], [-1, 7, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241081,"parameters":{"vectors":[[1,0],[0,1],[1,-7]],"coefficients":[-1,7,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,7,1],\"vectors\":[[1,0],[0,1],[1,-7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0082","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-6]],\"coefficients\":[-1,6,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -6]], [-1, 6, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -6]], [-1, 6, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241082,"parameters":{"vectors":[[1,0],[0,1],[1,-6]],"coefficients":[-1,6,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,6,1],\"vectors\":[[1,0],[0,1],[1,-6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0083","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-5]],\"coefficients\":[-1,5,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -5]], [-1, 5, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -5]], [-1, 5, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241083,"parameters":{"vectors":[[1,0],[0,1],[1,-5]],"coefficients":[-1,5,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,5,1],\"vectors\":[[1,0],[0,1],[1,-5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0084","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-4]],\"coefficients\":[-1,4,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -4]], [-1, 4, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -4]], [-1, 4, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241084,"parameters":{"vectors":[[1,0],[0,1],[1,-4]],"coefficients":[-1,4,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,4,1],\"vectors\":[[1,0],[0,1],[1,-4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0085","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-3]],\"coefficients\":[-1,3,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -3]], [-1, 3, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -3]], [-1, 3, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241085,"parameters":{"vectors":[[1,0],[0,1],[1,-3]],"coefficients":[-1,3,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,3,1],\"vectors\":[[1,0],[0,1],[1,-3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0086","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-2]],\"coefficients\":[-1,2,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -2]], [-1, 2, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -2]], [-1, 2, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241086,"parameters":{"vectors":[[1,0],[0,1],[1,-2]],"coefficients":[-1,2,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,2,1],\"vectors\":[[1,0],[0,1],[1,-2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0087","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,-1]],\"coefficients\":[-1,1,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, -1]], [-1, 1, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, -1]], [-1, 1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241087,"parameters":{"vectors":[[1,0],[0,1],[1,-1]],"coefficients":[-1,1,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,1,1],\"vectors\":[[1,0],[0,1],[1,-1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0088","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,0]],\"coefficients\":[-1,0,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 0]], [-1, 0, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 0]], [-1, 0, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241088,"parameters":{"vectors":[[1,0],[0,1],[1,0]],"coefficients":[-1,0,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,0,1],\"vectors\":[[1,0],[0,1],[1,0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0089","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-1,-1,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-1, -1, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 1]], [-1, -1, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241089,"parameters":{"vectors":[[1,0],[0,1],[1,1]],"coefficients":[-1,-1,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-1,1],\"vectors\":[[1,0],[0,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0090","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,2]],\"coefficients\":[-1,-2,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 2]], [-1, -2, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 2]], [-1, -2, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241090,"parameters":{"vectors":[[1,0],[0,1],[1,2]],"coefficients":[-1,-2,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-2,1],\"vectors\":[[1,0],[0,1],[1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0091","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,3]],\"coefficients\":[-1,-3,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 3]], [-1, -3, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 3]], [-1, -3, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241091,"parameters":{"vectors":[[1,0],[0,1],[1,3]],"coefficients":[-1,-3,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-3,1],\"vectors\":[[1,0],[0,1],[1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0092","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,4]],\"coefficients\":[-1,-4,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 4]], [-1, -4, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 4]], [-1, -4, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241092,"parameters":{"vectors":[[1,0],[0,1],[1,4]],"coefficients":[-1,-4,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-4,1],\"vectors\":[[1,0],[0,1],[1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0093","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,5]],\"coefficients\":[-1,-5,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 5]], [-1, -5, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 5]], [-1, -5, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241093,"parameters":{"vectors":[[1,0],[0,1],[1,5]],"coefficients":[-1,-5,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-5,1],\"vectors\":[[1,0],[0,1],[1,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0094","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,6]],\"coefficients\":[-1,-6,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 6]], [-1, -6, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 6]], [-1, -6, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241094,"parameters":{"vectors":[[1,0],[0,1],[1,6]],"coefficients":[-1,-6,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-6,1],\"vectors\":[[1,0],[0,1],[1,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0095","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,7]],\"coefficients\":[-1,-7,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 7]], [-1, -7, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 7]], [-1, -7, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241095,"parameters":{"vectors":[[1,0],[0,1],[1,7]],"coefficients":[-1,-7,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-7,1],\"vectors\":[[1,0],[0,1],[1,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0096","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,8]],\"coefficients\":[-1,-8,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 8]], [-1, -8, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 8]], [-1, -8, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241096,"parameters":{"vectors":[[1,0],[0,1],[1,8]],"coefficients":[-1,-8,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-8,1],\"vectors\":[[1,0],[0,1],[1,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0097","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,9]],\"coefficients\":[-1,-9,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 9]], [-1, -9, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 9]], [-1, -9, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241097,"parameters":{"vectors":[[1,0],[0,1],[1,9]],"coefficients":[-1,-9,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-9,1],\"vectors\":[[1,0],[0,1],[1,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0098","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,10]],\"coefficients\":[-1,-10,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 10]], [-1, -10, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 10]], [-1, -10, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241098,"parameters":{"vectors":[[1,0],[0,1],[1,10]],"coefficients":[-1,-10,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-10,1],\"vectors\":[[1,0],[0,1],[1,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0099","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,11]],\"coefficients\":[-1,-11,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241099,"parameters":{"vectors":[[1,0],[0,1],[1,11]],"coefficients":[-1,-11,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-11,1],\"vectors\":[[1,0],[0,1],[1,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r01-linear-independence-basis-0100","batch_id":"batch-20261013-advance-20261024-r01-linear-independence-basis-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-independence-basis"],"title_zh":"線性獨立與基底","stem_zh":"明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,12]],\"coefficients\":[-1,-12,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 12]], [-1, -12, 1], [0, 0], 是, 是, 是)","explanation_zh":"依賴結構核對=([[1, 0], [0, 1], [1, 12]], [-1, -12, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性獨立與基底"]},"variant":{"family_id":"qf-mod24-linbasis-dependence-witness-audit","seed":202610241100,"parameters":{"vectors":[[1,0],[0,1],[1,12]],"coefficients":[-1,-12,1],"domain":"Exact_finite_coordinate_basis","operation":"linbasis-dependence-witness-audit","semantic_template":"qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-12,1],\"vectors\":[[1,0],[0,1],[1,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}