{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0001","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"vector\":[-4,-2,-3]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-12, -3, -5], [-2, 12, -6]], [-4, -2, -3], [69, 2])","explanation_zh":"Wave4根節點核對=([[-12, -3, -5], [-2, 12, -6]], [-4, -2, -3], [69, 2])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275001,"parameters":{"matrix":[[-12,-3,-5],[-2,12,-6]],"vector":[-4,-2,-3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"vector\":[-4,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0002","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"vector\":[-3,-1,-2]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-11, -2, -2], [-1, 11, -1]], [-3, -1, -2], [39, -6])","explanation_zh":"Wave4根節點核對=([[-11, -2, -2], [-1, 11, -1]], [-3, -1, -2], [39, -6])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275002,"parameters":{"matrix":[[-11,-2,-2],[-1,11,-1]],"vector":[-3,-1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"vector\":[-3,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0003","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-10,-1,1],[0,10,4]],\"vector\":[-2,0,-1]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-10, -1, 1], [0, 10, 4]], [-2, 0, -1], [19, -4])","explanation_zh":"Wave4根節點核對=([[-10, -1, 1], [0, 10, 4]], [-2, 0, -1], [19, -4])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275003,"parameters":{"matrix":[[-10,-1,1],[0,10,4]],"vector":[-2,0,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-10,-1,1],[0,10,4]],\"vector\":[-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0004","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-9,0,4],[1,9,-4]],\"vector\":[-1,1,0]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-9, 0, 4], [1, 9, -4]], [-1, 1, 0], [9, 8])","explanation_zh":"Wave4根節點核對=([[-9, 0, 4], [1, 9, -4]], [-1, 1, 0], [9, 8])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275004,"parameters":{"matrix":[[-9,0,4],[1,9,-4]],"vector":[-1,1,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-9,0,4],[1,9,-4]],\"vector\":[-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0005","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-8,1,-4],[2,8,1]],\"vector\":[0,2,1]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-8, 1, -4], [2, 8, 1]], [0, 2, 1], [-2, 17])","explanation_zh":"Wave4根節點核對=([[-8, 1, -4], [2, 8, 1]], [0, 2, 1], [-2, 17])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275005,"parameters":{"matrix":[[-8,1,-4],[2,8,1]],"vector":[0,2,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-8,1,-4],[2,8,1]],\"vector\":[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0006","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-7,2,-1],[-2,7,6]],\"vector\":[1,-2,2]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-7, 2, -1], [-2, 7, 6]], [1, -2, 2], [-13, -4])","explanation_zh":"Wave4根節點核對=([[-7, 2, -1], [-2, 7, 6]], [1, -2, 2], [-13, -4])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275006,"parameters":{"matrix":[[-7,2,-1],[-2,7,6]],"vector":[1,-2,2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-7,2,-1],[-2,7,6]],\"vector\":[1,-2,2]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0007","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-6,3,2],[-1,6,-2]],\"vector\":[2,-1,3]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-6, 3, 2], [-1, 6, -2]], [2, -1, 3], [-9, -14])","explanation_zh":"Wave4根節點核對=([[-6, 3, 2], [-1, 6, -2]], [2, -1, 3], [-9, -14])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275007,"parameters":{"matrix":[[-6,3,2],[-1,6,-2]],"vector":[2,-1,3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-6,3,2],[-1,6,-2]],\"vector\":[2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0008","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-5,-3,5],[0,5,3]],\"vector\":[3,0,-3]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-5, -3, 5], [0, 5, 3]], [3, 0, -3], [-30, -9])","explanation_zh":"Wave4根節點核對=([[-5, -3, 5], [0, 5, 3]], [3, 0, -3], [-30, -9])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275008,"parameters":{"matrix":[[-5,-3,5],[0,5,3]],"vector":[3,0,-3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-5,-3,5],[0,5,3]],\"vector\":[3,0,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0009","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-4,-2,-3],[1,4,-5]],\"vector\":[4,1,-2]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-4, -2, -3], [1, 4, -5]], [4, 1, -2], [-12, 18])","explanation_zh":"Wave4根節點核對=([[-4, -2, -3], [1, 4, -5]], [4, 1, -2], [-12, 18])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275009,"parameters":{"matrix":[[-4,-2,-3],[1,4,-5]],"vector":[4,1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-4,-2,-3],[1,4,-5]],\"vector\":[4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0010","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-3,-1,0],[2,3,0]],\"vector\":[-4,2,-1]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-3, -1, 0], [2, 3, 0]], [-4, 2, -1], [10, -2])","explanation_zh":"Wave4根節點核對=([[-3, -1, 0], [2, 3, 0]], [-4, 2, -1], [10, -2])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275010,"parameters":{"matrix":[[-3,-1,0],[2,3,0]],"vector":[-4,2,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-3,-1,0],[2,3,0]],\"vector\":[-4,2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0011","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-2,0,3],[-2,2,5]],\"vector\":[-3,-2,0]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-2, 0, 3], [-2, 2, 5]], [-3, -2, 0], [6, 2])","explanation_zh":"Wave4根節點核對=([[-2, 0, 3], [-2, 2, 5]], [-3, -2, 0], [6, 2])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275011,"parameters":{"matrix":[[-2,0,3],[-2,2,5]],"vector":[-3,-2,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-2,0,3],[-2,2,5]],\"vector\":[-3,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0012","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[-1,1,-5],[-1,1,-3]],\"vector\":[-2,-1,1]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1, -5], [-1, 1, -3]], [-2, -1, 1], [-4, -2])","explanation_zh":"Wave4根節點核對=([[-1, 1, -5], [-1, 1, -3]], [-2, -1, 1], [-4, -2])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275012,"parameters":{"matrix":[[-1,1,-5],[-1,1,-3]],"vector":[-2,-1,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[-1,1,-5],[-1,1,-3]],\"vector\":[-2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0013","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[0,2,-2],[0,0,2]],\"vector\":[-1,0,2]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, -2], [0, 0, 2]], [-1, 0, 2], [-4, 4])","explanation_zh":"Wave4根節點核對=([[0, 2, -2], [0, 0, 2]], [-1, 0, 2], [-4, 4])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275013,"parameters":{"matrix":[[0,2,-2],[0,0,2]],"vector":[-1,0,2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[0,2,-2],[0,0,2]],\"vector\":[-1,0,2]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0014","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[1,3,1],[1,-1,-6]],\"vector\":[0,1,3]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[1, 3, 1], [1, -1, -6]], [0, 1, 3], [6, -19])","explanation_zh":"Wave4根節點核對=([[1, 3, 1], [1, -1, -6]], [0, 1, 3], [6, -19])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275014,"parameters":{"matrix":[[1,3,1],[1,-1,-6]],"vector":[0,1,3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[1,3,1],[1,-1,-6]],\"vector\":[0,1,3]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0015","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[2,-3,4],[2,-2,-1]],\"vector\":[1,2,-3]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[2, -3, 4], [2, -2, -1]], [1, 2, -3], [-16, 1])","explanation_zh":"Wave4根節點核對=([[2, -3, 4], [2, -2, -1]], [1, 2, -3], [-16, 1])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275015,"parameters":{"matrix":[[2,-3,4],[2,-2,-1]],"vector":[1,2,-3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[2,-3,4],[2,-2,-1]],\"vector\":[1,2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0016","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[3,-2,-4],[-2,-3,4]],\"vector\":[2,-2,-2]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[3, -2, -4], [-2, -3, 4]], [2, -2, -2], [18, -6])","explanation_zh":"Wave4根節點核對=([[3, -2, -4], [-2, -3, 4]], [2, -2, -2], [18, -6])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275016,"parameters":{"matrix":[[3,-2,-4],[-2,-3,4]],"vector":[2,-2,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[3,-2,-4],[-2,-3,4]],\"vector\":[2,-2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0017","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"vector\":[3,-1,-1]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[4, -1, -1], [-1, -4, -4]], [3, -1, -1], [14, 5])","explanation_zh":"Wave4根節點核對=([[4, -1, -1], [-1, -4, -4]], [3, -1, -1], [14, 5])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275017,"parameters":{"matrix":[[4,-1,-1],[-1,-4,-4]],"vector":[3,-1,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"vector\":[3,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0018","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[5,0,2],[0,-5,1]],\"vector\":[4,0,0]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[5, 0, 2], [0, -5, 1]], [4, 0, 0], [20, 0])","explanation_zh":"Wave4根節點核對=([[5, 0, 2], [0, -5, 1]], [4, 0, 0], [20, 0])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275018,"parameters":{"matrix":[[5,0,2],[0,-5,1]],"vector":[4,0,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[5,0,2],[0,-5,1]],\"vector\":[4,0,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0019","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[6,1,5],[1,-6,6]],\"vector\":[-4,1,1]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[6, 1, 5], [1, -6, 6]], [-4, 1, 1], [-18, -4])","explanation_zh":"Wave4根節點核對=([[6, 1, 5], [1, -6, 6]], [-4, 1, 1], [-18, -4])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275019,"parameters":{"matrix":[[6,1,5],[1,-6,6]],"vector":[-4,1,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[6,1,5],[1,-6,6]],\"vector\":[-4,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0020","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[7,2,-3],[2,-7,-2]],\"vector\":[-3,2,2]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[7, 2, -3], [2, -7, -2]], [-3, 2, 2], [-23, -24])","explanation_zh":"Wave4根節點核對=([[7, 2, -3], [2, -7, -2]], [-3, 2, 2], [-23, -24])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275020,"parameters":{"matrix":[[7,2,-3],[2,-7,-2]],"vector":[-3,2,2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[7,2,-3],[2,-7,-2]],\"vector\":[-3,2,2]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0021","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[8,3,0],[-2,-8,3]],\"vector\":[-2,-2,3]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[8, 3, 0], [-2, -8, 3]], [-2, -2, 3], [-22, 29])","explanation_zh":"Wave4根節點核對=([[8, 3, 0], [-2, -8, 3]], [-2, -2, 3], [-22, 29])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275021,"parameters":{"matrix":[[8,3,0],[-2,-8,3]],"vector":[-2,-2,3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[8,3,0],[-2,-8,3]],\"vector\":[-2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0022","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[9,-3,3],[-1,-9,-5]],\"vector\":[-1,-1,-3]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[9, -3, 3], [-1, -9, -5]], [-1, -1, -3], [-15, 25])","explanation_zh":"Wave4根節點核對=([[9, -3, 3], [-1, -9, -5]], [-1, -1, -3], [-15, 25])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275022,"parameters":{"matrix":[[9,-3,3],[-1,-9,-5]],"vector":[-1,-1,-3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[9,-3,3],[-1,-9,-5]],\"vector\":[-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0023","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[10,-2,-5],[0,-10,0]],\"vector\":[0,0,-2]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[10, -2, -5], [0, -10, 0]], [0, 0, -2], [10, 0])","explanation_zh":"Wave4根節點核對=([[10, -2, -5], [0, -10, 0]], [0, 0, -2], [10, 0])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275023,"parameters":{"matrix":[[10,-2,-5],[0,-10,0]],"vector":[0,0,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[10,-2,-5],[0,-10,0]],\"vector\":[0,0,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0024","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[11,-1,-2],[1,-11,5]],\"vector\":[1,1,-1]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[11, -1, -2], [1, -11, 5]], [1, 1, -1], [12, -15])","explanation_zh":"Wave4根節點核對=([[11, -1, -2], [1, -11, 5]], [1, 1, -1], [12, -15])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275024,"parameters":{"matrix":[[11,-1,-2],[1,-11,5]],"vector":[1,1,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[11,-1,-2],[1,-11,5]],\"vector\":[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0025","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-matrix-evaluate；資料={\"matrix\":[[12,0,1],[2,-12,-3]],\"vector\":[2,2,0]}。請依 [matrix,vector,image] 欄序回答。","answer":{"kind":"expression","value":"([[12, 0, 1], [2, -12, -3]], [2, 2, 0], [24, -20])","explanation_zh":"Wave4根節點核對=([[12, 0, 1], [2, -12, -3]], [2, 2, 0], [24, -20])；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-matrix-evaluate","seed":202610275025,"parameters":{"matrix":[[12,0,1],[2,-12,-3]],"vector":[2,2,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-matrix-evaluate","semantic_template":"qf-mod27-linearmap-matrix-evaluate/{\"matrix\":[[12,0,1],[2,-12,-3]],\"vector\":[2,2,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0026","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"bias\":[0,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-12, -3, -5], [-2, 12, -6]], [0, 0], [0, 0], 是, matrix_linear_certificate)","explanation_zh":"Wave4根節點核對=([[-12, -3, -5], [-2, 12, -6]], [0, 0], [0, 0], 是, matrix_linear_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275026,"parameters":{"matrix":[[-12,-3,-5],[-2,12,-6]],"bias":[0,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[0,0],\"matrix\":[[-12,-3,-5],[-2,12,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0027","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"bias\":[2,-2]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-11, -2, -2], [-1, 11, -1]], [2, -2], [2, -2], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-11, -2, -2], [-1, 11, -1]], [2, -2], [2, -2], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275027,"parameters":{"matrix":[[-11,-2,-2],[-1,11,-1]],"bias":[2,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[2,-2],\"matrix\":[[-11,-2,-2],[-1,11,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0028","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-10,-1,1],[0,10,4]],\"bias\":[3,-1]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-10, -1, 1], [0, 10, 4]], [3, -1], [3, -1], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-10, -1, 1], [0, 10, 4]], [3, -1], [3, -1], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275028,"parameters":{"matrix":[[-10,-1,1],[0,10,4]],"bias":[3,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[3,-1],\"matrix\":[[-10,-1,1],[0,10,4]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0029","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-9,0,4],[1,9,-4]],\"bias\":[4,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-9, 0, 4], [1, 9, -4]], [4, 0], [4, 0], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-9, 0, 4], [1, 9, -4]], [4, 0], [4, 0], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275029,"parameters":{"matrix":[[-9,0,4],[1,9,-4]],"bias":[4,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[4,0],\"matrix\":[[-9,0,4],[1,9,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0030","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-8,1,-4],[2,8,1]],\"bias\":[0,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-8, 1, -4], [2, 8, 1]], [0, 0], [0, 0], 是, matrix_linear_certificate)","explanation_zh":"Wave4根節點核對=([[-8, 1, -4], [2, 8, 1]], [0, 0], [0, 0], 是, matrix_linear_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275030,"parameters":{"matrix":[[-8,1,-4],[2,8,1]],"bias":[0,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[0,0],\"matrix\":[[-8,1,-4],[2,8,1]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0031","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-7,2,-1],[-2,7,6]],\"bias\":[1,2]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-7, 2, -1], [-2, 7, 6]], [1, 2], [1, 2], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-7, 2, -1], [-2, 7, 6]], [1, 2], [1, 2], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275031,"parameters":{"matrix":[[-7,2,-1],[-2,7,6]],"bias":[1,2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[1,2],\"matrix\":[[-7,2,-1],[-2,7,6]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0032","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-6,3,2],[-1,6,-2]],\"bias\":[2,3]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-6, 3, 2], [-1, 6, -2]], [2, 3], [2, 3], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-6, 3, 2], [-1, 6, -2]], [2, 3], [2, 3], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275032,"parameters":{"matrix":[[-6,3,2],[-1,6,-2]],"bias":[2,3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[2,3],\"matrix\":[[-6,3,2],[-1,6,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0033","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-5,-3,5],[0,5,3]],\"bias\":[3,-3]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-5, -3, 5], [0, 5, 3]], [3, -3], [3, -3], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-5, -3, 5], [0, 5, 3]], [3, -3], [3, -3], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275033,"parameters":{"matrix":[[-5,-3,5],[0,5,3]],"bias":[3,-3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[3,-3],\"matrix\":[[-5,-3,5],[0,5,3]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0034","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-4,-2,-3],[1,4,-5]],\"bias\":[0,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-4, -2, -3], [1, 4, -5]], [0, 0], [0, 0], 是, matrix_linear_certificate)","explanation_zh":"Wave4根節點核對=([[-4, -2, -3], [1, 4, -5]], [0, 0], [0, 0], 是, matrix_linear_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275034,"parameters":{"matrix":[[-4,-2,-3],[1,4,-5]],"bias":[0,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[0,0],\"matrix\":[[-4,-2,-3],[1,4,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0035","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-3,-1,0],[2,3,0]],\"bias\":[5,-1]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-3, -1, 0], [2, 3, 0]], [5, -1], [5, -1], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-3, -1, 0], [2, 3, 0]], [5, -1], [5, -1], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275035,"parameters":{"matrix":[[-3,-1,0],[2,3,0]],"bias":[5,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[5,-1],\"matrix\":[[-3,-1,0],[2,3,0]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0036","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-2,0,3],[-2,2,5]],\"bias\":[1,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-2, 0, 3], [-2, 2, 5]], [1, 0], [1, 0], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-2, 0, 3], [-2, 2, 5]], [1, 0], [1, 0], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275036,"parameters":{"matrix":[[-2,0,3],[-2,2,5]],"bias":[1,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[1,0],\"matrix\":[[-2,0,3],[-2,2,5]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0037","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[-1,1,-5],[-1,1,-3]],\"bias\":[2,1]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1, -5], [-1, 1, -3]], [2, 1], [2, 1], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[-1, 1, -5], [-1, 1, -3]], [2, 1], [2, 1], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275037,"parameters":{"matrix":[[-1,1,-5],[-1,1,-3]],"bias":[2,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[2,1],\"matrix\":[[-1,1,-5],[-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0038","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[0,2,-2],[0,0,2]],\"bias\":[0,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, -2], [0, 0, 2]], [0, 0], [0, 0], 是, matrix_linear_certificate)","explanation_zh":"Wave4根節點核對=([[0, 2, -2], [0, 0, 2]], [0, 0], [0, 0], 是, matrix_linear_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275038,"parameters":{"matrix":[[0,2,-2],[0,0,2]],"bias":[0,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[0,0],\"matrix\":[[0,2,-2],[0,0,2]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0039","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[1,3,1],[1,-1,-6]],\"bias\":[4,3]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[1, 3, 1], [1, -1, -6]], [4, 3], [4, 3], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[1, 3, 1], [1, -1, -6]], [4, 3], [4, 3], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275039,"parameters":{"matrix":[[1,3,1],[1,-1,-6]],"bias":[4,3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[4,3],\"matrix\":[[1,3,1],[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0040","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[2,-3,4],[2,-2,-1]],\"bias\":[5,-3]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[2, -3, 4], [2, -2, -1]], [5, -3], [5, -3], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[2, -3, 4], [2, -2, -1]], [5, -3], [5, -3], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275040,"parameters":{"matrix":[[2,-3,4],[2,-2,-1]],"bias":[5,-3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[5,-3],\"matrix\":[[2,-3,4],[2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0041","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[3,-2,-4],[-2,-3,4]],\"bias\":[1,-2]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[3, -2, -4], [-2, -3, 4]], [1, -2], [1, -2], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[3, -2, -4], [-2, -3, 4]], [1, -2], [1, -2], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275041,"parameters":{"matrix":[[3,-2,-4],[-2,-3,4]],"bias":[1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[1,-2],\"matrix\":[[3,-2,-4],[-2,-3,4]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0042","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"bias\":[0,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[4, -1, -1], [-1, -4, -4]], [0, 0], [0, 0], 是, matrix_linear_certificate)","explanation_zh":"Wave4根節點核對=([[4, -1, -1], [-1, -4, -4]], [0, 0], [0, 0], 是, matrix_linear_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275042,"parameters":{"matrix":[[4,-1,-1],[-1,-4,-4]],"bias":[0,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[0,0],\"matrix\":[[4,-1,-1],[-1,-4,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0043","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[5,0,2],[0,-5,1]],\"bias\":[3,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[5, 0, 2], [0, -5, 1]], [3, 0], [3, 0], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[5, 0, 2], [0, -5, 1]], [3, 0], [3, 0], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275043,"parameters":{"matrix":[[5,0,2],[0,-5,1]],"bias":[3,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[3,0],\"matrix\":[[5,0,2],[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0044","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[6,1,5],[1,-6,6]],\"bias\":[4,1]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[6, 1, 5], [1, -6, 6]], [4, 1], [4, 1], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[6, 1, 5], [1, -6, 6]], [4, 1], [4, 1], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275044,"parameters":{"matrix":[[6,1,5],[1,-6,6]],"bias":[4,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[4,1],\"matrix\":[[6,1,5],[1,-6,6]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0045","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[7,2,-3],[2,-7,-2]],\"bias\":[5,2]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[7, 2, -3], [2, -7, -2]], [5, 2], [5, 2], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[7, 2, -3], [2, -7, -2]], [5, 2], [5, 2], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275045,"parameters":{"matrix":[[7,2,-3],[2,-7,-2]],"bias":[5,2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[5,2],\"matrix\":[[7,2,-3],[2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0046","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[8,3,0],[-2,-8,3]],\"bias\":[0,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[8, 3, 0], [-2, -8, 3]], [0, 0], [0, 0], 是, matrix_linear_certificate)","explanation_zh":"Wave4根節點核對=([[8, 3, 0], [-2, -8, 3]], [0, 0], [0, 0], 是, matrix_linear_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275046,"parameters":{"matrix":[[8,3,0],[-2,-8,3]],"bias":[0,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[0,0],\"matrix\":[[8,3,0],[-2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0047","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[9,-3,3],[-1,-9,-5]],\"bias\":[2,-3]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[9, -3, 3], [-1, -9, -5]], [2, -3], [2, -3], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[9, -3, 3], [-1, -9, -5]], [2, -3], [2, -3], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275047,"parameters":{"matrix":[[9,-3,3],[-1,-9,-5]],"bias":[2,-3],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[2,-3],\"matrix\":[[9,-3,3],[-1,-9,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0048","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[10,-2,-5],[0,-10,0]],\"bias\":[3,-2]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[10, -2, -5], [0, -10, 0]], [3, -2], [3, -2], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[10, -2, -5], [0, -10, 0]], [3, -2], [3, -2], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275048,"parameters":{"matrix":[[10,-2,-5],[0,-10,0]],"bias":[3,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[3,-2],\"matrix\":[[10,-2,-5],[0,-10,0]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0049","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[11,-1,-2],[1,-11,5]],\"bias\":[4,-1]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[11, -1, -2], [1, -11, 5]], [4, -1], [4, -1], 否, nonzero_translation_is_not_linear)","explanation_zh":"Wave4根節點核對=([[11, -1, -2], [1, -11, 5]], [4, -1], [4, -1], 否, nonzero_translation_is_not_linear)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275049,"parameters":{"matrix":[[11,-1,-2],[1,-11,5]],"bias":[4,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[4,-1],\"matrix\":[[11,-1,-2],[1,-11,5]]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0050","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-affine-qualification；資料={\"matrix\":[[12,0,1],[2,-12,-3]],\"bias\":[0,0]}。請依 [matrix,bias,T_zero,linear,boundary] 欄序回答。","answer":{"kind":"expression","value":"([[12, 0, 1], [2, -12, -3]], [0, 0], [0, 0], 是, matrix_linear_certificate)","explanation_zh":"Wave4根節點核對=([[12, 0, 1], [2, -12, -3]], [0, 0], [0, 0], 是, matrix_linear_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-affine-qualification","seed":202610275050,"parameters":{"matrix":[[12,0,1],[2,-12,-3]],"bias":[0,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-affine-qualification","semantic_template":"qf-mod27-linearmap-affine-qualification/{\"bias\":[0,0],\"matrix\":[[12,0,1],[2,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0051","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"bias\":[0,0],\"u\":[1,-2,-3],\"v\":[-5,1,-2]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -2, -3], [-5, 1, -2], [0, 0], [76, 26], [76, 26], [0, 0], 是)","explanation_zh":"Wave4根節點核對=([1, -2, -3], [-5, 1, -2], [0, 0], [76, 26], [76, 26], [0, 0], 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275051,"parameters":{"matrix":[[-12,-3,-5],[-2,12,-6]],"bias":[0,0],"u":[1,-2,-3],"v":[-5,1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[0,0],\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"u\":[1,-2,-3],\"v\":[-5,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0052","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"bias\":[2,-2],\"u\":[1,-1,-2],\"v\":[-4,1,-1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -1, -2], [-4, 1, -1], [2, -2], [41, 4], [43, 2], [-2, 2], 否)","explanation_zh":"Wave4根節點核對=([1, -1, -2], [-4, 1, -1], [2, -2], [41, 4], [43, 2], [-2, 2], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275052,"parameters":{"matrix":[[-11,-2,-2],[-1,11,-1]],"bias":[2,-2],"u":[1,-1,-2],"v":[-4,1,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[2,-2],\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"u\":[1,-1,-2],\"v\":[-4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0053","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-10,-1,1],[0,10,4]],\"bias\":[3,-1],\"u\":[1,0,-1],\"v\":[-3,1,0]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, -1], [-3, 1, 0], [3, -1], [21, 5], [24, 4], [-3, 1], 否)","explanation_zh":"Wave4根節點核對=([1, 0, -1], [-3, 1, 0], [3, -1], [21, 5], [24, 4], [-3, 1], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275053,"parameters":{"matrix":[[-10,-1,1],[0,10,4]],"bias":[3,-1],"u":[1,0,-1],"v":[-3,1,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[3,-1],\"matrix\":[[-10,-1,1],[0,10,4]],\"u\":[1,0,-1],\"v\":[-3,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0054","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-9,0,4],[1,9,-4]],\"bias\":[4,0],\"u\":[1,1,0],\"v\":[-2,1,1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0], [-2, 1, 1], [4, 0], [17, 13], [21, 13], [-4, 0], 否)","explanation_zh":"Wave4根節點核對=([1, 1, 0], [-2, 1, 1], [4, 0], [17, 13], [21, 13], [-4, 0], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275054,"parameters":{"matrix":[[-9,0,4],[1,9,-4]],"bias":[4,0],"u":[1,1,0],"v":[-2,1,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[4,0],\"matrix\":[[-9,0,4],[1,9,-4]],\"u\":[1,1,0],\"v\":[-2,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0055","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-8,1,-4],[2,8,1]],\"bias\":[0,0],\"u\":[1,2,1],\"v\":[-1,1,-2]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 1], [-1, 1, -2], [0, 0], [7, 23], [7, 23], [0, 0], 是)","explanation_zh":"Wave4根節點核對=([1, 2, 1], [-1, 1, -2], [0, 0], [7, 23], [7, 23], [0, 0], 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275055,"parameters":{"matrix":[[-8,1,-4],[2,8,1]],"bias":[0,0],"u":[1,2,1],"v":[-1,1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[0,0],\"matrix\":[[-8,1,-4],[2,8,1]],\"u\":[1,2,1],\"v\":[-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0056","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-7,2,-1],[-2,7,6]],\"bias\":[1,2],\"u\":[1,-2,2],\"v\":[0,1,-1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -2, 2], [0, 1, -1], [1, 2], [-9, -1], [-8, 1], [-1, -2], 否)","explanation_zh":"Wave4根節點核對=([1, -2, 2], [0, 1, -1], [1, 2], [-9, -1], [-8, 1], [-1, -2], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275056,"parameters":{"matrix":[[-7,2,-1],[-2,7,6]],"bias":[1,2],"u":[1,-2,2],"v":[0,1,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[1,2],\"matrix\":[[-7,2,-1],[-2,7,6]],\"u\":[1,-2,2],\"v\":[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0057","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-6,3,2],[-1,6,-2]],\"bias\":[2,3],\"u\":[1,-1,3],\"v\":[1,1,0]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -1, 3], [1, 1, 0], [2, 3], [-4, -5], [-2, -2], [-2, -3], 否)","explanation_zh":"Wave4根節點核對=([1, -1, 3], [1, 1, 0], [2, 3], [-4, -5], [-2, -2], [-2, -3], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275057,"parameters":{"matrix":[[-6,3,2],[-1,6,-2]],"bias":[2,3],"u":[1,-1,3],"v":[1,1,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[2,3],\"matrix\":[[-6,3,2],[-1,6,-2]],\"u\":[1,-1,3],\"v\":[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0058","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-5,-3,5],[0,5,3]],\"bias\":[3,-3],\"u\":[1,0,-3],\"v\":[2,1,1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, -3], [2, 1, 1], [3, -3], [-25, -4], [-22, -7], [-3, 3], 否)","explanation_zh":"Wave4根節點核對=([1, 0, -3], [2, 1, 1], [3, -3], [-25, -4], [-22, -7], [-3, 3], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275058,"parameters":{"matrix":[[-5,-3,5],[0,5,3]],"bias":[3,-3],"u":[1,0,-3],"v":[2,1,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[3,-3],\"matrix\":[[-5,-3,5],[0,5,3]],\"u\":[1,0,-3],\"v\":[2,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0059","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-4,-2,-3],[1,4,-5]],\"bias\":[0,0],\"u\":[1,1,-2],\"v\":[3,1,-2]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, -2], [3, 1, -2], [0, 0], [-8, 32], [-8, 32], [0, 0], 是)","explanation_zh":"Wave4根節點核對=([1, 1, -2], [3, 1, -2], [0, 0], [-8, 32], [-8, 32], [0, 0], 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275059,"parameters":{"matrix":[[-4,-2,-3],[1,4,-5]],"bias":[0,0],"u":[1,1,-2],"v":[3,1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[0,0],\"matrix\":[[-4,-2,-3],[1,4,-5]],\"u\":[1,1,-2],\"v\":[3,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0060","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-3,-1,0],[2,3,0]],\"bias\":[5,-1],\"u\":[1,2,-1],\"v\":[4,1,-1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, -1], [4, 1, -1], [5, -1], [-13, 18], [-8, 17], [-5, 1], 否)","explanation_zh":"Wave4根節點核對=([1, 2, -1], [4, 1, -1], [5, -1], [-13, 18], [-8, 17], [-5, 1], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275060,"parameters":{"matrix":[[-3,-1,0],[2,3,0]],"bias":[5,-1],"u":[1,2,-1],"v":[4,1,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[5,-1],\"matrix\":[[-3,-1,0],[2,3,0]],\"u\":[1,2,-1],\"v\":[4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0061","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-2,0,3],[-2,2,5]],\"bias\":[1,0],\"u\":[1,-2,0],\"v\":[5,1,0]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -2, 0], [5, 1, 0], [1, 0], [-11, -14], [-10, -14], [-1, 0], 否)","explanation_zh":"Wave4根節點核對=([1, -2, 0], [5, 1, 0], [1, 0], [-11, -14], [-10, -14], [-1, 0], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275061,"parameters":{"matrix":[[-2,0,3],[-2,2,5]],"bias":[1,0],"u":[1,-2,0],"v":[5,1,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[1,0],\"matrix\":[[-2,0,3],[-2,2,5]],\"u\":[1,-2,0],\"v\":[5,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0062","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[-1,1,-5],[-1,1,-3]],\"bias\":[2,1],\"u\":[1,-1,1],\"v\":[-5,1,1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -1, 1], [-5, 1, 1], [2, 1], [-4, -1], [-2, 0], [-2, -1], 否)","explanation_zh":"Wave4根節點核對=([1, -1, 1], [-5, 1, 1], [2, 1], [-4, -1], [-2, 0], [-2, -1], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275062,"parameters":{"matrix":[[-1,1,-5],[-1,1,-3]],"bias":[2,1],"u":[1,-1,1],"v":[-5,1,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[2,1],\"matrix\":[[-1,1,-5],[-1,1,-3]],\"u\":[1,-1,1],\"v\":[-5,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0063","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[0,2,-2],[0,0,2]],\"bias\":[0,0],\"u\":[1,0,2],\"v\":[-4,1,-2]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 2], [-4, 1, -2], [0, 0], [2, 0], [2, 0], [0, 0], 是)","explanation_zh":"Wave4根節點核對=([1, 0, 2], [-4, 1, -2], [0, 0], [2, 0], [2, 0], [0, 0], 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275063,"parameters":{"matrix":[[0,2,-2],[0,0,2]],"bias":[0,0],"u":[1,0,2],"v":[-4,1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[0,0],\"matrix\":[[0,2,-2],[0,0,2]],\"u\":[1,0,2],\"v\":[-4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0064","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[1,3,1],[1,-1,-6]],\"bias\":[4,3],\"u\":[1,1,3],\"v\":[-3,1,-1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3], [-3, 1, -1], [4, 3], [10, -13], [14, -10], [-4, -3], 否)","explanation_zh":"Wave4根節點核對=([1, 1, 3], [-3, 1, -1], [4, 3], [10, -13], [14, -10], [-4, -3], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275064,"parameters":{"matrix":[[1,3,1],[1,-1,-6]],"bias":[4,3],"u":[1,1,3],"v":[-3,1,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[4,3],\"matrix\":[[1,3,1],[1,-1,-6]],\"u\":[1,1,3],\"v\":[-3,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0065","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[2,-3,4],[2,-2,-1]],\"bias\":[5,-3],\"u\":[1,2,-3],\"v\":[-2,1,0]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, -3], [-2, 1, 0], [5, -3], [-18, -8], [-13, -11], [-5, 3], 否)","explanation_zh":"Wave4根節點核對=([1, 2, -3], [-2, 1, 0], [5, -3], [-18, -8], [-13, -11], [-5, 3], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275065,"parameters":{"matrix":[[2,-3,4],[2,-2,-1]],"bias":[5,-3],"u":[1,2,-3],"v":[-2,1,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[5,-3],\"matrix\":[[2,-3,4],[2,-2,-1]],\"u\":[1,2,-3],\"v\":[-2,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0066","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[3,-2,-4],[-2,-3,4]],\"bias\":[1,-2],\"u\":[1,-2,-2],\"v\":[-1,1,1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -2, -2], [-1, 1, 1], [1, -2], [7, -3], [8, -5], [-1, 2], 否)","explanation_zh":"Wave4根節點核對=([1, -2, -2], [-1, 1, 1], [1, -2], [7, -3], [8, -5], [-1, 2], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275066,"parameters":{"matrix":[[3,-2,-4],[-2,-3,4]],"bias":[1,-2],"u":[1,-2,-2],"v":[-1,1,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[1,-2],\"matrix\":[[3,-2,-4],[-2,-3,4]],\"u\":[1,-2,-2],\"v\":[-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0067","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"bias\":[0,0],\"u\":[1,-1,-1],\"v\":[0,1,-2]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -1, -1], [0, 1, -2], [0, 0], [7, 11], [7, 11], [0, 0], 是)","explanation_zh":"Wave4根節點核對=([1, -1, -1], [0, 1, -2], [0, 0], [7, 11], [7, 11], [0, 0], 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275067,"parameters":{"matrix":[[4,-1,-1],[-1,-4,-4]],"bias":[0,0],"u":[1,-1,-1],"v":[0,1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[0,0],\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"u\":[1,-1,-1],\"v\":[0,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0068","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[5,0,2],[0,-5,1]],\"bias\":[3,0],\"u\":[1,0,0],\"v\":[1,1,-1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0], [1, 1, -1], [3, 0], [11, -6], [14, -6], [-3, 0], 否)","explanation_zh":"Wave4根節點核對=([1, 0, 0], [1, 1, -1], [3, 0], [11, -6], [14, -6], [-3, 0], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275068,"parameters":{"matrix":[[5,0,2],[0,-5,1]],"bias":[3,0],"u":[1,0,0],"v":[1,1,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[3,0],\"matrix\":[[5,0,2],[0,-5,1]],\"u\":[1,0,0],\"v\":[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0069","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[6,1,5],[1,-6,6]],\"bias\":[4,1],\"u\":[1,1,1],\"v\":[2,1,0]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1], [2, 1, 0], [4, 1], [29, -2], [33, -1], [-4, -1], 否)","explanation_zh":"Wave4根節點核對=([1, 1, 1], [2, 1, 0], [4, 1], [29, -2], [33, -1], [-4, -1], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275069,"parameters":{"matrix":[[6,1,5],[1,-6,6]],"bias":[4,1],"u":[1,1,1],"v":[2,1,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[4,1],\"matrix\":[[6,1,5],[1,-6,6]],\"u\":[1,1,1],\"v\":[2,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0070","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[7,2,-3],[2,-7,-2]],\"bias\":[5,2],\"u\":[1,2,2],\"v\":[3,1,1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2], [3, 1, 1], [5, 2], [30, -17], [35, -15], [-5, -2], 否)","explanation_zh":"Wave4根節點核對=([1, 2, 2], [3, 1, 1], [5, 2], [30, -17], [35, -15], [-5, -2], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275070,"parameters":{"matrix":[[7,2,-3],[2,-7,-2]],"bias":[5,2],"u":[1,2,2],"v":[3,1,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[5,2],\"matrix\":[[7,2,-3],[2,-7,-2]],\"u\":[1,2,2],\"v\":[3,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0071","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[8,3,0],[-2,-8,3]],\"bias\":[0,0],\"u\":[1,-2,3],\"v\":[4,1,-2]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -2, 3], [4, 1, -2], [0, 0], [37, 1], [37, 1], [0, 0], 是)","explanation_zh":"Wave4根節點核對=([1, -2, 3], [4, 1, -2], [0, 0], [37, 1], [37, 1], [0, 0], 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275071,"parameters":{"matrix":[[8,3,0],[-2,-8,3]],"bias":[0,0],"u":[1,-2,3],"v":[4,1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[0,0],\"matrix\":[[8,3,0],[-2,-8,3]],\"u\":[1,-2,3],\"v\":[4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0072","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[9,-3,3],[-1,-9,-5]],\"bias\":[2,-3],\"u\":[1,-1,-3],\"v\":[5,1,-1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, -1, -3], [5, 1, -1], [2, -3], [44, 11], [46, 8], [-2, 3], 否)","explanation_zh":"Wave4根節點核對=([1, -1, -3], [5, 1, -1], [2, -3], [44, 11], [46, 8], [-2, 3], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275072,"parameters":{"matrix":[[9,-3,3],[-1,-9,-5]],"bias":[2,-3],"u":[1,-1,-3],"v":[5,1,-1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[2,-3],\"matrix\":[[9,-3,3],[-1,-9,-5]],\"u\":[1,-1,-3],\"v\":[5,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0073","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[10,-2,-5],[0,-10,0]],\"bias\":[3,-2],\"u\":[1,0,-2],\"v\":[-5,1,0]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, -2], [-5, 1, 0], [3, -2], [-29, -12], [-26, -14], [-3, 2], 否)","explanation_zh":"Wave4根節點核對=([1, 0, -2], [-5, 1, 0], [3, -2], [-29, -12], [-26, -14], [-3, 2], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275073,"parameters":{"matrix":[[10,-2,-5],[0,-10,0]],"bias":[3,-2],"u":[1,0,-2],"v":[-5,1,0],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[3,-2],\"matrix\":[[10,-2,-5],[0,-10,0]],\"u\":[1,0,-2],\"v\":[-5,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0074","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[11,-1,-2],[1,-11,5]],\"bias\":[4,-1],\"u\":[1,1,-1],\"v\":[-4,1,1]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, -1], [-4, 1, 1], [4, -1], [-31, -26], [-27, -27], [-4, 1], 否)","explanation_zh":"Wave4根節點核對=([1, 1, -1], [-4, 1, 1], [4, -1], [-31, -26], [-27, -27], [-4, 1], 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275074,"parameters":{"matrix":[[11,-1,-2],[1,-11,5]],"bias":[4,-1],"u":[1,1,-1],"v":[-4,1,1],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[4,-1],\"matrix\":[[11,-1,-2],[1,-11,5]],\"u\":[1,1,-1],\"v\":[-4,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0075","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-additivity-witness；資料={\"matrix\":[[12,0,1],[2,-12,-3]],\"bias\":[0,0],\"u\":[1,2,0],\"v\":[-3,1,-2]}。請依 [u,v,bias,T_u_plus_v,T_u_plus_T_v,difference,additive] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 0], [-3, 1, -2], [0, 0], [-26, -34], [-26, -34], [0, 0], 是)","explanation_zh":"Wave4根節點核對=([1, 2, 0], [-3, 1, -2], [0, 0], [-26, -34], [-26, -34], [0, 0], 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-additivity-witness","seed":202610275075,"parameters":{"matrix":[[12,0,1],[2,-12,-3]],"bias":[0,0],"u":[1,2,0],"v":[-3,1,-2],"domain":"Exact_matrix_real_linear_map","operation":"linearmap-additivity-witness","semantic_template":"qf-mod27-linearmap-additivity-witness/{\"bias\":[0,0],\"matrix\":[[12,0,1],[2,-12,-3]],\"u\":[1,2,0],\"v\":[-3,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0076","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"u\":[1,-2,-3],\"v\":[-5,1,-2],\"alpha\":-2,\"beta\":-3}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-2, -3, [13, 1, 12], [-219, -86], [-219, -86], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-2, -3, [13, 1, 12], [-219, -86], [-219, -86], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275076,"parameters":{"matrix":[[-12,-3,-5],[-2,12,-6]],"u":[1,-2,-3],"v":[-5,1,-2],"alpha":-2,"beta":-3,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-2,\"beta\":-3,\"matrix\":[[-12,-3,-5],[-2,12,-6]],\"u\":[1,-2,-3],\"v\":[-5,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0077","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"u\":[1,-1,-2],\"v\":[-4,1,-1],\"alpha\":-1,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-1, -2, [7, -1, 4], [-83, -22], [-83, -22], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-1, -2, [7, -1, 4], [-83, -22], [-83, -22], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275077,"parameters":{"matrix":[[-11,-2,-2],[-1,11,-1]],"u":[1,-1,-2],"v":[-4,1,-1],"alpha":-1,"beta":-2,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-1,\"beta\":-2,\"matrix\":[[-11,-2,-2],[-1,11,-1]],\"u\":[1,-1,-2],\"v\":[-4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0078","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-10,-1,1],[0,10,4]],\"u\":[1,0,-1],\"v\":[-3,1,0],\"alpha\":0,\"beta\":-1}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, -1, [3, -1, 0], [-29, -10], [-29, -10], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(0, -1, [3, -1, 0], [-29, -10], [-29, -10], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275078,"parameters":{"matrix":[[-10,-1,1],[0,10,4]],"u":[1,0,-1],"v":[-3,1,0],"alpha":0,"beta":-1,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":0,\"beta\":-1,\"matrix\":[[-10,-1,1],[0,10,4]],\"u\":[1,0,-1],\"v\":[-3,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0079","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-9,0,4],[1,9,-4]],\"u\":[1,1,0],\"v\":[-2,1,1],\"alpha\":1,\"beta\":0}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [1, 1, 0], [-9, 10], [-9, 10], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(1, 0, [1, 1, 0], [-9, 10], [-9, 10], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275079,"parameters":{"matrix":[[-9,0,4],[1,9,-4]],"u":[1,1,0],"v":[-2,1,1],"alpha":1,"beta":0,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":1,\"beta\":0,\"matrix\":[[-9,0,4],[1,9,-4]],\"u\":[1,1,0],\"v\":[-2,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0080","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-8,1,-4],[2,8,1]],\"u\":[1,2,1],\"v\":[-1,1,-2],\"alpha\":2,\"beta\":1}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, [1, 5, 0], [-3, 42], [-3, 42], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(2, 1, [1, 5, 0], [-3, 42], [-3, 42], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275080,"parameters":{"matrix":[[-8,1,-4],[2,8,1]],"u":[1,2,1],"v":[-1,1,-2],"alpha":2,"beta":1,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":2,\"beta\":1,\"matrix\":[[-8,1,-4],[2,8,1]],\"u\":[1,2,1],\"v\":[-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0081","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-7,2,-1],[-2,7,6]],\"u\":[1,-2,2],\"v\":[0,1,-1],\"alpha\":-2,\"beta\":2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-2, 2, [-2, 6, -6], [32, 10], [32, 10], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-2, 2, [-2, 6, -6], [32, 10], [32, 10], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275081,"parameters":{"matrix":[[-7,2,-1],[-2,7,6]],"u":[1,-2,2],"v":[0,1,-1],"alpha":-2,"beta":2,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-2,\"beta\":2,\"matrix\":[[-7,2,-1],[-2,7,6]],\"u\":[1,-2,2],\"v\":[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0082","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-6,3,2],[-1,6,-2]],\"u\":[1,-1,3],\"v\":[1,1,0],\"alpha\":-1,\"beta\":3}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-1, 3, [2, 4, -3], [-6, 28], [-6, 28], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-1, 3, [2, 4, -3], [-6, 28], [-6, 28], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275082,"parameters":{"matrix":[[-6,3,2],[-1,6,-2]],"u":[1,-1,3],"v":[1,1,0],"alpha":-1,"beta":3,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-1,\"beta\":3,\"matrix\":[[-6,3,2],[-1,6,-2]],\"u\":[1,-1,3],\"v\":[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0083","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-5,-3,5],[0,5,3]],\"u\":[1,0,-3],\"v\":[2,1,1],\"alpha\":0,\"beta\":-3}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, -3, [-6, -3, -3], [24, -24], [24, -24], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(0, -3, [-6, -3, -3], [24, -24], [24, -24], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275083,"parameters":{"matrix":[[-5,-3,5],[0,5,3]],"u":[1,0,-3],"v":[2,1,1],"alpha":0,"beta":-3,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":0,\"beta\":-3,\"matrix\":[[-5,-3,5],[0,5,3]],\"u\":[1,0,-3],\"v\":[2,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0084","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-4,-2,-3],[1,4,-5]],\"u\":[1,1,-2],\"v\":[3,1,-2],\"alpha\":1,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, -2, [-5, -1, 2], [16, -19], [16, -19], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(1, -2, [-5, -1, 2], [16, -19], [16, -19], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275084,"parameters":{"matrix":[[-4,-2,-3],[1,4,-5]],"u":[1,1,-2],"v":[3,1,-2],"alpha":1,"beta":-2,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":1,\"beta\":-2,\"matrix\":[[-4,-2,-3],[1,4,-5]],\"u\":[1,1,-2],\"v\":[3,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0085","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-3,-1,0],[2,3,0]],\"u\":[1,2,-1],\"v\":[4,1,-1],\"alpha\":2,\"beta\":-1}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, -1, [-2, 3, -1], [3, 5], [3, 5], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(2, -1, [-2, 3, -1], [3, 5], [3, 5], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275085,"parameters":{"matrix":[[-3,-1,0],[2,3,0]],"u":[1,2,-1],"v":[4,1,-1],"alpha":2,"beta":-1,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":2,\"beta\":-1,\"matrix\":[[-3,-1,0],[2,3,0]],\"u\":[1,2,-1],\"v\":[4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0086","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-2,0,3],[-2,2,5]],\"u\":[1,-2,0],\"v\":[5,1,0],\"alpha\":-2,\"beta\":0}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-2, 0, [-2, 4, 0], [4, 12], [4, 12], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-2, 0, [-2, 4, 0], [4, 12], [4, 12], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275086,"parameters":{"matrix":[[-2,0,3],[-2,2,5]],"u":[1,-2,0],"v":[5,1,0],"alpha":-2,"beta":0,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-2,\"beta\":0,\"matrix\":[[-2,0,3],[-2,2,5]],\"u\":[1,-2,0],\"v\":[5,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0087","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-1,1,-5],[-1,1,-3]],\"u\":[1,-1,1],\"v\":[-5,1,1],\"alpha\":-1,\"beta\":1}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-1, 1, [-6, 2, 0], [8, 8], [8, 8], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-1, 1, [-6, 2, 0], [8, 8], [8, 8], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275087,"parameters":{"matrix":[[-1,1,-5],[-1,1,-3]],"u":[1,-1,1],"v":[-5,1,1],"alpha":-1,"beta":1,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-1,\"beta\":1,\"matrix\":[[-1,1,-5],[-1,1,-3]],\"u\":[1,-1,1],\"v\":[-5,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0088","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[0,2,-2],[0,0,2]],\"u\":[1,0,2],\"v\":[-4,1,-2],\"alpha\":0,\"beta\":2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 2, [-8, 2, -4], [12, -8], [12, -8], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(0, 2, [-8, 2, -4], [12, -8], [12, -8], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275088,"parameters":{"matrix":[[0,2,-2],[0,0,2]],"u":[1,0,2],"v":[-4,1,-2],"alpha":0,"beta":2,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":0,\"beta\":2,\"matrix\":[[0,2,-2],[0,0,2]],\"u\":[1,0,2],\"v\":[-4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0089","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[1,3,1],[1,-1,-6]],\"u\":[1,1,3],\"v\":[-3,1,-1],\"alpha\":1,\"beta\":3}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 3, [-8, 4, 0], [4, -12], [4, -12], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(1, 3, [-8, 4, 0], [4, -12], [4, -12], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275089,"parameters":{"matrix":[[1,3,1],[1,-1,-6]],"u":[1,1,3],"v":[-3,1,-1],"alpha":1,"beta":3,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":1,\"beta\":3,\"matrix\":[[1,3,1],[1,-1,-6]],\"u\":[1,1,3],\"v\":[-3,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0090","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[2,-3,4],[2,-2,-1]],\"u\":[1,2,-3],\"v\":[-2,1,0],\"alpha\":2,\"beta\":-3}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, -3, [8, 1, -6], [-11, 20], [-11, 20], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(2, -3, [8, 1, -6], [-11, 20], [-11, 20], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275090,"parameters":{"matrix":[[2,-3,4],[2,-2,-1]],"u":[1,2,-3],"v":[-2,1,0],"alpha":2,"beta":-3,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":2,\"beta\":-3,\"matrix\":[[2,-3,4],[2,-2,-1]],\"u\":[1,2,-3],\"v\":[-2,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0091","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[3,-2,-4],[-2,-3,4]],\"u\":[1,-2,-2],\"v\":[-1,1,1],\"alpha\":-2,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-2, -2, [0, 2, 2], [-12, 2], [-12, 2], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-2, -2, [0, 2, 2], [-12, 2], [-12, 2], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275091,"parameters":{"matrix":[[3,-2,-4],[-2,-3,4]],"u":[1,-2,-2],"v":[-1,1,1],"alpha":-2,"beta":-2,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-2,\"beta\":-2,\"matrix\":[[3,-2,-4],[-2,-3,4]],\"u\":[1,-2,-2],\"v\":[-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0092","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"u\":[1,-1,-1],\"v\":[0,1,-2],\"alpha\":-1,\"beta\":-1}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-1, -1, [-1, 0, 3], [-7, -11], [-7, -11], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-1, -1, [-1, 0, 3], [-7, -11], [-7, -11], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275092,"parameters":{"matrix":[[4,-1,-1],[-1,-4,-4]],"u":[1,-1,-1],"v":[0,1,-2],"alpha":-1,"beta":-1,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-1,\"beta\":-1,\"matrix\":[[4,-1,-1],[-1,-4,-4]],\"u\":[1,-1,-1],\"v\":[0,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0093","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[5,0,2],[0,-5,1]],\"u\":[1,0,0],\"v\":[1,1,-1],\"alpha\":0,\"beta\":0}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, [0, 0, 0], [0, 0], [0, 0], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(0, 0, [0, 0, 0], [0, 0], [0, 0], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275093,"parameters":{"matrix":[[5,0,2],[0,-5,1]],"u":[1,0,0],"v":[1,1,-1],"alpha":0,"beta":0,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":0,\"beta\":0,\"matrix\":[[5,0,2],[0,-5,1]],\"u\":[1,0,0],\"v\":[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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0094","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[6,1,5],[1,-6,6]],\"u\":[1,1,1],\"v\":[2,1,0],\"alpha\":1,\"beta\":1}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [3, 2, 1], [25, -3], [25, -3], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(1, 1, [3, 2, 1], [25, -3], [25, -3], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275094,"parameters":{"matrix":[[6,1,5],[1,-6,6]],"u":[1,1,1],"v":[2,1,0],"alpha":1,"beta":1,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":1,\"beta\":1,\"matrix\":[[6,1,5],[1,-6,6]],\"u\":[1,1,1],\"v\":[2,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0095","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[7,2,-3],[2,-7,-2]],\"u\":[1,2,2],\"v\":[3,1,1],\"alpha\":2,\"beta\":2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [8, 6, 6], [50, -38], [50, -38], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(2, 2, [8, 6, 6], [50, -38], [50, -38], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275095,"parameters":{"matrix":[[7,2,-3],[2,-7,-2]],"u":[1,2,2],"v":[3,1,1],"alpha":2,"beta":2,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":2,\"beta\":2,\"matrix\":[[7,2,-3],[2,-7,-2]],\"u\":[1,2,2],\"v\":[3,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0096","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[8,3,0],[-2,-8,3]],\"u\":[1,-2,3],\"v\":[4,1,-2],\"alpha\":-2,\"beta\":3}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-2, 3, [10, 7, -12], [101, -112], [101, -112], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-2, 3, [10, 7, -12], [101, -112], [101, -112], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275096,"parameters":{"matrix":[[8,3,0],[-2,-8,3]],"u":[1,-2,3],"v":[4,1,-2],"alpha":-2,"beta":3,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-2,\"beta\":3,\"matrix\":[[8,3,0],[-2,-8,3]],\"u\":[1,-2,3],\"v\":[4,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0097","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[9,-3,3],[-1,-9,-5]],\"u\":[1,-1,-3],\"v\":[5,1,-1],\"alpha\":-1,\"beta\":-3}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(-1, -3, [-16, -2, 6], [-120, 4], [-120, 4], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(-1, -3, [-16, -2, 6], [-120, 4], [-120, 4], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275097,"parameters":{"matrix":[[9,-3,3],[-1,-9,-5]],"u":[1,-1,-3],"v":[5,1,-1],"alpha":-1,"beta":-3,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":-1,\"beta\":-3,\"matrix\":[[9,-3,3],[-1,-9,-5]],\"u\":[1,-1,-3],\"v\":[5,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0098","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[10,-2,-5],[0,-10,0]],\"u\":[1,0,-2],\"v\":[-5,1,0],\"alpha\":0,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, -2, [10, -2, 0], [104, 20], [104, 20], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(0, -2, [10, -2, 0], [104, 20], [104, 20], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275098,"parameters":{"matrix":[[10,-2,-5],[0,-10,0]],"u":[1,0,-2],"v":[-5,1,0],"alpha":0,"beta":-2,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":0,\"beta\":-2,\"matrix\":[[10,-2,-5],[0,-10,0]],\"u\":[1,0,-2],\"v\":[-5,1,0]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0099","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[11,-1,-2],[1,-11,5]],\"u\":[1,1,-1],\"v\":[-4,1,1],\"alpha\":1,\"beta\":-1}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, -1, [5, 0, -2], [59, -5], [59, -5], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(1, -1, [5, 0, -2], [59, -5], [59, -5], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275099,"parameters":{"matrix":[[11,-1,-2],[1,-11,5]],"u":[1,1,-1],"v":[-4,1,1],"alpha":1,"beta":-1,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":1,\"beta\":-1,\"matrix\":[[11,-1,-2],[1,-11,5]],\"u\":[1,1,-1],\"v\":[-4,1,1]}"}},"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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261027-r05-general-linear-map-0100","batch_id":"batch-20261014-advance-20261027-r05-general-linear-map-foundations","language":"zh-Hant","source_mko_ids":["mko-general-linear-map"],"title_zh":"實線性映射","stem_zh":"來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[12,0,1],[2,-12,-3]],\"u\":[1,2,0],\"v\":[-3,1,-2],\"alpha\":2,\"beta\":0}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [2, 4, 0], [24, -44], [24, -44], 是, matrix_formula_universal_certificate)","explanation_zh":"Wave4根節點核對=(2, 0, [2, 4, 0], [24, -44], [24, -44], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實線性映射"]},"variant":{"family_id":"qf-mod27-linearmap-combination-image","seed":202610275100,"parameters":{"matrix":[[12,0,1],[2,-12,-3]],"u":[1,2,0],"v":[-3,1,-2],"alpha":2,"beta":0,"domain":"Exact_matrix_real_linear_map","operation":"linearmap-combination-image","semantic_template":"qf-mod27-linearmap-combination-image/{\"alpha\":2,\"beta\":0,\"matrix\":[[12,0,1],[2,-12,-3]],\"u\":[1,2,0],\"v\":[-3,1,-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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}