{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0001","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[0, 0], [0, 0]]、v=[0, 0]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([0, 0], [0, 0], [0, 0], [0, 0])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163001,"parameters":{"M":[[0,0],[0,0]],"v":[0,0],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[0,0],[0,0]],\"v\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0002","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 2], [3, 4]]、v=[0, 0]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([1, 3], [2, 4], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([1, 3], [2, 4], [0, 0], [0, 0])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163002,"parameters":{"M":[[1,2],[3,4]],"v":[0,0],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[1,2],[3,4]],\"v\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0003","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-10, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-10, 2], [1, -1], [-11, 3], [-11, 3])","explanation_zh":"有限結構核對=([-10, 2], [1, -1], [-11, 3], [-11, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163003,"parameters":{"M":[[-10,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-10,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0004","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-9, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-9, 2], [1, -1], [-10, 3], [-10, 3])","explanation_zh":"有限結構核對=([-9, 2], [1, -1], [-10, 3], [-10, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163004,"parameters":{"M":[[-9,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-9,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0005","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-8, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-8, 2], [1, -1], [-9, 3], [-9, 3])","explanation_zh":"有限結構核對=([-8, 2], [1, -1], [-9, 3], [-9, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163005,"parameters":{"M":[[-8,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-8,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0006","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-7, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-7, 2], [1, -1], [-8, 3], [-8, 3])","explanation_zh":"有限結構核對=([-7, 2], [1, -1], [-8, 3], [-8, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163006,"parameters":{"M":[[-7,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-7,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0007","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-6, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-6, 2], [1, -1], [-7, 3], [-7, 3])","explanation_zh":"有限結構核對=([-6, 2], [1, -1], [-7, 3], [-7, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163007,"parameters":{"M":[[-6,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-6,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0008","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-5, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-5, 2], [1, -1], [-6, 3], [-6, 3])","explanation_zh":"有限結構核對=([-5, 2], [1, -1], [-6, 3], [-6, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163008,"parameters":{"M":[[-5,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-5,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0009","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-4, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-4, 2], [1, -1], [-5, 3], [-5, 3])","explanation_zh":"有限結構核對=([-4, 2], [1, -1], [-5, 3], [-5, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163009,"parameters":{"M":[[-4,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-4,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0010","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-3, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-3, 2], [1, -1], [-4, 3], [-4, 3])","explanation_zh":"有限結構核對=([-3, 2], [1, -1], [-4, 3], [-4, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163010,"parameters":{"M":[[-3,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-3,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0011","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-2, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-2, 2], [1, -1], [-3, 3], [-3, 3])","explanation_zh":"有限結構核對=([-2, 2], [1, -1], [-3, 3], [-3, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163011,"parameters":{"M":[[-2,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-2,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0012","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-1, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([-1, 2], [1, -1], [-2, 3], [-2, 3])","explanation_zh":"有限結構核對=([-1, 2], [1, -1], [-2, 3], [-2, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163012,"parameters":{"M":[[-1,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[-1,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0013","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[0, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([0, 2], [1, -1], [-1, 3], [-1, 3])","explanation_zh":"有限結構核對=([0, 2], [1, -1], [-1, 3], [-1, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163013,"parameters":{"M":[[0,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[0,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0014","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([1, 2], [1, -1], [0, 3], [0, 3])","explanation_zh":"有限結構核對=([1, 2], [1, -1], [0, 3], [0, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163014,"parameters":{"M":[[1,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[1,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0015","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[2, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([2, 2], [1, -1], [1, 3], [1, 3])","explanation_zh":"有限結構核對=([2, 2], [1, -1], [1, 3], [1, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163015,"parameters":{"M":[[2,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[2,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0016","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[3, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([3, 2], [1, -1], [2, 3], [2, 3])","explanation_zh":"有限結構核對=([3, 2], [1, -1], [2, 3], [2, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163016,"parameters":{"M":[[3,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[3,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0017","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[4, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([4, 2], [1, -1], [3, 3], [3, 3])","explanation_zh":"有限結構核對=([4, 2], [1, -1], [3, 3], [3, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163017,"parameters":{"M":[[4,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[4,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0018","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[5, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([5, 2], [1, -1], [4, 3], [4, 3])","explanation_zh":"有限結構核對=([5, 2], [1, -1], [4, 3], [4, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163018,"parameters":{"M":[[5,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[5,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0019","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[6, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([6, 2], [1, -1], [5, 3], [5, 3])","explanation_zh":"有限結構核對=([6, 2], [1, -1], [5, 3], [5, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163019,"parameters":{"M":[[6,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[6,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0020","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[7, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([7, 2], [1, -1], [6, 3], [6, 3])","explanation_zh":"有限結構核對=([7, 2], [1, -1], [6, 3], [6, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163020,"parameters":{"M":[[7,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[7,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0021","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[8, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([8, 2], [1, -1], [7, 3], [7, 3])","explanation_zh":"有限結構核對=([8, 2], [1, -1], [7, 3], [7, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163021,"parameters":{"M":[[8,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[8,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0022","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[9, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([9, 2], [1, -1], [8, 3], [8, 3])","explanation_zh":"有限結構核對=([9, 2], [1, -1], [8, 3], [8, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163022,"parameters":{"M":[[9,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[9,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0023","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[10, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([10, 2], [1, -1], [9, 3], [9, 3])","explanation_zh":"有限結構核對=([10, 2], [1, -1], [9, 3], [9, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163023,"parameters":{"M":[[10,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[10,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0024","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[11, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([11, 2], [1, -1], [10, 3], [10, 3])","explanation_zh":"有限結構核對=([11, 2], [1, -1], [10, 3], [10, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163024,"parameters":{"M":[[11,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[11,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0025","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"基底像與列向量重建","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[12, 1], [2, -1]]、v=[1, -1]。依序求 (第一列向量M*e1, 第二列向量M*e2, Mv, v1*column1+v2*column2)。","answer":{"kind":"expression","value":"([12, 2], [1, -1], [11, 3], [11, 3])","explanation_zh":"有限結構核對=([12, 2], [1, -1], [11, 3], [11, 3])。讀出兩個基底像，再以給定分量組合；另按行乘列求Mv。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基底像與列向量重建"]},"variant":{"family_id":"qf-density16-linear-basis-column-reconstruction","seed":202610163025,"parameters":{"M":[[12,1],[2,-1]],"v":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-basis-column-reconstruction","semantic_template":"qf-density16-linear-basis-column-reconstruction/{\"M\":[[12,1],[2,-1]],\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0026","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[0, 0], [0, 0]]、u=[0, 0]、v=[0, 0]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([0, 0], [0, 0], [0, 0], [0, 0], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163026,"parameters":{"M":[[0,0],[0,0]],"u":[0,0],"v":[0,0],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[0,0],[0,0]],\"u\":[0,0],\"v\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0027","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 1]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([1, 0], [0, 1], [1, 1], [1, 1], [0, 0])","explanation_zh":"有限結構核對=([1, 0], [0, 1], [1, 1], [1, 1], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163027,"parameters":{"M":[[1,0],[0,1]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[1,0],[0,1]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0028","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-10, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-10, 0], [1, 2], [-9, 2], [-9, 2], [0, 0])","explanation_zh":"有限結構核對=([-10, 0], [1, 2], [-9, 2], [-9, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163028,"parameters":{"M":[[-10,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-10,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0029","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-9, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-9, 0], [1, 2], [-8, 2], [-8, 2], [0, 0])","explanation_zh":"有限結構核對=([-9, 0], [1, 2], [-8, 2], [-8, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163029,"parameters":{"M":[[-9,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-9,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0030","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-8, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-8, 0], [1, 2], [-7, 2], [-7, 2], [0, 0])","explanation_zh":"有限結構核對=([-8, 0], [1, 2], [-7, 2], [-7, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163030,"parameters":{"M":[[-8,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-8,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0031","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-7, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-7, 0], [1, 2], [-6, 2], [-6, 2], [0, 0])","explanation_zh":"有限結構核對=([-7, 0], [1, 2], [-6, 2], [-6, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163031,"parameters":{"M":[[-7,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-7,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0032","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-6, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-6, 0], [1, 2], [-5, 2], [-5, 2], [0, 0])","explanation_zh":"有限結構核對=([-6, 0], [1, 2], [-5, 2], [-5, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163032,"parameters":{"M":[[-6,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-6,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0033","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-5, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-5, 0], [1, 2], [-4, 2], [-4, 2], [0, 0])","explanation_zh":"有限結構核對=([-5, 0], [1, 2], [-4, 2], [-4, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163033,"parameters":{"M":[[-5,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-5,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0034","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-4, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-4, 0], [1, 2], [-3, 2], [-3, 2], [0, 0])","explanation_zh":"有限結構核對=([-4, 0], [1, 2], [-3, 2], [-3, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163034,"parameters":{"M":[[-4,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-4,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0035","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-3, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-3, 0], [1, 2], [-2, 2], [-2, 2], [0, 0])","explanation_zh":"有限結構核對=([-3, 0], [1, 2], [-2, 2], [-2, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163035,"parameters":{"M":[[-3,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-3,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0036","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-2, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-2, 0], [1, 2], [-1, 2], [-1, 2], [0, 0])","explanation_zh":"有限結構核對=([-2, 0], [1, 2], [-1, 2], [-1, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163036,"parameters":{"M":[[-2,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-2,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0037","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[-1, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([-1, 0], [1, 2], [0, 2], [0, 2], [0, 0])","explanation_zh":"有限結構核對=([-1, 0], [1, 2], [0, 2], [0, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163037,"parameters":{"M":[[-1,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[-1,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0038","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[0, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([0, 0], [1, 2], [1, 2], [1, 2], [0, 0])","explanation_zh":"有限結構核對=([0, 0], [1, 2], [1, 2], [1, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163038,"parameters":{"M":[[0,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[0,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0039","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([1, 0], [1, 2], [2, 2], [2, 2], [0, 0])","explanation_zh":"有限結構核對=([1, 0], [1, 2], [2, 2], [2, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163039,"parameters":{"M":[[1,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[1,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0040","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[2, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([2, 0], [1, 2], [3, 2], [3, 2], [0, 0])","explanation_zh":"有限結構核對=([2, 0], [1, 2], [3, 2], [3, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163040,"parameters":{"M":[[2,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[2,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0041","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[3, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([3, 0], [1, 2], [4, 2], [4, 2], [0, 0])","explanation_zh":"有限結構核對=([3, 0], [1, 2], [4, 2], [4, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163041,"parameters":{"M":[[3,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[3,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0042","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[4, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([4, 0], [1, 2], [5, 2], [5, 2], [0, 0])","explanation_zh":"有限結構核對=([4, 0], [1, 2], [5, 2], [5, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163042,"parameters":{"M":[[4,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[4,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0043","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[5, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([5, 0], [1, 2], [6, 2], [6, 2], [0, 0])","explanation_zh":"有限結構核對=([5, 0], [1, 2], [6, 2], [6, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163043,"parameters":{"M":[[5,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[5,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0044","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[6, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([6, 0], [1, 2], [7, 2], [7, 2], [0, 0])","explanation_zh":"有限結構核對=([6, 0], [1, 2], [7, 2], [7, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163044,"parameters":{"M":[[6,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[6,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0045","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[7, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([7, 0], [1, 2], [8, 2], [8, 2], [0, 0])","explanation_zh":"有限結構核對=([7, 0], [1, 2], [8, 2], [8, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163045,"parameters":{"M":[[7,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[7,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0046","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[8, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([8, 0], [1, 2], [9, 2], [9, 2], [0, 0])","explanation_zh":"有限結構核對=([8, 0], [1, 2], [9, 2], [9, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163046,"parameters":{"M":[[8,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[8,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0047","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[9, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([9, 0], [1, 2], [10, 2], [10, 2], [0, 0])","explanation_zh":"有限結構核對=([9, 0], [1, 2], [10, 2], [10, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163047,"parameters":{"M":[[9,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[9,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0048","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[10, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([10, 0], [1, 2], [11, 2], [11, 2], [0, 0])","explanation_zh":"有限結構核對=([10, 0], [1, 2], [11, 2], [11, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163048,"parameters":{"M":[[10,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[10,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0049","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[11, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([11, 0], [1, 2], [12, 2], [12, 2], [0, 0])","explanation_zh":"有限結構核對=([11, 0], [1, 2], [12, 2], [12, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163049,"parameters":{"M":[[11,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[11,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0050","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的加法兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[12, 1], [0, 2]]、u=[1, 0]、v=[0, 1]。求 (Mu, Mv, M(u+v), Mu+Mv, 後兩側之差)。","answer":{"kind":"expression","value":"([12, 0], [1, 2], [13, 2], [13, 2], [0, 0])","explanation_zh":"有限結構核對=([12, 0], [1, 2], [13, 2], [13, 2], [0, 0])。先後兩種求值都逐分量計算，核明示矩陣模型的實際殘差。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的加法兩側值"]},"variant":{"family_id":"qf-density16-linear-additivity-values","seed":202610163050,"parameters":{"M":[[12,1],[0,2]],"u":[1,0],"v":[0,1],"domain":"R2_column_integer_slice","operation":"linear-additivity-values","semantic_template":"qf-density16-linear-additivity-values/{\"M\":[[12,1],[0,2]],\"u\":[1,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0051","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[0, 0], [0, 0]]、v=[0, 0]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([0, 0], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163051,"parameters":{"M":[[0,0],[0,0]],"v":[0,0],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[0,0],[0,0]],\"c\":0,\"v\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0052","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 1]]、v=[1, -1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([1, -1], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([1, -1], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163052,"parameters":{"M":[[1,0],[0,1]],"v":[1,-1],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,1]],\"c\":0,\"v\":[1,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0053","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-10, 1]、c=1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-10, 2], [-10, 2], [-10, 2], [0, 0])","explanation_zh":"有限結構核對=([-10, 2], [-10, 2], [-10, 2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163053,"parameters":{"M":[[1,0],[0,2]],"v":[-10,1],"c":1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[-10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0054","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-9, 1]、c=-1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-9, 2], [9, -2], [9, -2], [0, 0])","explanation_zh":"有限結構核對=([-9, 2], [9, -2], [9, -2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163054,"parameters":{"M":[[1,0],[0,2]],"v":[-9,1],"c":-1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[-9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0055","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-8, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-8, 2], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([-8, 2], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163055,"parameters":{"M":[[1,0],[0,2]],"v":[-8,1],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0056","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-7, 1]、c=1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-7, 2], [-7, 2], [-7, 2], [0, 0])","explanation_zh":"有限結構核對=([-7, 2], [-7, 2], [-7, 2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163056,"parameters":{"M":[[1,0],[0,2]],"v":[-7,1],"c":1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[-7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0057","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-6, 1]、c=-1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-6, 2], [6, -2], [6, -2], [0, 0])","explanation_zh":"有限結構核對=([-6, 2], [6, -2], [6, -2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163057,"parameters":{"M":[[1,0],[0,2]],"v":[-6,1],"c":-1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[-6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0058","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-5, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-5, 2], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([-5, 2], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163058,"parameters":{"M":[[1,0],[0,2]],"v":[-5,1],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0059","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-4, 1]、c=1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-4, 2], [-4, 2], [-4, 2], [0, 0])","explanation_zh":"有限結構核對=([-4, 2], [-4, 2], [-4, 2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163059,"parameters":{"M":[[1,0],[0,2]],"v":[-4,1],"c":1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0060","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-3, 1]、c=-1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-3, 2], [3, -2], [3, -2], [0, 0])","explanation_zh":"有限結構核對=([-3, 2], [3, -2], [3, -2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163060,"parameters":{"M":[[1,0],[0,2]],"v":[-3,1],"c":-1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0061","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-2, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-2, 2], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([-2, 2], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163061,"parameters":{"M":[[1,0],[0,2]],"v":[-2,1],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0062","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-1, 1]、c=1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([-1, 2], [-1, 2], [-1, 2], [0, 0])","explanation_zh":"有限結構核對=([-1, 2], [-1, 2], [-1, 2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163062,"parameters":{"M":[[1,0],[0,2]],"v":[-1,1],"c":1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0063","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[0, 1]、c=-1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([0, 2], [0, -2], [0, -2], [0, 0])","explanation_zh":"有限結構核對=([0, 2], [0, -2], [0, -2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163063,"parameters":{"M":[[1,0],[0,2]],"v":[0,1],"c":-1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0064","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[1, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([1, 2], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([1, 2], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163064,"parameters":{"M":[[1,0],[0,2]],"v":[1,1],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0065","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[2, 1]、c=1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([2, 2], [2, 2], [2, 2], [0, 0])","explanation_zh":"有限結構核對=([2, 2], [2, 2], [2, 2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163065,"parameters":{"M":[[1,0],[0,2]],"v":[2,1],"c":1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0066","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[3, 1]、c=-1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([3, 2], [-3, -2], [-3, -2], [0, 0])","explanation_zh":"有限結構核對=([3, 2], [-3, -2], [-3, -2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163066,"parameters":{"M":[[1,0],[0,2]],"v":[3,1],"c":-1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0067","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[4, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([4, 2], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([4, 2], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163067,"parameters":{"M":[[1,0],[0,2]],"v":[4,1],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0068","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[5, 1]、c=1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([5, 2], [5, 2], [5, 2], [0, 0])","explanation_zh":"有限結構核對=([5, 2], [5, 2], [5, 2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163068,"parameters":{"M":[[1,0],[0,2]],"v":[5,1],"c":1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0069","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[6, 1]、c=-1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([6, 2], [-6, -2], [-6, -2], [0, 0])","explanation_zh":"有限結構核對=([6, 2], [-6, -2], [-6, -2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163069,"parameters":{"M":[[1,0],[0,2]],"v":[6,1],"c":-1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0070","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[7, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([7, 2], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([7, 2], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163070,"parameters":{"M":[[1,0],[0,2]],"v":[7,1],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0071","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[8, 1]、c=1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([8, 2], [8, 2], [8, 2], [0, 0])","explanation_zh":"有限結構核對=([8, 2], [8, 2], [8, 2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163071,"parameters":{"M":[[1,0],[0,2]],"v":[8,1],"c":1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0072","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[9, 1]、c=-1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([9, 2], [-9, -2], [-9, -2], [0, 0])","explanation_zh":"有限結構核對=([9, 2], [-9, -2], [-9, -2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163072,"parameters":{"M":[[1,0],[0,2]],"v":[9,1],"c":-1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0073","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[10, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([10, 2], [0, 0], [0, 0], [0, 0])","explanation_zh":"有限結構核對=([10, 2], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163073,"parameters":{"M":[[1,0],[0,2]],"v":[10,1],"c":0,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0074","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[11, 1]、c=1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([11, 2], [11, 2], [11, 2], [0, 0])","explanation_zh":"有限結構核對=([11, 2], [11, 2], [11, 2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163074,"parameters":{"M":[[1,0],[0,2]],"v":[11,1],"c":1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":1,\"v\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0075","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣映射的純量兩側值","stem_zh":"R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[12, 1]、c=-1。求 (Mv, M(cv), cMv, 後兩側之差)。","answer":{"kind":"expression","value":"([12, 2], [-12, -2], [-12, -2], [0, 0])","explanation_zh":"有限結構核對=([12, 2], [-12, -2], [-12, -2], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣映射的純量兩側值"]},"variant":{"family_id":"qf-density16-linear-scalar-values","seed":202610163075,"parameters":{"M":[[1,0],[0,2]],"v":[12,1],"c":-1,"domain":"R2_column_integer_slice","operation":"linear-scalar-values","semantic_template":"qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":-1,\"v\":[12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0076","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[0, 0], [0, 0]]、v=[0, 0]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], 是)","explanation_zh":"有限結構核對=([0, 0], [0, 0], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163076,"parameters":{"M":[[0,0],[0,0]],"v":[0,0],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[0,0],[0,0]],\"b\":[0,0],\"v\":[0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0077","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[0, 0], [0, 0]]、v=[99, 1]、b=[1, -1]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([0, 0], [1, -1], [1, -1], 否)","explanation_zh":"有限結構核對=([0, 0], [1, -1], [1, -1], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163077,"parameters":{"M":[[0,0],[0,0]],"v":[99,1],"b":[1,-1],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[0,0],[0,0]],\"b\":[1,-1],\"v\":[99,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0078","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-10, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-10, 2], [-10, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([-10, 2], [-10, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163078,"parameters":{"M":[[1,0],[0,2]],"v":[-10,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0079","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-9, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-9, 2], [-8, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([-9, 2], [-8, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163079,"parameters":{"M":[[1,0],[0,2]],"v":[-9,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0080","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-8, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-8, 2], [-8, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([-8, 2], [-8, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163080,"parameters":{"M":[[1,0],[0,2]],"v":[-8,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0081","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-7, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-7, 2], [-6, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([-7, 2], [-6, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163081,"parameters":{"M":[[1,0],[0,2]],"v":[-7,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0082","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-6, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-6, 2], [-6, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([-6, 2], [-6, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163082,"parameters":{"M":[[1,0],[0,2]],"v":[-6,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0083","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-5, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-5, 2], [-4, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([-5, 2], [-4, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163083,"parameters":{"M":[[1,0],[0,2]],"v":[-5,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0084","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-4, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-4, 2], [-4, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([-4, 2], [-4, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163084,"parameters":{"M":[[1,0],[0,2]],"v":[-4,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0085","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-3, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-3, 2], [-2, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([-3, 2], [-2, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163085,"parameters":{"M":[[1,0],[0,2]],"v":[-3,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0086","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-2, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-2, 2], [-2, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([-2, 2], [-2, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163086,"parameters":{"M":[[1,0],[0,2]],"v":[-2,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0087","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[-1, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([-1, 2], [0, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([-1, 2], [0, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163087,"parameters":{"M":[[1,0],[0,2]],"v":[-1,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0088","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[0, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([0, 2], [0, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([0, 2], [0, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163088,"parameters":{"M":[[1,0],[0,2]],"v":[0,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0089","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[1, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([1, 2], [2, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([1, 2], [2, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163089,"parameters":{"M":[[1,0],[0,2]],"v":[1,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0090","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[2, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([2, 2], [2, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([2, 2], [2, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163090,"parameters":{"M":[[1,0],[0,2]],"v":[2,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0091","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[3, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([3, 2], [4, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([3, 2], [4, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163091,"parameters":{"M":[[1,0],[0,2]],"v":[3,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0092","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[4, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([4, 2], [4, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([4, 2], [4, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163092,"parameters":{"M":[[1,0],[0,2]],"v":[4,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0093","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[5, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([5, 2], [6, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([5, 2], [6, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163093,"parameters":{"M":[[1,0],[0,2]],"v":[5,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0094","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[6, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([6, 2], [6, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([6, 2], [6, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163094,"parameters":{"M":[[1,0],[0,2]],"v":[6,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0095","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[7, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([7, 2], [8, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([7, 2], [8, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163095,"parameters":{"M":[[1,0],[0,2]],"v":[7,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0096","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[8, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([8, 2], [8, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([8, 2], [8, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163096,"parameters":{"M":[[1,0],[0,2]],"v":[8,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0097","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[9, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([9, 2], [10, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([9, 2], [10, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163097,"parameters":{"M":[[1,0],[0,2]],"v":[9,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0098","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[10, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([10, 2], [10, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([10, 2], [10, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163098,"parameters":{"M":[[1,0],[0,2]],"v":[10,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0099","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[11, 1]、b=[1, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([11, 2], [12, 2], [1, 0], 否)","explanation_zh":"有限結構核對=([11, 2], [12, 2], [1, 0], 否)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163099,"parameters":{"M":[[1,0],[0,2]],"v":[11,1],"b":[1,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[1,0],\"v\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261016-r03-linear-transformation-0100","batch_id":"batch-20261012-advance-20261016-r03-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射模型的零點阻擋","stem_zh":"R²到R²列向量，明示2×2矩陣模型。宣告全R²的F(v)=Mv+b，M=[[1, 0], [0, 2]]、v=[12, 1]、b=[0, 0]。求 (Mv, Mv+b, F(0)=b, 此明示仿射模型為線性資格)。","answer":{"kind":"expression","value":"([12, 2], [12, 2], [0, 0], 是)","explanation_zh":"有限結構核對=([12, 2], [12, 2], [0, 0], 是)。零點像就是b；依此已宣告的矩陣仿射模型核b是否兩分量皆零。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射模型的零點阻擋"]},"variant":{"family_id":"qf-density16-linear-affine-zero-obstruction","seed":202610163100,"parameters":{"M":[[1,0],[0,2]],"v":[12,1],"b":[0,0],"domain":"R2_column_integer_slice","operation":"linear-affine-zero-obstruction","semantic_template":"qf-density16-linear-affine-zero-obstruction/{\"M\":[[1,0],[0,2]],\"b\":[0,0],\"v\":[12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}