{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0001","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 0], [0, 0]]，T(x)=Mx。求 T(v)，v=[0, 0]。","answer":{"kind":"expression","value":"[0, 0]","explanation_zh":"Mv=[0, 0]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072001,"parameters":{"M":[[0,0],[0,0]],"v":[0,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[\"zero_input\"],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0002","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[1, 2], [2, 4]]，T(x)=Mx。求 T(v)，v=[1, 0]。","answer":{"kind":"expression","value":"[1, 2]","explanation_zh":"Mv=[1, 2]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072002,"parameters":{"M":[[1,2],[2,4]],"v":[1,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"1\",null],[\"2\",null]],[\"1\",\"0\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0003","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 1], [1, 0]]，T(x)=Mx。求 T(v)，v=[-2, 1]。","answer":{"kind":"expression","value":"[1, -2]","explanation_zh":"Mv=[1, -2]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072003,"parameters":{"M":[[0,1],[1,0]],"v":[-2,1],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],[\"-2\",\"1\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0004","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-9, 4], [-2, 4]]，T(x)=Mx。求 T(v)，v=[0, 3]。","answer":{"kind":"expression","value":"[12, 12]","explanation_zh":"Mv=[12, 12]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072004,"parameters":{"M":[[-9,4],[-2,4]],"v":[0,3],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"4\"],[null,\"4\"]],[\"0\",\"3\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0005","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-8, 1], [-2, 5]]，T(x)=Mx。求 T(v)，v=[4, 0]。","answer":{"kind":"expression","value":"[-32, -8]","explanation_zh":"Mv=[-32, -8]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072005,"parameters":{"M":[[-8,1],[-2,5]],"v":[4,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-8\",null],[\"-2\",null]],[\"4\",\"0\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0006","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-7, 2], [-2, 6]]，T(x)=Mx。求 T(v)，v=[-5, 1]。","answer":{"kind":"expression","value":"[37, 16]","explanation_zh":"Mv=[37, 16]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072006,"parameters":{"M":[[-7,2],[-2,6]],"v":[-5,1],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],[\"-5\",\"1\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0007","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-6, 3], [-2, 7]]，T(x)=Mx。求 T(v)，v=[0, 6]。","answer":{"kind":"expression","value":"[18, 42]","explanation_zh":"Mv=[18, 42]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072007,"parameters":{"M":[[-6,3],[-2,7]],"v":[0,6],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"3\"],[null,\"7\"]],[\"0\",\"6\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0008","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-5, 4], [-2, 8]]，T(x)=Mx。求 T(v)，v=[7, 0]。","answer":{"kind":"expression","value":"[-35, -14]","explanation_zh":"Mv=[-35, -14]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072008,"parameters":{"M":[[-5,4],[-2,8]],"v":[7,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-5\",null],[\"-2\",null]],[\"7\",\"0\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0009","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-4, 1], [-2, 9]]，T(x)=Mx。求 T(v)，v=[-8, 1]。","answer":{"kind":"expression","value":"[33, 25]","explanation_zh":"Mv=[33, 25]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072009,"parameters":{"M":[[-4,1],[-2,9]],"v":[-8,1],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],[\"-8\",\"1\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0010","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-3, 2], [-2, 10]]，T(x)=Mx。求 T(v)，v=[0, 9]。","answer":{"kind":"expression","value":"[18, 90]","explanation_zh":"Mv=[18, 90]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072010,"parameters":{"M":[[-3,2],[-2,10]],"v":[0,9],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"2\"],[null,\"10\"]],[\"0\",\"9\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0011","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-2, 3], [-2, 11]]，T(x)=Mx。求 T(v)，v=[10, 0]。","answer":{"kind":"expression","value":"[-20, -20]","explanation_zh":"Mv=[-20, -20]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072011,"parameters":{"M":[[-2,3],[-2,11]],"v":[10,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-2\",null],[\"-2\",null]],[\"10\",\"0\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0012","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-1, 4], [-2, 12]]，T(x)=Mx。求 T(v)，v=[-11, 1]。","answer":{"kind":"expression","value":"[15, 34]","explanation_zh":"Mv=[15, 34]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072012,"parameters":{"M":[[-1,4],[-2,12]],"v":[-11,1],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],[\"-11\",\"1\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0013","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 1], [-2, 13]]，T(x)=Mx。求 T(v)，v=[0, 12]。","answer":{"kind":"expression","value":"[12, 156]","explanation_zh":"Mv=[12, 156]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072013,"parameters":{"M":[[0,1],[-2,13]],"v":[0,12],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"1\"],[null,\"13\"]],[\"0\",\"12\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0014","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[1, 2], [-2, 14]]，T(x)=Mx。求 T(v)，v=[13, 0]。","answer":{"kind":"expression","value":"[13, -26]","explanation_zh":"Mv=[13, -26]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072014,"parameters":{"M":[[1,2],[-2,14]],"v":[13,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"1\",null],[\"-2\",null]],[\"13\",\"0\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0015","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[2, 3], [-2, 15]]，T(x)=Mx。求 T(v)，v=[-14, 1]。","answer":{"kind":"expression","value":"[-25, 43]","explanation_zh":"Mv=[-25, 43]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072015,"parameters":{"M":[[2,3],[-2,15]],"v":[-14,1],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"2\",\"3\"],[\"-2\",\"15\"]],[\"-14\",\"1\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0016","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[3, 4], [-2, 16]]，T(x)=Mx。求 T(v)，v=[0, 15]。","answer":{"kind":"expression","value":"[60, 240]","explanation_zh":"Mv=[60, 240]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072016,"parameters":{"M":[[3,4],[-2,16]],"v":[0,15],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"4\"],[null,\"16\"]],[\"0\",\"15\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0017","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[4, 1], [-2, 17]]，T(x)=Mx。求 T(v)，v=[16, 0]。","answer":{"kind":"expression","value":"[64, -32]","explanation_zh":"Mv=[64, -32]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072017,"parameters":{"M":[[4,1],[-2,17]],"v":[16,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"4\",null],[\"-2\",null]],[\"16\",\"0\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0018","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[5, 2], [-2, 18]]，T(x)=Mx。求 T(v)，v=[-17, 1]。","answer":{"kind":"expression","value":"[-83, 52]","explanation_zh":"Mv=[-83, 52]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072018,"parameters":{"M":[[5,2],[-2,18]],"v":[-17,1],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"5\",\"2\"],[\"-2\",\"18\"]],[\"-17\",\"1\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0019","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[6, 3], [-2, 19]]，T(x)=Mx。求 T(v)，v=[0, 18]。","answer":{"kind":"expression","value":"[54, 342]","explanation_zh":"Mv=[54, 342]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072019,"parameters":{"M":[[6,3],[-2,19]],"v":[0,18],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"3\"],[null,\"19\"]],[\"0\",\"18\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0020","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[7, 4], [-2, 20]]，T(x)=Mx。求 T(v)，v=[19, 0]。","answer":{"kind":"expression","value":"[133, -38]","explanation_zh":"Mv=[133, -38]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072020,"parameters":{"M":[[7,4],[-2,20]],"v":[19,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"7\",null],[\"-2\",null]],[\"19\",\"0\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0021","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[8, 1], [-2, 21]]，T(x)=Mx。求 T(v)，v=[-20, 1]。","answer":{"kind":"expression","value":"[-159, 61]","explanation_zh":"Mv=[-159, 61]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072021,"parameters":{"M":[[8,1],[-2,21]],"v":[-20,1],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"8\",\"1\"],[\"-2\",\"21\"]],[\"-20\",\"1\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0022","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[9, 2], [-2, 22]]，T(x)=Mx。求 T(v)，v=[0, 21]。","answer":{"kind":"expression","value":"[42, 462]","explanation_zh":"Mv=[42, 462]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072022,"parameters":{"M":[[9,2],[-2,22]],"v":[0,21],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"2\"],[null,\"22\"]],[\"0\",\"21\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0023","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[10, 3], [-2, 23]]，T(x)=Mx。求 T(v)，v=[22, 0]。","answer":{"kind":"expression","value":"[220, -44]","explanation_zh":"Mv=[220, -44]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072023,"parameters":{"M":[[10,3],[-2,23]],"v":[22,0],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"10\",null],[\"-2\",null]],[\"22\",\"0\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0024","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[11, 4], [-2, 24]]，T(x)=Mx。求 T(v)，v=[-23, 1]。","answer":{"kind":"expression","value":"[-249, 70]","explanation_zh":"Mv=[-249, 70]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072024,"parameters":{"M":[[11,4],[-2,24]],"v":[-23,1],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],[\"-23\",\"1\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0025","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"二階矩陣的向量像","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[12, 1], [-2, 25]]，T(x)=Mx。求 T(v)，v=[0, 24]。","answer":{"kind":"expression","value":"[24, 600]","explanation_zh":"Mv=[24, 600]。按固定基底的矩陣與直行向量相乘；零輸入的像為零，不混淆點與向量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩陣的向量像"]},"variant":{"family_id":"qf-linear-map-image","seed":202610072025,"parameters":{"M":[[12,1],[-2,25]],"v":[0,24],"operation":"linear_image","domain":"reals","semantic_template":"qf-linear-map-image/{\"inputs\":[[[null,\"1\"],[null,\"25\"]],[\"0\",\"24\"]],\"operation\":\"linear_image\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0026","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 0], [0, 0]]，T(x)=Mx。另設 F(x)=Mx+b，b=[0, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"是","explanation_zh":"F(0)=[0, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072026,"parameters":{"M":[[0,0],[0,0]],"b":[0,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"0\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0027","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[1, 2], [2, 4]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-1, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-1, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072027,"parameters":{"M":[[1,2],[2,4]],"b":[-1,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-1\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0028","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 1], [1, 0]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-2, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-2, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072028,"parameters":{"M":[[0,1],[1,0]],"b":[-2,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-2\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0029","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-9, 4], [-2, 4]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-3, 5]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-3, 5]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072029,"parameters":{"M":[[-9,4],[-2,4]],"b":[-3,5],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-3\",\"5\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0030","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-8, 1], [-2, 5]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-4, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-4, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072030,"parameters":{"M":[[-8,1],[-2,5]],"b":[-4,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-4\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0031","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-7, 2], [-2, 6]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-5, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-5, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072031,"parameters":{"M":[[-7,2],[-2,6]],"b":[-5,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-5\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0032","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-6, 3], [-2, 7]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-6, 8]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-6, 8]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072032,"parameters":{"M":[[-6,3],[-2,7]],"b":[-6,8],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-6\",\"8\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0033","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-5, 4], [-2, 8]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-7, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-7, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072033,"parameters":{"M":[[-5,4],[-2,8]],"b":[-7,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-7\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0034","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-4, 1], [-2, 9]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-8, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-8, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072034,"parameters":{"M":[[-4,1],[-2,9]],"b":[-8,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-8\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0035","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-3, 2], [-2, 10]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-9, 11]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-9, 11]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072035,"parameters":{"M":[[-3,2],[-2,10]],"b":[-9,11],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-9\",\"11\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0036","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-2, 3], [-2, 11]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-10, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-10, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072036,"parameters":{"M":[[-2,3],[-2,11]],"b":[-10,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-10\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0037","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-1, 4], [-2, 12]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-11, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-11, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072037,"parameters":{"M":[[-1,4],[-2,12]],"b":[-11,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-11\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0038","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 1], [-2, 13]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-12, 14]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-12, 14]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072038,"parameters":{"M":[[0,1],[-2,13]],"b":[-12,14],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-12\",\"14\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0039","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[1, 2], [-2, 14]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-13, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-13, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072039,"parameters":{"M":[[1,2],[-2,14]],"b":[-13,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-13\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0040","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[2, 3], [-2, 15]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-14, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-14, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072040,"parameters":{"M":[[2,3],[-2,15]],"b":[-14,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-14\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0041","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[3, 4], [-2, 16]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-15, 17]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-15, 17]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072041,"parameters":{"M":[[3,4],[-2,16]],"b":[-15,17],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-15\",\"17\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0042","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[4, 1], [-2, 17]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-16, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-16, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072042,"parameters":{"M":[[4,1],[-2,17]],"b":[-16,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-16\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0043","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[5, 2], [-2, 18]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-17, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-17, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072043,"parameters":{"M":[[5,2],[-2,18]],"b":[-17,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-17\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0044","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[6, 3], [-2, 19]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-18, 20]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-18, 20]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072044,"parameters":{"M":[[6,3],[-2,19]],"b":[-18,20],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-18\",\"20\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0045","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[7, 4], [-2, 20]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-19, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-19, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072045,"parameters":{"M":[[7,4],[-2,20]],"b":[-19,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-19\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0046","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[8, 1], [-2, 21]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-20, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-20, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072046,"parameters":{"M":[[8,1],[-2,21]],"b":[-20,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-20\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0047","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[9, 2], [-2, 22]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-21, 23]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-21, 23]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072047,"parameters":{"M":[[9,2],[-2,22]],"b":[-21,23],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-21\",\"23\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0048","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[10, 3], [-2, 23]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-22, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-22, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072048,"parameters":{"M":[[10,3],[-2,23]],"b":[-22,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-22\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0049","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[11, 4], [-2, 24]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-23, 0]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-23, 0]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072049,"parameters":{"M":[[11,4],[-2,24]],"b":[-23,0],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-23\",\"0\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0050","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"仿射偏移是否仍為線性","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[12, 1], [-2, 25]]，T(x)=Mx。另設 F(x)=Mx+b，b=[-24, 26]。F 在全 R² 上是否線性？","answer":{"kind":"short_text","value":"否","explanation_zh":"F(0)=[-24, 26]。此明示仿射形式在且僅在 b=0 時線性；必要性看 F(0)，並非只試一對非零輸入。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射偏移是否仍為線性"]},"variant":{"family_id":"qf-linear-map-affine-test","seed":202610072050,"parameters":{"M":[[12,1],[-2,25]],"b":[-24,26],"operation":"affine_linearity","domain":"reals","semantic_template":"qf-linear-map-affine-test/{\"inputs\":[\"-24\",\"26\"],\"operation\":\"affine_linearity\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0051","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 0], [0, 0]]，T(x)=Mx。u=[0, 1]、v=[0, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072051,"parameters":{"M":[[0,0],[0,0]],"u":[0,1],"v":[0,2],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"0\",\"0\"],[\"0\",\"0\"]],[[\"0\",\"1\"],[\"0\",\"2\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0052","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[1, 2], [2, 4]]，T(x)=Mx。u=[1, 1]、v=[-1, 1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[4, 8] = [4, 8]","explanation_zh":"兩側值=[4, 8] = [4, 8]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072052,"parameters":{"M":[[1,2],[2,4]],"u":[1,1],"v":[-1,1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"1\",\"2\"],[\"2\",\"4\"]],[[\"-1\",\"1\"],[\"1\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0053","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 1], [1, 0]]，T(x)=Mx。u=[2, 1]、v=[-2, 0]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[1, 0] = [1, 0]","explanation_zh":"兩側值=[1, 0] = [1, 0]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072053,"parameters":{"M":[[0,1],[1,0]],"u":[2,1],"v":[-2,0],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],[[\"-2\",\"0\"],[\"2\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0054","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-9, 4], [-2, 4]]，T(x)=Mx。u=[3, 1]、v=[-3, -1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072054,"parameters":{"M":[[-9,4],[-2,4]],"u":[3,1],"v":[-3,-1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-9\",\"4\"],[\"-2\",\"4\"]],[[\"-3\",\"-1\"],[\"3\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0055","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-8, 1], [-2, 5]]，T(x)=Mx。u=[4, 1]、v=[-4, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 15] = [3, 15]","explanation_zh":"兩側值=[3, 15] = [3, 15]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072055,"parameters":{"M":[[-8,1],[-2,5]],"u":[4,1],"v":[-4,2],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-8\",\"1\"],[\"-2\",\"5\"]],[[\"-4\",\"2\"],[\"4\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0056","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-7, 2], [-2, 6]]，T(x)=Mx。u=[5, 1]、v=[-5, 1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[4, 12] = [4, 12]","explanation_zh":"兩側值=[4, 12] = [4, 12]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072056,"parameters":{"M":[[-7,2],[-2,6]],"u":[5,1],"v":[-5,1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],[[\"-5\",\"1\"],[\"5\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0057","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-6, 3], [-2, 7]]，T(x)=Mx。u=[6, 1]、v=[-6, 0]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 7] = [3, 7]","explanation_zh":"兩側值=[3, 7] = [3, 7]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072057,"parameters":{"M":[[-6,3],[-2,7]],"u":[6,1],"v":[-6,0],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-6\",\"3\"],[\"-2\",\"7\"]],[[\"-6\",\"0\"],[\"6\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0058","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-5, 4], [-2, 8]]，T(x)=Mx。u=[7, 1]、v=[-7, -1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072058,"parameters":{"M":[[-5,4],[-2,8]],"u":[7,1],"v":[-7,-1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-5\",\"4\"],[\"-2\",\"8\"]],[[\"-7\",\"-1\"],[\"7\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0059","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-4, 1], [-2, 9]]，T(x)=Mx。u=[8, 1]、v=[-8, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 27] = [3, 27]","explanation_zh":"兩側值=[3, 27] = [3, 27]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072059,"parameters":{"M":[[-4,1],[-2,9]],"u":[8,1],"v":[-8,2],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],[[\"-8\",\"2\"],[\"8\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0060","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-3, 2], [-2, 10]]，T(x)=Mx。u=[9, 1]、v=[-9, 1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[4, 20] = [4, 20]","explanation_zh":"兩側值=[4, 20] = [4, 20]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072060,"parameters":{"M":[[-3,2],[-2,10]],"u":[9,1],"v":[-9,1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-3\",\"2\"],[\"-2\",\"10\"]],[[\"-9\",\"1\"],[\"9\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0061","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-2, 3], [-2, 11]]，T(x)=Mx。u=[10, 1]、v=[-10, 0]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 11] = [3, 11]","explanation_zh":"兩側值=[3, 11] = [3, 11]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072061,"parameters":{"M":[[-2,3],[-2,11]],"u":[10,1],"v":[-10,0],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],[[\"-10\",\"0\"],[\"10\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0062","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-1, 4], [-2, 12]]，T(x)=Mx。u=[11, 1]、v=[-11, -1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072062,"parameters":{"M":[[-1,4],[-2,12]],"u":[11,1],"v":[-11,-1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],[[\"-11\",\"-1\"],[\"11\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0063","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 1], [-2, 13]]，T(x)=Mx。u=[12, 1]、v=[-12, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 39] = [3, 39]","explanation_zh":"兩側值=[3, 39] = [3, 39]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072063,"parameters":{"M":[[0,1],[-2,13]],"u":[12,1],"v":[-12,2],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"0\",\"1\"],[\"-2\",\"13\"]],[[\"-12\",\"2\"],[\"12\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0064","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[1, 2], [-2, 14]]，T(x)=Mx。u=[13, 1]、v=[-13, 1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[4, 28] = [4, 28]","explanation_zh":"兩側值=[4, 28] = [4, 28]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072064,"parameters":{"M":[[1,2],[-2,14]],"u":[13,1],"v":[-13,1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"1\",\"2\"],[\"-2\",\"14\"]],[[\"-13\",\"1\"],[\"13\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0065","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[2, 3], [-2, 15]]，T(x)=Mx。u=[14, 1]、v=[-14, 0]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 15] = [3, 15]","explanation_zh":"兩側值=[3, 15] = [3, 15]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072065,"parameters":{"M":[[2,3],[-2,15]],"u":[14,1],"v":[-14,0],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"2\",\"3\"],[\"-2\",\"15\"]],[[\"-14\",\"0\"],[\"14\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0066","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[3, 4], [-2, 16]]，T(x)=Mx。u=[15, 1]、v=[-15, -1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072066,"parameters":{"M":[[3,4],[-2,16]],"u":[15,1],"v":[-15,-1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"3\",\"4\"],[\"-2\",\"16\"]],[[\"-15\",\"-1\"],[\"15\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0067","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[4, 1], [-2, 17]]，T(x)=Mx。u=[16, 1]、v=[-16, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 51] = [3, 51]","explanation_zh":"兩側值=[3, 51] = [3, 51]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072067,"parameters":{"M":[[4,1],[-2,17]],"u":[16,1],"v":[-16,2],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"4\",\"1\"],[\"-2\",\"17\"]],[[\"-16\",\"2\"],[\"16\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0068","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[5, 2], [-2, 18]]，T(x)=Mx。u=[17, 1]、v=[-17, 1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[4, 36] = [4, 36]","explanation_zh":"兩側值=[4, 36] = [4, 36]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072068,"parameters":{"M":[[5,2],[-2,18]],"u":[17,1],"v":[-17,1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"5\",\"2\"],[\"-2\",\"18\"]],[[\"-17\",\"1\"],[\"17\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0069","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[6, 3], [-2, 19]]，T(x)=Mx。u=[18, 1]、v=[-18, 0]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 19] = [3, 19]","explanation_zh":"兩側值=[3, 19] = [3, 19]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072069,"parameters":{"M":[[6,3],[-2,19]],"u":[18,1],"v":[-18,0],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"6\",\"3\"],[\"-2\",\"19\"]],[[\"-18\",\"0\"],[\"18\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0070","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[7, 4], [-2, 20]]，T(x)=Mx。u=[19, 1]、v=[-19, -1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072070,"parameters":{"M":[[7,4],[-2,20]],"u":[19,1],"v":[-19,-1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"7\",\"4\"],[\"-2\",\"20\"]],[[\"-19\",\"-1\"],[\"19\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0071","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[8, 1], [-2, 21]]，T(x)=Mx。u=[20, 1]、v=[-20, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 63] = [3, 63]","explanation_zh":"兩側值=[3, 63] = [3, 63]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072071,"parameters":{"M":[[8,1],[-2,21]],"u":[20,1],"v":[-20,2],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"8\",\"1\"],[\"-2\",\"21\"]],[[\"-20\",\"2\"],[\"20\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0072","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[9, 2], [-2, 22]]，T(x)=Mx。u=[21, 1]、v=[-21, 1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[4, 44] = [4, 44]","explanation_zh":"兩側值=[4, 44] = [4, 44]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072072,"parameters":{"M":[[9,2],[-2,22]],"u":[21,1],"v":[-21,1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"9\",\"2\"],[\"-2\",\"22\"]],[[\"-21\",\"1\"],[\"21\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0073","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[10, 3], [-2, 23]]，T(x)=Mx。u=[22, 1]、v=[-22, 0]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 23] = [3, 23]","explanation_zh":"兩側值=[3, 23] = [3, 23]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072073,"parameters":{"M":[[10,3],[-2,23]],"u":[22,1],"v":[-22,0],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"10\",\"3\"],[\"-2\",\"23\"]],[[\"-22\",\"0\"],[\"22\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0074","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[11, 4], [-2, 24]]，T(x)=Mx。u=[23, 1]、v=[-23, -1]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072074,"parameters":{"M":[[11,4],[-2,24]],"u":[23,1],"v":[-23,-1],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],[[\"-23\",\"-1\"],[\"23\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0075","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型加法兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[12, 1], [-2, 25]]，T(x)=Mx。u=[24, 1]、v=[-24, 2]，分別求 T(u+v) 與 T(u)+T(v)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[3, 75] = [3, 75]","explanation_zh":"兩側值=[3, 75] = [3, 75]。計算明示矩陣模型的兩個具體值；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型加法兩側值"]},"variant":{"family_id":"qf-linear-map-additivity-values","seed":202610072075,"parameters":{"M":[[12,1],[-2,25]],"u":[24,1],"v":[-24,2],"operation":"additivity_values","domain":"reals","semantic_template":"qf-linear-map-additivity-values/{\"inputs\":[[[\"12\",\"1\"],[\"-2\",\"25\"]],[[\"-24\",\"2\"],[\"24\",\"1\"]]],\"operation\":\"additivity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0076","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 0], [0, 0]]，T(x)=Mx。u=[-12, 1]、c=0，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072076,"parameters":{"M":[[0,0],[0,0]],"u":[-12,1],"c":0,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[\"zero_scalar\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0077","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[1, 2], [2, 4]]，T(x)=Mx。u=[0, 0]、c=-2，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[0, 0] = [0, 0]","explanation_zh":"兩側值=[0, 0] = [0, 0]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072077,"parameters":{"M":[[1,2],[2,4]],"u":[0,0],"c":-2,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[\"zero_input\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0078","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 1], [1, 0]]，T(x)=Mx。u=[-10, 3]、c=3，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[9, -30] = [9, -30]","explanation_zh":"兩側值=[9, -30] = [9, -30]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072078,"parameters":{"M":[[0,1],[1,0]],"u":[-10,3],"c":3,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"0\",\"1\"],[\"1\",\"0\"]],[\"-10\",\"3\"],\"3\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0079","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-9, 4], [-2, 4]]，T(x)=Mx。u=[-9, 4]、c=-4，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-388, -136] = [-388, -136]","explanation_zh":"兩側值=[-388, -136] = [-388, -136]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072079,"parameters":{"M":[[-9,4],[-2,4]],"u":[-9,4],"c":-4,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-9\",\"4\"],[\"-2\",\"4\"]],[\"-9\",\"4\"],\"-4\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0080","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-8, 1], [-2, 5]]，T(x)=Mx。u=[-8, 5]、c=5，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[345, 205] = [345, 205]","explanation_zh":"兩側值=[345, 205] = [345, 205]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072080,"parameters":{"M":[[-8,1],[-2,5]],"u":[-8,5],"c":5,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-8\",\"1\"],[\"-2\",\"5\"]],[\"-8\",\"5\"],\"5\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0081","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-7, 2], [-2, 6]]，T(x)=Mx。u=[-7, 6]、c=-1，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-61, -50] = [-61, -50]","explanation_zh":"兩側值=[-61, -50] = [-61, -50]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072081,"parameters":{"M":[[-7,2],[-2,6]],"u":[-7,6],"c":-1,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-7\",\"2\"],[\"-2\",\"6\"]],[\"-7\",\"6\"],\"-1\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0082","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-6, 3], [-2, 7]]，T(x)=Mx。u=[-6, 7]、c=2，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[114, 122] = [114, 122]","explanation_zh":"兩側值=[114, 122] = [114, 122]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072082,"parameters":{"M":[[-6,3],[-2,7]],"u":[-6,7],"c":2,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-6\",\"3\"],[\"-2\",\"7\"]],[\"-6\",\"7\"],\"2\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0083","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-5, 4], [-2, 8]]，T(x)=Mx。u=[-5, 8]、c=-3，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-171, -222] = [-171, -222]","explanation_zh":"兩側值=[-171, -222] = [-171, -222]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072083,"parameters":{"M":[[-5,4],[-2,8]],"u":[-5,8],"c":-3,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-5\",\"4\"],[\"-2\",\"8\"]],[\"-5\",\"8\"],\"-3\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0084","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-4, 1], [-2, 9]]，T(x)=Mx。u=[-4, 9]、c=4，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[100, 356] = [100, 356]","explanation_zh":"兩側值=[100, 356] = [100, 356]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072084,"parameters":{"M":[[-4,1],[-2,9]],"u":[-4,9],"c":4,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-4\",\"1\"],[\"-2\",\"9\"]],[\"-4\",\"9\"],\"4\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0085","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-3, 2], [-2, 10]]，T(x)=Mx。u=[-3, 10]、c=-5，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-145, -530] = [-145, -530]","explanation_zh":"兩側值=[-145, -530] = [-145, -530]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072085,"parameters":{"M":[[-3,2],[-2,10]],"u":[-3,10],"c":-5,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-3\",\"2\"],[\"-2\",\"10\"]],[\"-3\",\"10\"],\"-5\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0086","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-2, 3], [-2, 11]]，T(x)=Mx。u=[-2, 11]、c=1，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[37, 125] = [37, 125]","explanation_zh":"兩側值=[37, 125] = [37, 125]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072086,"parameters":{"M":[[-2,3],[-2,11]],"u":[-2,11],"c":1,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],[\"-2\",\"11\"],\"1\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0087","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-1, 4], [-2, 12]]，T(x)=Mx。u=[-1, 12]、c=-2，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-98, -292] = [-98, -292]","explanation_zh":"兩側值=[-98, -292] = [-98, -292]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072087,"parameters":{"M":[[-1,4],[-2,12]],"u":[-1,12],"c":-2,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-1\",\"4\"],[\"-2\",\"12\"]],[\"-1\",\"12\"],\"-2\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0088","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[0, 1], [-2, 13]]，T(x)=Mx。u=[0, 13]、c=3，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[39, 507] = [39, 507]","explanation_zh":"兩側值=[39, 507] = [39, 507]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072088,"parameters":{"M":[[0,1],[-2,13]],"u":[0,13],"c":3,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"0\",\"1\"],[\"-2\",\"13\"]],[\"0\",\"13\"],\"3\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0089","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[1, 2], [-2, 14]]，T(x)=Mx。u=[1, 14]、c=-4，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-116, -776] = [-116, -776]","explanation_zh":"兩側值=[-116, -776] = [-116, -776]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072089,"parameters":{"M":[[1,2],[-2,14]],"u":[1,14],"c":-4,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"1\",\"2\"],[\"-2\",\"14\"]],[\"1\",\"14\"],\"-4\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0090","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[2, 3], [-2, 15]]，T(x)=Mx。u=[2, 15]、c=5，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[245, 1105] = [245, 1105]","explanation_zh":"兩側值=[245, 1105] = [245, 1105]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072090,"parameters":{"M":[[2,3],[-2,15]],"u":[2,15],"c":5,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"2\",\"3\"],[\"-2\",\"15\"]],[\"2\",\"15\"],\"5\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0091","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[3, 4], [-2, 16]]，T(x)=Mx。u=[3, 16]、c=-1，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-73, -250] = [-73, -250]","explanation_zh":"兩側值=[-73, -250] = [-73, -250]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072091,"parameters":{"M":[[3,4],[-2,16]],"u":[3,16],"c":-1,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"3\",\"4\"],[\"-2\",\"16\"]],[\"3\",\"16\"],\"-1\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0092","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[4, 1], [-2, 17]]，T(x)=Mx。u=[4, 17]、c=2，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[66, 562] = [66, 562]","explanation_zh":"兩側值=[66, 562] = [66, 562]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072092,"parameters":{"M":[[4,1],[-2,17]],"u":[4,17],"c":2,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"4\",\"1\"],[\"-2\",\"17\"]],[\"4\",\"17\"],\"2\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0093","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[5, 2], [-2, 18]]，T(x)=Mx。u=[5, 18]、c=-3，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-183, -942] = [-183, -942]","explanation_zh":"兩側值=[-183, -942] = [-183, -942]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072093,"parameters":{"M":[[5,2],[-2,18]],"u":[5,18],"c":-3,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"5\",\"2\"],[\"-2\",\"18\"]],[\"5\",\"18\"],\"-3\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0094","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[6, 3], [-2, 19]]，T(x)=Mx。u=[6, 19]、c=4，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[372, 1396] = [372, 1396]","explanation_zh":"兩側值=[372, 1396] = [372, 1396]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072094,"parameters":{"M":[[6,3],[-2,19]],"u":[6,19],"c":4,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"6\",\"3\"],[\"-2\",\"19\"]],[\"6\",\"19\"],\"4\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0095","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[7, 4], [-2, 20]]，T(x)=Mx。u=[7, 20]、c=-5，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-645, -1930] = [-645, -1930]","explanation_zh":"兩側值=[-645, -1930] = [-645, -1930]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072095,"parameters":{"M":[[7,4],[-2,20]],"u":[7,20],"c":-5,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"7\",\"4\"],[\"-2\",\"20\"]],[\"7\",\"20\"],\"-5\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0096","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[8, 1], [-2, 21]]，T(x)=Mx。u=[8, 21]、c=1，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[85, 425] = [85, 425]","explanation_zh":"兩側值=[85, 425] = [85, 425]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072096,"parameters":{"M":[[8,1],[-2,21]],"u":[8,21],"c":1,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"8\",\"1\"],[\"-2\",\"21\"]],[\"8\",\"21\"],\"1\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0097","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[9, 2], [-2, 22]]，T(x)=Mx。u=[9, 22]、c=-2，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-250, -932] = [-250, -932]","explanation_zh":"兩側值=[-250, -932] = [-250, -932]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072097,"parameters":{"M":[[9,2],[-2,22]],"u":[9,22],"c":-2,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"9\",\"2\"],[\"-2\",\"22\"]],[\"9\",\"22\"],\"-2\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0098","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[10, 3], [-2, 23]]，T(x)=Mx。u=[10, 23]、c=3，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[507, 1527] = [507, 1527]","explanation_zh":"兩側值=[507, 1527] = [507, 1527]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072098,"parameters":{"M":[[10,3],[-2,23]],"u":[10,23],"c":3,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"10\",\"3\"],[\"-2\",\"23\"]],[\"10\",\"23\"],\"3\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0099","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[11, 4], [-2, 24]]，T(x)=Mx。u=[11, 24]、c=-4，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[-868, -2216] = [-868, -2216]","explanation_zh":"兩側值=[-868, -2216] = [-868, -2216]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072099,"parameters":{"M":[[11,4],[-2,24]],"u":[11,24],"c":-4,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"11\",\"4\"],[\"-2\",\"24\"]],[\"11\",\"24\"],\"-4\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261006-advance-20261007-r02-linear-transformation-0100","batch_id":"batch-20261006-advance-20261007-r02-linear-transformation-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-transformation"],"title_zh":"矩陣模型純量兩側值","stem_zh":"實平面線性模型，固定標準基底，以直行向量記輸入，M=[[12, 1], [-2, 25]]，T(x)=Mx。u=[12, 25]、c=5，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。","answer":{"kind":"expression","value":"[845, 3005] = [845, 3005]","explanation_zh":"兩側值=[845, 3005] = [845, 3005]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["矩陣模型純量兩側值"]},"variant":{"family_id":"qf-linear-map-homogeneity-values","seed":202610072100,"parameters":{"M":[[12,1],[-2,25]],"u":[12,25],"c":5,"operation":"homogeneity_values","domain":"reals","semantic_template":"qf-linear-map-homogeneity-values/{\"inputs\":[[[\"12\",\"1\"],[\"-2\",\"25\"]],[\"12\",\"25\"],\"5\"],\"operation\":\"homogeneity_values\"}"}},"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":"2026-10-06","contract_version":"ocme-question-factory-v0.1"}}