{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261006-advance-20261007-r02-linear-transformation-0029", "pilot_number": "R0064-32829", "number": 32829, "family_id": "qf-linear-map-affine-test", "title": { "zh-Hant": "第 32829 題：仿射偏移是否仍為線性", "en": "Question 32829: Linearity of an Explicit Affine Map" }, "original_question": { "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" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261006-advance-20261007-r02-linear-transformation-0029", "batch_id": "batch-20261006-advance-20261007-r02-linear-transformation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "0cb4b8271de0d7540704edb841d0f86fc38fdae9313b0c14d08d7cec5e1a7a7a", "family_id": "qf-linear-map-affine-test", "seed": 202610072029, "semantic_template": "qf-linear-map-affine-test/{\"inputs\":[\"-3\",\"5\"],\"operation\":\"affine_linearity\"}" }, "fields": { "title": "Linearity of an Explicit Affine Map", "stem": "Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-3, 5]. Is F linear on the whole R²?", "answer": { "kind": "short_text", "value": "No", "explanation": "Here F(0)=b=[-3, 5]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs." }, "skills": [ "Linearity of an Explicit Affine Map" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "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？", "en": "Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-3, 5]. Is F linear on the whole R²?" } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-linear-map-affine-test, affine_linearity, reals, []]", "after": "derived = [[zero_offset_linearity_YesNo_STRING]]", "relation": "implies", "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles", "conditions": [], "explanation": { "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。", "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product := mul(-9, 0)", "after": "Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product，不以小數近似替代。", "en": "Recompute Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-9", "0" ], "output": "0", "label": "Affine ORIGINAL matrix at zero ORIGINAL row0 column0 product" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product := mul(4, 0)", "after": "Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product，不以小數近似替代。", "en": "Recompute Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "0" ], "output": "0", "label": "Affine ORIGINAL matrix at zero ORIGINAL row0 column1 product" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum := sum(0, 0)", "after": "Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum，不以小數近似替代。", "en": "Recompute Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "Affine ORIGINAL matrix at zero ORIGINAL row0 two products sum" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [Affine ORIGINAL matrix at zero, 0, [-9, 4], [0, 0]]", "after": "derived = [[0, 0], 0]", "relation": "implies", "rule_id": "LinearMap_EACH_row_by_column_component_ALL_products", "conditions": [], "explanation": { "zh-Hant": "固定標準基底用row-by-column，兩個product都加；derived向量分量可超SAFE，不能再當raw vector重驗。", "en": "In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product := mul(-2, 0)", "after": "Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product，不以小數近似替代。", "en": "Recompute Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "0" ], "output": "0", "label": "Affine ORIGINAL matrix at zero ORIGINAL row1 column0 product" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product := mul(4, 0)", "after": "Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product，不以小數近似替代。", "en": "Recompute Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "0" ], "output": "0", "label": "Affine ORIGINAL matrix at zero ORIGINAL row1 column1 product" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum := sum(0, 0)", "after": "Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum，不以小數近似替代。", "en": "Recompute Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "Affine ORIGINAL matrix at zero ORIGINAL row1 two products sum" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [Affine ORIGINAL matrix at zero, 1, [-2, 4], [0, 0]]", "after": "derived = [[0, 0], 0]", "relation": "implies", "rule_id": "LinearMap_EACH_row_by_column_component_ALL_products", "conditions": [], "explanation": { "zh-Hant": "固定標準基底用row-by-column，兩個product都加；derived向量分量可超SAFE，不能再當raw vector重驗。", "en": "In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Affine ORIGINAL M0 plus offset ORIGINAL component0 := sum(0, -3)", "after": "Affine ORIGINAL M0 plus offset ORIGINAL component0 = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine ORIGINAL M0 plus offset ORIGINAL component0，不以小數近似替代。", "en": "Recompute Affine ORIGINAL M0 plus offset ORIGINAL component0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-3" ], "output": "-3", "label": "Affine ORIGINAL M0 plus offset ORIGINAL component0" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Affine ORIGINAL M0 plus offset ORIGINAL component1 := sum(0, 5)", "after": "Affine ORIGINAL M0 plus offset ORIGINAL component1 = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Affine ORIGINAL M0 plus offset ORIGINAL component1，不以小數近似替代。", "en": "Recompute Affine ORIGINAL M0 plus offset ORIGINAL component1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "5" ], "output": "5", "label": "Affine ORIGINAL M0 plus offset ORIGINAL component1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [[[-9, 4], [-2, 4]], [-3, 5]]", "after": "derived = [[-3, 5], 否]", "relation": "implies", "rule_id": "LinearMap_AFFINE_whole_form_linear_IFF_zero_offset_NOT_det_gate", "conditions": [], "explanation": { "zh-Hant": "對已給全R² affine法則，F(0)=b：b非0違反線性必要條件，b0則Mx由分配律線性。M可奇異或全0，不需det／可逆，也不是非零點抽測。", "en": "For the GIVEN whole affine law on R²,F(0)=b.Nonzero b violates necessary linearity;at b0,distributivity makes Mx linear.M may be singular/zero;no determinant/invertibility gate or nonzero-point sampling is needed." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": "否" }, "conclusion": { "canonical_value": "否", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261006-advance-20261007-r02-linear-transformation-foundations", "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r02-linear-transformation-foundations/questions.jsonl", "batch_sha256": "8aa7124b4f367aa227816eaa77df8620070a5f7ef0a9cf180a0decb92ac73796", "original_line": 29, "original_line_sha256": "5c677a52803d0ab7ee74f8dff9eae089f0827a9c7f3af31aef2110734dc8ab3f", "question_compact_sha256": "0cb4b8271de0d7540704edb841d0f86fc38fdae9313b0c14d08d7cec5e1a7a7a", "english_row_compact_sha256": "db77dbe6bee655205307074b4a59007ab4f80b37695550b037ef0d79f8dcb6cd", "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r02-linear-transformation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r02-linear-transformation-foundations.en.jsonl" }, "mko_bindings": { "mko-linear-transformation": { "sha256": "59ce4d8fac03871ea16eb62a76c9fae255470dc391ff7312ad4da120bafbf132", "english_sha256": "d95d3ac4155053ddc0d186089ae6ec2d7e4e20468aa14880d371ac260c3cd0a3", "original_path": "public/data/mko/linear-transformation.json", "english_path": "public/data/i18n/editions/mko-linear-transformation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "528186aa825c3054db15f4ce0a03ad58f72be7e7eb5bb236c4cc7ef34ff5a470", "original_candidate_sha256": "0e32340e164681405da5e89b6a581696e10575b1fad359b6d67d27eafa2a2203", "original_candidate_line_sha256": "0e32340e164681405da5e89b6a581696e10575b1fad359b6d67d27eafa2a2203" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-011-p65", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b", "question_first_witness_sha256": "066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "bca71e82a3b603522a7c8b63fcbc81f6ac763288e5b122c1663da40e54865d65", "arithmetic_steps": 8 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }