{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261027-r05-general-linear-map-0084", "pilot_number": "R0025-13059", "number": 13059, "family_id": "qf-mod27-linearmap-combination-image", "title": { "zh-Hant": "第 13059 題：實線性映射", "en": "Question 13059: The Matrix Formula Preserves a Whole Linear Combination" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261027-r05-general-linear-map-0084", "batch_id": "batch-20261014-advance-20261027-r05-general-linear-map-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-general-linear-map" ], "title_zh": "實線性映射", "stem_zh": "來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-4,-2,-3],[1,4,-5]],\"u\":[1,1,-2],\"v\":[3,1,-2],\"alpha\":1,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "(1, -2, [-5, -1, 2], [16, -19], [16, -19], 是, matrix_formula_universal_certificate)", "explanation_zh": "Wave4根節點核對=(1, -2, [-5, -1, 2], [16, -19], [16, -19], 是, matrix_formula_universal_certificate)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣分配律提供全稱座標證書；非零平移的仿射映射不線性且有限樣本不足。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "實線性映射" ] }, "variant": { "family_id": "qf-mod27-linearmap-combination-image", "seed": 202610275084, "parameters": { "matrix": [ [ -4, -2, -3 ], [ 1, 4, -5 ] ], "u": [ 1, 1, -2 ], "v": [ 3, 1, -2 ], "alpha": 1, "beta": -2, "domain": "Exact_matrix_real_linear_map", "operation": "linearmap-combination-image", "semantic_template": "qf-mod27-linearmap-combination-image/{\"alpha\":1,\"beta\":-2,\"matrix\":[[-4,-2,-3],[1,4,-5]],\"u\":[1,1,-2],\"v\":[3,1,-2]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261027-r05-general-linear-map-0084", "batch_id": "batch-20261014-advance-20261027-r05-general-linear-map-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "2bd7c61a1a7007ac9bf5c63e8e871c9a293956f7a314ec9ea5e0e7ae15243ef1", "family_id": "qf-mod27-linearmap-combination-image", "seed": 202610275084, "semantic_template": "qf-mod27-linearmap-combination-image/{\"alpha\":1,\"beta\":-2,\"matrix\":[[-4,-2,-3],[1,4,-5]],\"u\":[1,1,-2],\"v\":[3,1,-2]}" }, "fields": { "title": "The Matrix Formula Preserves a Whole Linear Combination", "stem": "Use the complete rectangular matrix A=[[-4, -2, -3], [1, 4, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 1, -2], v=[3, 1, -2], alpha=1, beta=-2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).", "answer": { "kind": "expression", "value": "(1, -2, [-5, -1, 2], [16, -19], [16, -19], Yes, matrix_formula_universal_certificate)", "explanation": "Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -2, [-5, -1, 2], [16, -19], [16, -19], Yes, matrix_formula_universal_certificate)." }, "skills": [ "The Matrix Formula Preserves a Whole Linear Combination" ], "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": "來源與目標皆為明示實向量空間。操作=linearmap-combination-image；資料={\"matrix\":[[-4,-2,-3],[1,4,-5]],\"u\":[1,1,-2],\"v\":[3,1,-2],\"alpha\":1,\"beta\":-2}。請依 [alpha,beta,combined,lhs,rhs,equal,certificate] 欄序回答。", "en": "Use the complete rectangular matrix A=[[-4, -2, -3], [1, 4, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 1, -2], v=[3, 1, -2], alpha=1, beta=-2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "input combination coordinate0 first scaled := mul(1, 1)", "after": "input combination coordinate0 first scaled = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate0 first scaled，不以小數近似替代。", "en": "Recompute input combination coordinate0 first scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "input combination coordinate0 first scaled" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "input combination coordinate0 second scaled := mul(-2, 3)", "after": "input combination coordinate0 second scaled = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate0 second scaled，不以小數近似替代。", "en": "Recompute input combination coordinate0 second scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "3" ], "output": "-6", "label": "input combination coordinate0 second scaled" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "input combination coordinate0 sum := sum(1, -6)", "after": "input combination coordinate0 sum = -5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate0 sum，不以小數近似替代。", "en": "Recompute input combination coordinate0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "-6" ], "output": "-5", "label": "input combination coordinate0 sum" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "input combination coordinate1 first scaled := mul(1, 1)", "after": "input combination coordinate1 first scaled = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate1 first scaled，不以小數近似替代。", "en": "Recompute input combination coordinate1 first scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "input combination coordinate1 first scaled" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "input combination coordinate1 second scaled := mul(-2, 1)", "after": "input combination coordinate1 second scaled = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate1 second scaled，不以小數近似替代。", "en": "Recompute input combination coordinate1 second scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "input combination coordinate1 second scaled" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "input combination coordinate1 sum := sum(1, -2)", "after": "input combination coordinate1 sum = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate1 sum，不以小數近似替代。", "en": "Recompute input combination coordinate1 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "-2" ], "output": "-1", "label": "input combination coordinate1 sum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "input combination coordinate2 first scaled := mul(1, -2)", "after": "input combination coordinate2 first scaled = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate2 first scaled，不以小數近似替代。", "en": "Recompute input combination coordinate2 first scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-2" ], "output": "-2", "label": "input combination coordinate2 first scaled" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "input combination coordinate2 second scaled := mul(-2, -2)", "after": "input combination coordinate2 second scaled = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate2 second scaled，不以小數近似替代。", "en": "Recompute input combination coordinate2 second scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "input combination coordinate2 second scaled" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "input combination coordinate2 sum := sum(-2, 4)", "after": "input combination coordinate2 sum = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 input combination coordinate2 sum，不以小數近似替代。", "en": "Recompute input combination coordinate2 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "4" ], "output": "2", "label": "input combination coordinate2 sum" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "combined image:matrix=[[-4, -2, -3], [1, 4, -5]];input=[-5, -1, 2]", "after": "R^3->R^2;each output is its original row dot input", "relation": "implies", "rule_id": "rectangular_linear_map_source_target_roles", "conditions": [], "explanation": { "zh-Hant": "輸入維度等於欄數，輸出維度等於列數，矩陣可長方形；不交換行列或補不存在的座標。", "en": "Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "combined image row0 column0 term := mul(-4, -5)", "after": "combined image row0 column0 term = 20", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined image row0 column0 term，不以小數近似替代。", "en": "Recompute combined image row0 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-4", "-5" ], "output": "20", "label": "combined image row0 column0 term" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "combined image row0 column1 term := mul(-2, -1)", "after": "combined image row0 column1 term = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined image row0 column1 term，不以小數近似替代。", "en": "Recompute combined image row0 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-1" ], "output": "2", "label": "combined image row0 column1 term" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "combined image row0 column2 term := mul(-3, 2)", "after": "combined image row0 column2 term = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined image row0 column2 term，不以小數近似替代。", "en": "Recompute combined image row0 column2 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "2" ], "output": "-6", "label": "combined image row0 column2 term" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "combined image row0 sum := sum(20, 2, -6)", "after": "combined image row0 sum = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined image row0 sum，不以小數近似替代。", "en": "Recompute combined image row0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "20", "2", "-6" ], "output": "16", "label": "combined image row0 sum" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "combined image row1 column0 term := mul(1, -5)", "after": "combined image row1 column0 term = -5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined image row1 column0 term，不以小數近似替代。", "en": "Recompute combined image row1 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-5" ], "output": "-5", "label": "combined image row1 column0 term" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "combined image row1 column1 term := mul(4, -1)", "after": "combined image row1 column1 term = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined image row1 column1 term，不以小數近似替代。", "en": "Recompute combined image row1 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "-1" ], "output": "-4", "label": "combined image row1 column1 term" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "combined image row1 column2 term := mul(-5, 2)", "after": "combined image row1 column2 term = -10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined image row1 column2 term，不以小數近似替代。", "en": "Recompute combined image row1 column2 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-5", "2" ], "output": "-10", "label": "combined image row1 column2 term" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "combined image row1 sum := sum(-5, -4, -10)", "after": "combined image row1 sum = -19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined image row1 sum，不以小數近似替代。", "en": "Recompute combined image row1 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-5", "-4", "-10" ], "output": "-19", "label": "combined image row1 sum" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "u image:matrix=[[-4, -2, -3], [1, 4, -5]];input=[1, 1, -2]", "after": "R^3->R^2;each output is its original row dot input", "relation": "implies", "rule_id": "rectangular_linear_map_source_target_roles", "conditions": [], "explanation": { "zh-Hant": "輸入維度等於欄數，輸出維度等於列數，矩陣可長方形；不交換行列或補不存在的座標。", "en": "Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "u image row0 column0 term := mul(-4, 1)", "after": "u image row0 column0 term = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u image row0 column0 term，不以小數近似替代。", "en": "Recompute u image row0 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-4", "1" ], "output": "-4", "label": "u image row0 column0 term" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "u image row0 column1 term := mul(-2, 1)", "after": "u image row0 column1 term = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u image row0 column1 term，不以小數近似替代。", "en": "Recompute u image row0 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "u image row0 column1 term" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "u image row0 column2 term := mul(-3, -2)", "after": "u image row0 column2 term = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u image row0 column2 term，不以小數近似替代。", "en": "Recompute u image row0 column2 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "-2" ], "output": "6", "label": "u image row0 column2 term" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "u image row0 sum := sum(-4, -2, 6)", "after": "u image row0 sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u image row0 sum，不以小數近似替代。", "en": "Recompute u image row0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "-2", "6" ], "output": "0", "label": "u image row0 sum" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "u image row1 column0 term := mul(1, 1)", "after": "u image row1 column0 term = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u image row1 column0 term，不以小數近似替代。", "en": "Recompute u image row1 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "u image row1 column0 term" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "u image row1 column1 term := mul(4, 1)", "after": "u image row1 column1 term = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u image row1 column1 term，不以小數近似替代。", "en": "Recompute u image row1 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "u image row1 column1 term" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "u image row1 column2 term := mul(-5, -2)", "after": "u image row1 column2 term = 10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u image row1 column2 term，不以小數近似替代。", "en": "Recompute u image row1 column2 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-5", "-2" ], "output": "10", "label": "u image row1 column2 term" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "u image row1 sum := sum(1, 4, 10)", "after": "u image row1 sum = 15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u image row1 sum，不以小數近似替代。", "en": "Recompute u image row1 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "4", "10" ], "output": "15", "label": "u image row1 sum" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "v image:matrix=[[-4, -2, -3], [1, 4, -5]];input=[3, 1, -2]", "after": "R^3->R^2;each output is its original row dot input", "relation": "implies", "rule_id": "rectangular_linear_map_source_target_roles", "conditions": [], "explanation": { "zh-Hant": "輸入維度等於欄數，輸出維度等於列數，矩陣可長方形；不交換行列或補不存在的座標。", "en": "Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "v image row0 column0 term := mul(-4, 3)", "after": "v image row0 column0 term = -12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v image row0 column0 term，不以小數近似替代。", "en": "Recompute v image row0 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-4", "3" ], "output": "-12", "label": "v image row0 column0 term" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "v image row0 column1 term := mul(-2, 1)", "after": "v image row0 column1 term = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v image row0 column1 term，不以小數近似替代。", "en": "Recompute v image row0 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "v image row0 column1 term" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "v image row0 column2 term := mul(-3, -2)", "after": "v image row0 column2 term = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v image row0 column2 term，不以小數近似替代。", "en": "Recompute v image row0 column2 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "-2" ], "output": "6", "label": "v image row0 column2 term" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "v image row0 sum := sum(-12, -2, 6)", "after": "v image row0 sum = -8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v image row0 sum，不以小數近似替代。", "en": "Recompute v image row0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-12", "-2", "6" ], "output": "-8", "label": "v image row0 sum" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "v image row1 column0 term := mul(1, 3)", "after": "v image row1 column0 term = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v image row1 column0 term，不以小數近似替代。", "en": "Recompute v image row1 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "v image row1 column0 term" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "v image row1 column1 term := mul(4, 1)", "after": "v image row1 column1 term = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v image row1 column1 term，不以小數近似替代。", "en": "Recompute v image row1 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "v image row1 column1 term" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "v image row1 column2 term := mul(-5, -2)", "after": "v image row1 column2 term = 10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v image row1 column2 term，不以小數近似替代。", "en": "Recompute v image row1 column2 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-5", "-2" ], "output": "10", "label": "v image row1 column2 term" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "v image row1 sum := sum(3, 4, 10)", "after": "v image row1 sum = 17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v image row1 sum，不以小數近似替代。", "en": "Recompute v image row1 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3", "4", "10" ], "output": "17", "label": "v image row1 sum" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "output combination coordinate0 first scaled := mul(1, 0)", "after": "output combination coordinate0 first scaled = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 output combination coordinate0 first scaled，不以小數近似替代。", "en": "Recompute output combination coordinate0 first scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "0" ], "output": "0", "label": "output combination coordinate0 first scaled" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "output combination coordinate0 second scaled := mul(-2, -8)", "after": "output combination coordinate0 second scaled = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 output combination coordinate0 second scaled，不以小數近似替代。", "en": "Recompute output combination coordinate0 second scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-8" ], "output": "16", "label": "output combination coordinate0 second scaled" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "output combination coordinate0 sum := sum(0, 16)", "after": "output combination coordinate0 sum = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 output combination coordinate0 sum，不以小數近似替代。", "en": "Recompute output combination coordinate0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "16" ], "output": "16", "label": "output combination coordinate0 sum" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "output combination coordinate1 first scaled := mul(1, 15)", "after": "output combination coordinate1 first scaled = 15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 output combination coordinate1 first scaled，不以小數近似替代。", "en": "Recompute output combination coordinate1 first scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "15" ], "output": "15", "label": "output combination coordinate1 first scaled" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "output combination coordinate1 second scaled := mul(-2, 17)", "after": "output combination coordinate1 second scaled = -34", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 output combination coordinate1 second scaled，不以小數近似替代。", "en": "Recompute output combination coordinate1 second scaled with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "17" ], "output": "-34", "label": "output combination coordinate1 second scaled" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "output combination coordinate1 sum := sum(15, -34)", "after": "output combination coordinate1 sum = -19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 output combination coordinate1 sum，不以小數近似替代。", "en": "Recompute output combination coordinate1 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "15", "-34" ], "output": "-19", "label": "output combination coordinate1 sum" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "matrix formula;alpha=1;beta=-2;lhs=[16, -19];rhs=[16, -19]", "after": "all row dot products distribute over all real linear combinations;instance equality=是", "relation": "implies", "rule_id": "matrix_combination_universal_coordinate_certificate", "conditions": [], "explanation": { "zh-Hant": "每個輸出座標的有限和可分配任意實純量，逐列同時成立即是全稱線性證書；本題再用完整座標重組回核實例。", "en": "Every output coordinate is a finite sum distributing over arbitrary real scalars.All rows together give a universal linearity certificate;the full coordinate calculation also checks this instance." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 1, -2, [ -5, -1, 2 ], [ 16, -19 ], [ 16, -19 ], true, "matrix_formula_universal_certificate" ] }, "conclusion": { "canonical_value": "(1, -2, [-5, -1, 2], [16, -19], [16, -19], 是, matrix_formula_universal_certificate)", "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-20261014-advance-20261027-r05-general-linear-map-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261027-r05-general-linear-map-foundations/questions.jsonl", "batch_sha256": "72939413540b8aa3948b02d6ef39fd207a7f87a539e65976ba2ac5acfd7861c5", "original_line": 84, "original_line_sha256": "9de4982cb0fa08a9c46a6fe69faeee01f767ff4d625ddee42ead3f54ed8c0762", "question_compact_sha256": "2bd7c61a1a7007ac9bf5c63e8e871c9a293956f7a314ec9ea5e0e7ae15243ef1", "english_row_compact_sha256": "cb0a9d1c72c644b2e52303a862574fab6202bdfc05266e382a67f66a38317c27", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261027-r05-general-linear-map-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261027-r05-general-linear-map-foundations.en.jsonl" }, "mko_bindings": { "mko-general-linear-map": { "sha256": "184a885b199aff8dc05ec273c0fe5b2dcdd38a9ef98c8ec790ce2668da9a07c7", "english_sha256": "efdc1e4563905094acbcc230800a69094676b62be7baef1646756189ac8456f0", "original_path": "public/data/mko/general-linear-map.json", "english_path": "public/data/i18n/editions/mko-general-linear-map.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a02aa2dccf790869cf864b821d0e60119686214561153daba232d921e07e7924", "original_candidate_sha256": "8d4cef8fd87b41dde4c6313a39315811292a94f76f0e5797beaabc828efb9d49", "original_candidate_line_sha256": "8d4cef8fd87b41dde4c6313a39315811292a94f76f0e5797beaabc828efb9d49" }, "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-20261026-007-p26", "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": "0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5", "question_first_witness_sha256": "ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 43, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "7cc21151e12f59fb318aabac983f3dc3ccb45206759590cc1f1c3c540636387e", "arithmetic_steps": 39 }, "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." } }