{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261102-r01-vector-space-dimension-0030", "pilot_number": "R0032-16930", "number": 16930, "family_id": "qf-mod33-dimension-exact-value", "title": { "zh-Hant": "第 16930 題：向量空間維度", "en": "Question 16930: Read Dimension from a Previously Qualified Basis" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261102-r01-vector-space-dimension-0030", "batch_id": "batch-20261014-advance-20261102-r01-vector-space-dimension-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-vector-space-dimension" ], "title_zh": "向量空間維度", "stem_zh": "R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[4,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)", "explanation_zh": "Wave5第二層核對=(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)；依完整明示資料重算；dimension只在真正basis成立後取basis cardinality；獨立族只給下界、生成族只給上界；basis invariance使用有限維定理。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "向量空間維度" ] }, "variant": { "family_id": "qf-mod33-dimension-exact-value", "seed": 202611021030, "parameters": { "dimension": 2, "vectors": [ [ 1, 0 ], [ 4, 1 ] ], "domain": "Exact_R2_R3_basis_dimension", "operation": "dimension-exact-value", "semantic_template": "qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[4,1]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean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-20261102-r01-vector-space-dimension-0030", "batch_id": "batch-20261014-advance-20261102-r01-vector-space-dimension-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "bb0dca38d07c3b357dc62a041119cd2b1dac6f96a8fd95ee782f3fe78cfa655b", "family_id": "qf-mod33-dimension-exact-value", "seed": 202611021030, "semantic_template": "qf-mod33-dimension-exact-value/{\"dimension\":2,\"vectors\":[[1,0],[4,1]]}" }, "fields": { "title": "Read Dimension from a Previously Qualified Basis", "stem": "Use the complete ordered real coordinate family vectors=[[1, 0], [4, 1]] in ambient R^2, limited to dimensions 2 or 3. Treat vectors as columns and calculate rank by exact rational Gaussian elimination, not floating-point samples. A basis is both independent and spanning. An independent family gives only a lower bound; a spanning family gives only an upper bound. Basis-cardinality invariance uses the finite-dimensional theorem, not a few examples. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).", "answer": { "kind": "expression", "value": "(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)", "explanation": "Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)." }, "skills": [ "Read Dimension from a Previously Qualified Basis" ], "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": "R²或R³完整座標族與精確rank證書。操作=dimension-exact-value；資料={\"dimension\":2,\"vectors\":[[1,0],[4,1]]}。請依 [ambient_dimension,basis_count,rank,dimension,certificate] 欄序回答。", "en": "Use the complete ordered real coordinate family vectors=[[1, 0], [4, 1]] in ambient R^2, limited to dimensions 2 or 3. Treat vectors as columns and calculate rank by exact rational Gaussian elimination, not floating-point samples. A basis is both independent and spanning. An independent family gives only a lower bound; a spanning family gives only an upper bound. Basis-cardinality invariance uses the finite-dimensional theorem, not a few examples. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "same usual real ambient R^2;original ORDERED vectors as columns=[[1, 0], [4, 1]]", "after": "complete row-coordinate matrix=[[1, 4], [0, 1]];vector count=2;zero/repeated vectors retained", "relation": "implies", "rule_id": "dimension_actual_columns_not_semantic_deduplication", "conditions": [], "explanation": { "zh-Hant": "欄是原向量，列是座標；來源允許零向量及重複向量，故不能拿semantic template去重後的族改掉數量或證書。用精確有理數列運算，視為實數上的可逆座標變換。", "en": "Columns are the original vectors and rows their coordinates.Zero and repeated vectors are allowed,so semantic-template deduplication cannot replace the original family or its count.Exact rational row operations are invertible coordinate changes over the reals." }, "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": "S002", "uses": [ "G001", "S001" ], "before": "dimension pivot row0 column0 normalized entry0 := div(1, 1)", "after": "dimension pivot row0 column0 normalized entry0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dimension pivot row0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute dimension pivot row0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "dimension pivot row0 column0 normalized entry0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "dimension pivot row0 column0 normalized entry1 := div(4, 1)", "after": "dimension pivot row0 column0 normalized entry1 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dimension pivot row0 column0 normalized entry1，不以小數近似替代。", "en": "Recompute dimension pivot row0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "4", "1" ], "output": "4", "label": "dimension pivot row0 column0 normalized entry1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "pivot row=0;pivot column=0;nonzero exact pivot=1;old row=[1, 4]", "after": "divide whole row by pivot;normalized row=[1, 4]", "relation": "implies", "rule_id": "dimension_nonzero_pivot_scaling_every_entry", "conditions": [], "explanation": { "zh-Hant": "以同一非零pivot除整列，逐格列出精確分數；不能只改pivot格或用浮點近似。逆向乘回pivot即可恢復。", "en": "Divide the whole row by the same nonzero pivot,showing every exact fraction.Do not change only the pivot cell or use floating-point approximations.Multiplying back by the pivot reverses the step." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "dimension pivot row1 column1 normalized entry0 := div(0, 1)", "after": "dimension pivot row1 column1 normalized entry0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dimension pivot row1 column1 normalized entry0，不以小數近似替代。", "en": "Recompute dimension pivot row1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "dimension pivot row1 column1 normalized entry0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "dimension pivot row1 column1 normalized entry1 := div(1, 1)", "after": "dimension pivot row1 column1 normalized entry1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dimension pivot row1 column1 normalized entry1，不以小數近似替代。", "en": "Recompute dimension pivot row1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "dimension pivot row1 column1 normalized entry1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "pivot row=1;pivot column=1;nonzero exact pivot=1;old row=[0, 1]", "after": "divide whole row by pivot;normalized row=[0, 1]", "relation": "implies", "rule_id": "dimension_nonzero_pivot_scaling_every_entry", "conditions": [], "explanation": { "zh-Hant": "以同一非零pivot除整列，逐格列出精確分數；不能只改pivot格或用浮點近似。逆向乘回pivot即可恢復。", "en": "Divide the whole row by the same nonzero pivot,showing every exact fraction.Do not change only the pivot cell or use floating-point approximations.Multiplying back by the pivot reverses the step." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "dimension eliminate row0 pivot1 column1 entry0 multiple := mul(4, 0)", "after": "dimension eliminate row0 pivot1 column1 entry0 multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dimension eliminate row0 pivot1 column1 entry0 multiple，不以小數近似替代。", "en": "Recompute dimension eliminate row0 pivot1 column1 entry0 multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "0" ], "output": "0", "label": "dimension eliminate row0 pivot1 column1 entry0 multiple" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "dimension eliminate row0 pivot1 column1 entry0 difference := sub(1, 0)", "after": "dimension eliminate row0 pivot1 column1 entry0 difference = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dimension eliminate row0 pivot1 column1 entry0 difference，不以小數近似替代。", "en": "Recompute dimension eliminate row0 pivot1 column1 entry0 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "dimension eliminate row0 pivot1 column1 entry0 difference" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "dimension eliminate row0 pivot1 column1 entry1 multiple := mul(4, 1)", "after": "dimension eliminate row0 pivot1 column1 entry1 multiple = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dimension eliminate row0 pivot1 column1 entry1 multiple，不以小數近似替代。", "en": "Recompute dimension eliminate row0 pivot1 column1 entry1 multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "dimension eliminate row0 pivot1 column1 entry1 multiple" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "dimension eliminate row0 pivot1 column1 entry1 difference := sub(4, 4)", "after": "dimension eliminate row0 pivot1 column1 entry1 difference = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。", "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dimension eliminate row0 pivot1 column1 entry1 difference，不以小數近似替代。", "en": "Recompute dimension eliminate row0 pivot1 column1 entry1 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "4", "4" ], "output": "0", "label": "dimension eliminate row0 pivot1 column1 entry1 difference" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "target row=0;old row=[1, 4];subtract factor=4 times normalized pivot row=[0, 1]", "after": "new target row=[1, 0];entry at pivot column1=0", "relation": "implies", "rule_id": "dimension_exact_row_subtraction_all_coordinates", "conditions": [], "explanation": { "zh-Hant": "對每格減去factor倍pivot列，才將此pivot欄消零；加回同倍pivot列可逆，前後欄係數方程等價。", "en": "Subtract the factor times the pivot row in every coordinate to clear this pivot column.Adding it back reverses the step,so column-coefficient equations remain equivalent." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "original column family count=2;ambient dimension=2;final exact matrix=[[1, 0], [0, 1]];pivot columns=[0, 1]", "after": "rank=2;independent=是;spans whole ambient=是;basis=是", "relation": "implies", "rule_id": "dimension_pivots_relations_and_whole_ambient_spanning", "conditions": [], "explanation": { "zh-Hant": "可逆列運算保留M·c=0的所有實係數解，所以pivot數為實秩，秩等欄數才獨立。左乘可逆E把原欄空間映為E作用後的空間，保維度及是否為全R^n，但不宣稱兩個實際欄空間逐點相同。秩等n才生成全ambient，兩者兼具才basis。", "en": "Invertible row operations preserve all real solutions of M*c=0.The pivot count is real rank,and independence requires rank equal to column count.Left multiplication by invertible E maps the original column span to its E-image,preserving dimension and whole-ambient spanning status,not literal equality of the two spans.Spanning requires rank=n;both conditions are needed for a basis." }, "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": "S014", "uses": [ "G001", "S013" ], "before": "same real ambient R^2;supplied family has2 columns;rank=2;independent=是;spanning=是", "after": "certified basis cardinality=2;dimension=2;finite_dimensional_basis_cardinality_invariance_theorem", "relation": "implies", "rule_id": "dimension_basis_cardinality_after_both_obligations", "conditions": [], "explanation": { "zh-Hant": "先有真正basis，才用有限維基底大小不變定理讀維度。這個實例沒有重新證明一般不變定理，也沒有產生Lean Evidence；不把相關或不生成的族長度當維度。", "en": "Only after a genuine basis is certified use the finite-dimensional basis-cardinality invariance theorem to read dimension.This instance neither reproves that universal theorem nor creates Lean Evidence.A dependent or nonspanning family's length is not its dimension certificate." }, "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": [ 2, 2, 2, 2, "finite_dimensional_basis_cardinality_invariance_theorem" ] }, "conclusion": { "canonical_value": "(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)", "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-20261102-r01-vector-space-dimension-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261102-r01-vector-space-dimension-foundations/questions.jsonl", "batch_sha256": "6e7b40b1d19d27dc7ae4d9148a072ef8c5445ec9215e01cd7c365cfbe97cc323", "original_line": 30, "original_line_sha256": "9317affdc8c18770b37de0d3ef0ea1f3c4668b610b9d5c55d1e8530ef9703a8b", "question_compact_sha256": "bb0dca38d07c3b357dc62a041119cd2b1dac6f96a8fd95ee782f3fe78cfa655b", "english_row_compact_sha256": "3a7d3c63d05bbea5342bd25f6cf224ea9ee8bb6047c8c163c0da6c1cc77b27b0", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261102-r01-vector-space-dimension-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261102-r01-vector-space-dimension-foundations.en.jsonl" }, "mko_bindings": { "mko-vector-space-dimension": { "sha256": "0d547787130d8aa91b60c2231053891922efe324656000b1e67b8cd7f2658dd8", "english_sha256": "4160447f99df20950bd70ea71b06a3a8a3bcf0dfe817b1224bc3e955bfad0ef0", "original_path": "public/data/mko/vector-space-dimension.json", "english_path": "public/data/i18n/editions/mko-vector-space-dimension.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "fe4c48735753bdd58c10db9b2ca3a2a3baeb31d0424c3fc09dea62317185ddae", "original_candidate_sha256": "e8126a58a5b66458247ae35c90e4b8c69a27bc3be6b1360ce449dc51e3376f82", "original_candidate_line_sha256": "e8126a58a5b66458247ae35c90e4b8c69a27bc3be6b1360ce449dc51e3376f82" }, "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-p33", "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": "d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e", "question_first_witness_sha256": "790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 14, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "e44941c55e3ceafddd363c7327cbf87520bae5440a76e98a83f41ec35687d946", "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." } }