{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r01-linear-independence-basis-0099",
  "pilot_number": "R0021-11274",
  "number": 11274,
  "family_id": "qf-mod24-linbasis-dependence-witness-audit",
  "title": {
    "zh-Hant": "第 11274 題：線性獨立與基底",
    "en": "Question 11274: A Zero Combination Must Also Be Nontrivial to Witness Dependence"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r01-linear-independence-basis-0099",
    "batch_id": "batch-20261013-advance-20261024-r01-linear-independence-basis-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-independence-basis"
    ],
    "title_zh": "線性獨立與基底",
    "stem_zh": "明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,11]],\"coefficients\":[-1,-11,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], 是, 是, 是)",
      "explanation_zh": "依賴結構核對=([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], 是, 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "線性獨立與基底"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-linbasis-dependence-witness-audit",
      "seed": 202610241099,
      "parameters": {
        "vectors": [
          [
            1,
            0
          ],
          [
            0,
            1
          ],
          [
            1,
            11
          ]
        ],
        "coefficients": [
          -1,
          -11,
          1
        ],
        "domain": "Exact_finite_coordinate_basis",
        "operation": "linbasis-dependence-witness-audit",
        "semantic_template": "qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-11,1],\"vectors\":[[1,0],[0,1],[1,11]]}"
      }
    },
    "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、一般無限定理、Markov長期性、PDE適定性、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261024-r01-linear-independence-basis-0099",
    "batch_id": "batch-20261013-advance-20261024-r01-linear-independence-basis-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "615519781cb5066fdf229d1550df02f2e9e74323a22c55bde5f1b76d38e6cd48",
      "family_id": "qf-mod24-linbasis-dependence-witness-audit",
      "seed": 202610241099,
      "semantic_template": "qf-mod24-linbasis-dependence-witness-audit/{\"coefficients\":[-1,-11,1],\"vectors\":[[1,0],[0,1],[1,11]]}"
    },
    "fields": {
      "title": "A Zero Combination Must Also Be Nontrivial to Witness Dependence",
      "stem": "Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 11]], coefficients=[-1, -11, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).",
      "answer": {
        "kind": "expression",
        "value": "([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], Yes, Yes, Yes)",
        "explanation": "Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], Yes, Yes, Yes)."
      },
      "skills": [
        "A Zero Combination Must Also Be Nontrivial to Witness Dependence"
      ],
      "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": "明示有限座標向量族。操作=linbasis-dependence-witness-audit；資料={\"vectors\":[[1,0],[0,1],[1,11]],\"coefficients\":[-1,-11,1]}。請依 [vectors,coefficients,sum,nontrivial,zero_sum,dependence_witness] 欄序回答。",
        "en": "Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 11]], coefficients=[-1, -11, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "dependence witness:ordered vectors=[[1, 0], [0, 1], [1, 11]];aligned weights=[-1, -11, 1]",
      "after": "dimension=2;preserve duplicates and order",
      "relation": "implies",
      "rule_id": "ordered_vector_weight_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留向量順序、重複項與係數配對；同一座標加完所有向量的貢獻。",
        "en": "Preserve vector order,duplicates and coefficient pairing.Sum every vector's contribution within the same coordinate."
      },
      "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": "dependence witness coordinate0 term0 := mul(-1, 1)",
      "after": "dependence witness coordinate0 term0 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dependence witness coordinate0 term0，不以小數近似替代。",
          "en": "Recompute dependence witness coordinate0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "dependence witness coordinate0 term0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "dependence witness coordinate0 term1 := mul(-11, 0)",
      "after": "dependence witness coordinate0 term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dependence witness coordinate0 term1，不以小數近似替代。",
          "en": "Recompute dependence witness coordinate0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-11",
          "0"
        ],
        "output": "0",
        "label": "dependence witness coordinate0 term1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "dependence witness coordinate0 term2 := mul(1, 1)",
      "after": "dependence witness coordinate0 term2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dependence witness coordinate0 term2，不以小數近似替代。",
          "en": "Recompute dependence witness coordinate0 term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "dependence witness coordinate0 term2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "dependence witness coordinate0 sum := sum(-1, 0, 1)",
      "after": "dependence witness coordinate0 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dependence witness coordinate0 sum，不以小數近似替代。",
          "en": "Recompute dependence witness coordinate0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "0",
          "1"
        ],
        "output": "0",
        "label": "dependence witness coordinate0 sum"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "dependence witness coordinate1 term0 := mul(-1, 0)",
      "after": "dependence witness coordinate1 term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dependence witness coordinate1 term0，不以小數近似替代。",
          "en": "Recompute dependence witness coordinate1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "dependence witness coordinate1 term0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "dependence witness coordinate1 term1 := mul(-11, 1)",
      "after": "dependence witness coordinate1 term1 = -11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dependence witness coordinate1 term1，不以小數近似替代。",
          "en": "Recompute dependence witness coordinate1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-11",
          "1"
        ],
        "output": "-11",
        "label": "dependence witness coordinate1 term1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "dependence witness coordinate1 term2 := mul(1, 11)",
      "after": "dependence witness coordinate1 term2 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dependence witness coordinate1 term2，不以小數近似替代。",
          "en": "Recompute dependence witness coordinate1 term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "11"
        ],
        "output": "11",
        "label": "dependence witness coordinate1 term2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "dependence witness coordinate1 sum := sum(0, -11, 11)",
      "after": "dependence witness coordinate1 sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 dependence witness coordinate1 sum，不以小數近似替代。",
          "en": "Recompute dependence witness coordinate1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-11",
          "11"
        ],
        "output": "0",
        "label": "dependence witness coordinate1 sum"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "given coefficients=[-1, -11, 1];weighted sum=[0, 0]",
      "after": "nontrivial=是;zero_sum=是;dependence_witness=是",
      "relation": "implies",
      "rule_id": "dependence_requires_nontrivial_zero_combination",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時有非零係數且向量和全零才證相依。全零係數的零和是平凡解；指定見證失敗不證線性獨立。",
        "en": "A dependence witness needs both some nonzero coefficient and a zero vector sum.All-zero coefficients give only the trivial solution.A failed designated witness does not prove independence."
      },
      "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,
          0
        ],
        [
          0,
          1
        ],
        [
          1,
          11
        ]
      ],
      [
        -1,
        -11,
        1
      ],
      [
        0,
        0
      ],
      true,
      true,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], 是, 是, 是)",
    "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-20261013-advance-20261024-r01-linear-independence-basis-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations/questions.jsonl",
      "batch_sha256": "9c09bb587cfb555cea8fbb6e53978a87ed1a994c6c16dd9c77a76f0a37571632",
      "original_line": 99,
      "original_line_sha256": "6c96293416a4223f8684892cda6ef04128eaaa186f86f42771af41f982e2bc19",
      "question_compact_sha256": "615519781cb5066fdf229d1550df02f2e9e74323a22c55bde5f1b76d38e6cd48",
      "english_row_compact_sha256": "a60a734e20e99ae6300bf94d7b8ced1e7e8e00413fdbd7a5e63abd9191c0a63d",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations.en.jsonl"
    },
    "mko_bindings": {
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        "english_sha256": "48c43771b8f6734ece79626851e05a5dca9bb984732eb3f69cffe2c6e364a912",
        "original_path": "public/data/mko/linear-independence-basis.json",
        "english_path": "public/data/i18n/editions/mko-linear-independence-basis.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "18bbb89127e14ce3466c1a8730f1c3c6e22a8f59a9148433b2abc7f6ea274ee5",
    "original_candidate_sha256": "790e7e7daa8b3f6e5855694b2604abb503e263052ad857bdd73e8bfc79581f59",
    "original_candidate_line_sha256": "790e7e7daa8b3f6e5855694b2604abb503e263052ad857bdd73e8bfc79581f59"
  },
  "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-20261026-006-p22",
    "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": "47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258",
      "question_first_witness_sha256": "a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "1837168cd4ae8fd40b0995f5bdd2d44144cb289e4b5929b55c0e6220a6f88434",
      "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."
  }
}