{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-independence-basis",
  "semantic_family_id": "sf-linear-independence-basis",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "線性獨立與基底"
  },
  "summary": {
    "zh-Hant": "線性獨立排除非平凡零和關係；基底同時線性獨立並生成已宣告空間。"
  },
  "statement": {
    "zh-Hant": "有限向量族線性獨立，若係數加權和為零只可能所有係數皆零。向量族是空間V的基底，必須同時線性獨立且span等於V；兩條責任不可互相代替。"
  },
  "formula": {
    "tex": "B(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>B</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f26780f0baf666ea25e0c82fb0dacd0beb8058d9e9c6d2cb7ca7b9523a8f443b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫限整數座標有限向量族；rank以精確有理消去計算。"
    },
    {
      "id": "a2",
      "zh-Hant": "行列式與座標唯一性只用於有序R²方形基底，不外推非方形族。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "basis_record",
      "role_zh": "基底核對紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "declared_vector_space",
      "role_zh": "宣告向量空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-linear-combination-span",
      "title_zh": "線性組合與生成",
      "reason_zh": "基底必須生成宣告空間。"
    },
    {
      "id": "mko-system-linear-equations",
      "title_zh": "二元一次聯立方程組",
      "reason_zh": "線性獨立與座標可化為齊次或非齊次系統。"
    },
    {
      "id": "mko-determinant",
      "title_zh": "行列式",
      "reason_zh": "R²方形族可用非零行列式核獨立與生成。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "獨立性核零向量表示是否只有平凡係數。",
      "基底是獨立性與生成性的合取。",
      "有序基底成立後，每個R²目標才有唯一座標。"
    ],
    "common_misconception_zh": "只因向量生成空間或只因彼此獨立，就單獨宣稱它們是基底。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "rank與determinant結論限明示有限座標模型。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入維度、基底變換與線性映射矩陣。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-independence-basis.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-24 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}