{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-combination-span",
  "semantic_family_id": "sf-linear-combination-span",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "線性組合與生成"
  },
  "summary": {
    "zh-Hant": "線性組合是有限個向量的純量加權和；生成空間收集全部有限線性組合。"
  },
  "statement": {
    "zh-Hant": "給實向量空間中的向量族，有限線性組合為有限個實係數加權和。該向量族的span是所有有限長度線性組合的集合，不包含未定義的無限級數。"
  },
  "formula": {
    "tex": "S(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>S</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "S",
        "glyph": "S"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e9998b7064b919043980363a57cdbb8287102e9031deff16b0178fae41d44ba3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用固定維度整數座標向量與整數或精確有理係數。"
    },
    {
      "id": "a2",
      "zh-Hant": "每題只核明示有限生成族與候選witness；不以一個固定arity窮盡抽象span定義。"
    }
  ],
  "symbols": [
    {
      "token": "S",
      "role": "span_operator",
      "role_zh": "生成空間算子",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "finite_generator_family",
      "role_zh": "有限生成族",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "線性組合使用向量加法與實純量乘法。"
    },
    {
      "id": "mko-system-linear-equations",
      "title_zh": "二元一次聯立方程組",
      "reason_zh": "座標membership可化為線性方程。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "加權和要求所有向量維度一致且係數數量匹配。",
      "給定係數可直接重建目標，形成membership witness。",
      "R²兩個獨立生成向量可用非零行列式求唯一座標。"
    ],
    "common_misconception_zh": "把無限和叫有限線性組合，或只看維度就宣稱某向量在span中。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核有限座標witness；一般span仍量化所有有限組合。",
    "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": "加入線性獨立、基底與維度。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-combination-span.md 整理；2026-10-13 脈衝 R02（使用者手動接續 2026-10-23 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"
  }
}