{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-quotient-vector-space",
  "semantic_family_id": "sf-quotient-vector-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "商向量空間"
  },
  "summary": {
    "zh-Hant": "V依子空間W的cosets形成商向量空間，運算須與代表元選擇無關。"
  },
  "statement": {
    "zh-Hant": "v與u代表同一coset恰當v-u在W；quotient元素是cosets而非唯一representatives。核[v]+[u]=[v+u]與lambda[v]=[lambda v]的代表元獨立性。有限維時dim(V/W)=dimV-dimW，零coset是W。"
  },
  "formula": {
    "tex": "Q(V,W)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>Q</mi><mo>(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "Q",
        "glyph": "Q"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        },
        {
          "type": "symbol",
          "name": "W",
          "glyph": "W"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3a04061588ab8a264cc835d904f8a5b5ea35e926dc362934e4c47829547ffd02"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "V=R^n且W由完整exact independent basis表示。"
    },
    {
      "id": "a2",
      "zh-Hant": "membership、rank與dimension certificate一律使用精確有理運算。"
    }
  ],
  "symbols": [
    {
      "token": "V/W",
      "role": "quotient_vector_space",
      "role_zh": "商向量空間",
      "scope": "representative_formula"
    },
    {
      "token": "W",
      "role": "vector_subspace",
      "role_zh": "被商子空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-vector-subspace",
      "title_zh": "實向量子空間",
      "reason_zh": "W的資格。"
    },
    {
      "id": "mko-quotient-set",
      "title_zh": "商集",
      "reason_zh": "coset與等價類載體。"
    },
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "逐coset運算。"
    },
    {
      "id": "mko-equivalence-class",
      "title_zh": "等價類",
      "reason_zh": "代表元不唯一。"
    },
    {
      "id": "mko-vector-space-dimension",
      "title_zh": "向量空間維度",
      "reason_zh": "有限維dimension formula。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "[v]=[u] iff v-u in W。",
      "well-defined要求換代表元不改結果coset。",
      "dim公式限有限維；零元素是整個W coset。"
    ],
    "common_misconception_zh": "把代表向量當quotient元素唯一名稱，或把商空間當數值除法。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限維實座標商空間；拓撲商與Banach quotient另列。",
    "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first isomorphism theorem與exact sequences。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-quotient-vector-space.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-11-06 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}