{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-general-linear-map",
  "semantic_family_id": "sf-general-linear-map",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實向量空間間線性映射"
  },
  "summary": {
    "zh-Hant": "一般實向量空間間映射若對所有向量與實純量保持線性組合，則為線性映射。"
  },
  "statement": {
    "zh-Hant": "T:V到W為實線性映射，若對所有u,v及實數alpha,beta有T(alpha u+beta v)=alpha T(u)+beta T(v)。有限維矩陣公式提供全稱座標證書；帶非零平移的仿射映射通常不線性。"
  },
  "formula": {
    "tex": "T(v)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>T</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "T",
        "glyph": "T"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "v",
          "glyph": "v"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "38b9407fa0d8a9b9b638c13531bfa6de70f73d0c71ca29e800173cf39cf9e735"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫明示R^n到R^m的矩陣或矩陣加bias公式。"
    },
    {
      "id": "a2",
      "zh-Hant": "矩陣分配律提供全稱證書；有限輸入抽樣不證任意黑箱映射線性。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "real_linear_map",
      "role_zh": "實線性映射",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "real_vector_space",
      "role_zh": "來源空間",
      "scope": "representative_formula"
    },
    {
      "token": "W",
      "role": "real_vector_space",
      "role_zh": "目標空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "來源與目標皆為實向量空間。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "映射",
      "reason_zh": "T須為明示總映射。"
    },
    {
      "id": "mko-linear-combination-span",
      "title_zh": "有限線性組合與生成",
      "reason_zh": "線性映射保持有限線性組合。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "單一等式同時涵蓋加法與齊次性。",
      "矩陣乘法在有限維座標中給線性映射。",
      "Ax+b只有b=0時才通過線性資格。"
    ],
    "common_misconception_zh": "把任意仿射映射、只通過幾個樣本的黑箱，或只滿足加法的函數稱為線性。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "座標矩陣證書只涵蓋明示有限維公式；不外推任意黑箱，也不宣稱有界、連續、單射或滿射。",
    "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": "在賦範空間上加入有界線性算子。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-general-linear-map.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-10-27 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"
  }
}