{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-transformation",
  "semantic_family_id": "sf-linear-transformation",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "矩陣表示的實平面線性映射"
  },
  "summary": {
    "zh-Hant": "先在 R²→R² 的矩陣切片驗證加法與純量相容，不把帶非零平移的仿射式叫線性映射。"
  },
  "statement": {
    "zh-Hant": "映射 T:R²→R² 為線性映射，當且僅當對所有向量 u、v 與實純量 c，T(u+v)=T(u)+T(v)、T(cv)=cT(v)。本輪 T(v)=Mv，M 是2×2實矩陣，向量以列向量約定。仿射式 F(v)=Mv+b 在全實平面上為線性映射，當且僅當 b 是零向量。"
  },
  "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": "明示全實平面的定義域、陪域、列向量與2×2矩陣，不泛化成任意維度或任意映射的有限抽樣判定。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫矩陣、向量及純量採安全整數，以精確運算求值。"
    },
    {
      "id": "a3",
      "zh-Hant": "零矩陣與零向量合法；線性式不等於必須可逆，也不要求斜率或行列式非零。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "real_plane_linear_map",
      "role_zh": "矩陣表示的實平面線性映射",
      "scope": "representative_formula"
    },
    {
      "token": "v",
      "role": "column_vector",
      "role_zh": "兩分量實列向量",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-matrix",
      "title_zh": "有限矩陣的尺寸、索引與轉置",
      "reason_zh": "固定2×2矩陣與索引。"
    },
    {
      "id": "mko-vector",
      "title_zh": "平面向量",
      "reason_zh": "兩分量向量的加法及純量運算。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "總函數的輸入與輸出宣告。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "Mv 的每個分量是該行與 v 的對應乘積和，不是逐項乘後仍輸出矩陣。加法相容與純量相容是所有合法輸入的條件，不由少數測點替任意函數背書。",
      "矩陣表示給出一個明示的線性模型；本輪題目計算等式兩側的實際值，不將這些有限計算另包裝成普遍 proof。",
      "若 b≠0，F(0)=b≠0，已不能線性；若 b=0 則回到 Mv。仿射判定不靠改無關的 M 元素湊變種；可逆、單射與等距另有條件。"
    ],
    "common_misconception_zh": "把圖上直線或仿射式的非零截距當成線性，或由有限測點判定任意函數線性。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "2×2整數切片可精確核對；全實數線性性與矩陣／LinearMap 橋未新增形式 Evidence。",
    "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": "橋接 Matrix.mulVecLin 與標準基底的 R² 加法、純量相容；不是微分線性化工作。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-transformation.md 整理；2026-10-06 脈衝 R02（使用者手動接續 2026-10-07 工作序列；實際日期仍為 2026-10-06），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-06"
  }
}