{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-rank-nullity-theorem-01",
  "mko_id": "mko-rank-nullity-theorem",
  "title": {
    "zh-Hant": "三個來源座標的秩零度",
    "en": "Rank and nullity for three source coordinates"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-rank-nullity-theorem-01",
    "source": {
      "mko_id": "mko-rank-nullity-theorem",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "e567ac132f3923949776cbd2866741a3564ba2c4d3e79d654abe99ba42b5ee0a",
      "english_sha256": "c3ab90f3622c1f66f461358b65aaa251f413745f3f1300b780355f32f83b5c59"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "T:R³→R²，A=[[1,2,3],[2,4,6]]有rank1、nullity2，3=1+2。",
      "en": "For T:R3 to R2, A=[[1,2,3],[2,4,6]] has rank 1 and nullity 2, with 3=1+2."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "A為2×3矩陣[[1,2,3],[2,4,6]]，T(x,y,z)=A(x,y,z)^T，來源R³、目標R²。",
          "en": "A is 2-by-3, [[1,2,3],[2,4,6]], T(x,y,z)=A(x,y,z)^T, source R3 and target R2."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "A\\in\\mathbb R^{2\\times3},\\quad T:\\mathbb R^3\\to\\mathbb R^2",
        "after_tex": "\\dim(\\operatorname{domain}T)=3",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "來源維度取column數3，不是row數2；三個標準單位向量獨立且生成R³。",
          "en": "The source dimension is the column count, 3, not the row count, 2; the three standard unit vectors are linearly independent and span R3."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "shape固定映射方向，不可用目標維度作左側。",
            "en": "Shape fixes map direction; target dimension cannot replace source dimension."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "\\begin{pmatrix}1&2&3\\\\2&4&6\\end{pmatrix}",
        "after_tex": "R_2\\leftarrow R_2-2R_1:\\quad\\begin{pmatrix}1&2&3\\\\0&0&0\\end{pmatrix}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "精確可逆消去給完整RREF，一個pivot與兩個自由變數。",
          "en": "Exact invertible elimination gives full RREF, one pivot and two free variables."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "下步另直接證column-space，避免只猜rank。",
            "en": "The next step directly certifies the column space rather than guessing rank."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "T(x,y,z)=(x+2y+3z)(1,2)",
        "after_tex": "\\operatorname{im}T=\\operatorname{span}\\{(1,2)\\},\\quad T(s,0,0)=s(1,2)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部輸出在這條線，每個實倍數亦有前像，證完整像。",
          "en": "All outputs lie on this line and every real multiple has a preimage, proving the full image."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "非零(1,2)獨立，是像的一項基底。",
            "en": "Nonzero (1,2) is independent and gives a one-vector image basis."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "T(x,y,z)=0",
        "after_tex": "x=-2y-3z,\\quad(x,y,z)=y(-2,1,0)+z(-3,0,1)",
        "relation": "equivalent",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "齊次方程給全部kernel元素的雙參數表示。",
          "en": "The homogeneous equation gives a two-parameter description of every kernel element."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "y、z為任意實數，不限有限網格。",
            "en": "y,z are arbitrary real numbers, not a finite grid."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04"
        ],
        "before_tex": "u=(-2,1,0),\\quad v=(-3,0,1)",
        "after_tex": "Au=Av=0,\\quad\\alpha u+\\beta v=0\\Rightarrow\\alpha=\\beta=0",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "反代為零且第二、三座標證獨立，連同S04生成給完整kernel basis。",
          "en": "Substitution gives zero and coordinates two and three prove independence; S04 supplies full spanning."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "不只數自由參數，而是已直接核基底資格。",
            "en": "The basis is directly qualified rather than merely counting parameters."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "\\mathcal B_{\\ker}=(u,v),\\quad\\mathcal B_{\\operatorname{im}}=((1,2))",
        "after_tex": "\\operatorname{nullity}T=2,\\quad\\operatorname{rank}T=1",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整基底給兩個維度，nullity不是核元素的個數。",
          "en": "Complete bases give both dimensions; nullity is not the kernel's number of elements."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "核的實參數給無限元素，但維度2。",
            "en": "Real kernel parameters give infinitely many elements but dimension 2."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S01",
          "S06"
        ],
        "before_tex": "\\dim\\mathbb R^3=3,\\quad\\operatorname{nullity}T=2,\\quad\\operatorname{rank}T=1",
        "after_tex": "3=2+1=\\dim\\ker T+\\dim\\operatorname{im}T",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "精確加總驗本矩陣的秩零度式，兩維度均有證書。",
          "en": "The exact sum verifies this matrix's rank-nullity identity, with certificates for both dimensions."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "不是數字巧合，S03與S05給完整像與核基底。",
            "en": "This is not a numerical coincidence; S03 and S05 give full image and kernel bases."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "(x,y,z)=s(1,0,0)+y(-2,1,0)+z(-3,0,1)",
        "after_tex": "s=x+2y+3z",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "來源由一個非核方向與兩個核方向唯一表示，具體解釋分拆。",
          "en": "The source uniquely splits into one nonkernel direction and two kernel directions, explaining the count."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "第二、三座標定y、z，第一座標再定s，故唯一。",
            "en": "Coordinates two and three fix y,z, then the first fixes s, proving uniqueness."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "T:R³→R²，A=[[1,2,3],[2,4,6]]有rank1、nullity2，3=1+2。",
      "en": "For T:R3 to R2, A=[[1,2,3],[2,4,6]] has rank 1 and nullity 2, with 3=1+2."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "2×3矩陣來源維度是2。",
          "en": "A 2-by-3 matrix has source dimension 2."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "輸入三座標，來源R³。",
          "en": "Its three-coordinate input belongs to R3."
        }
      },
      {
        "claim": {
          "zh-Hant": "核無限故nullity無限。",
          "en": "An infinite kernel means infinite nullity."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "完整兩向量基底給nullity2。",
          "en": "Its full two-vector basis gives nullity 2."
        }
      },
      {
        "claim": {
          "zh-Hant": "只測幾個核向量就證完整核。",
          "en": "A few tested kernel vectors prove the full kernel."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "需S04的全稱解與S05獨立。",
          "en": "The universal solution S04 and independence S05 are required."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設通常實數域與座標向量運算、矩陣乘法、線性獨立及生成的定義；所用消去均為可逆初等列運算。",
          "en": "Assume the real field, usual coordinate-vector arithmetic, matrix multiplication, and definitions of independence and span; elimination uses invertible elementary row operations."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "矩陣A為m×n並明示T:R^n→R^m。以exact RREF、kernel basis與column-space rank驗n=nullity+rank。nullity是kernel dimension而非解數；sampled vectors不足。",
          "en": "An explicit m-by-n matrix represents T:R^n→R^m. Exact RREF, a full kernel basis and column-space rank verify n=nullity+rank. Nullity is kernel dimension, not the number of solutions; sampled vectors do not establish the identity in general."
        },
        "source": {
          "mko_id": "mko-rank-nullity-theorem",
          "sha256": "e567ac132f3923949776cbd2866741a3564ba2c4d3e79d654abe99ba42b5ee0a",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "sum",
        "lhs": "2+1",
        "rhs": "3",
        "step": "S07"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "一個完整精確矩陣的kernel、image與來源維度證書，不取代一般證明。",
      "en": "Complete kernel, image and source-dimension certificates for one exact matrix, not a general proof."
    },
    "md_symbols": [
      [
        "u,v",
        "kernel基底",
        "kernel-basis vectors"
      ],
      [
        "s",
        "非核方向係數",
        "nonkernel coefficient"
      ]
    ],
    "md_error_controls": [
      {
        "title": "映射方向 / Map direction",
        "zh": "m×n矩陣來源Rⁿ，本例n=3。",
        "en": "An m-by-n matrix has source R^n, here n=3."
      },
      {
        "title": "維度非元素數 / Dimension not cardinality",
        "zh": "nullity計kernel basis大小，不計無限解數。",
        "en": "Nullity counts basis vectors, not infinitely many solutions."
      }
    ],
    "md_open_items": []
  },
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    "collection": "coverage-20261021",
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    "original_source_sha256": "8719bc547d83d493161e4c36002b5c8bd70cec444fe62f2391bb69b5b395450a",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
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    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
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}