{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $F$ be a field and let $A$ be an $m \\times n$ matrix over $F$.\nThen the dimension of the [[row space|069MCGX635]] of $A$ equals the dimension of the [[column space|03AJKXH1AV]] of $A$.\nEquivalently, $\\rank A^T = \\rank A$, where $A^T$ is the [[transpose|09DXWYRD43]].\n\n## Remarks\n\nOne proof uses row reduction: row operations keep the row space and the dimension of the column space.\nIn reduced row echelon form, both dimensions equal the number of pivots.\nAnother proof uses the [[annihilator|0ASHNYF7AG]]: the dual map has the transposed matrix and the same rank.\nOver $\\C$, also $\\rank A^* = \\rank A$.\n\n## Explanations\n\n### L1 human\n\nThe number of independent rows of a matrix always equals the number of independent columns.\nThis is surprising, because rows and columns live in different spaces.\nFor a $3 \\times 5$ matrix, it shows at once that the rank is at most $3$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "15A03"
    ],
    "name": "Row rank equals column rank",
    "role": "theorem",
    "sources": [
      {
        "locator": "Proofs that column rank = row rank",
        "url": "https://en.wikipedia.org/wiki/Rank_(linear_algebra)"
      }
    ],
    "tag": "0TZAG60ZYT",
    "uses": [
      "0EWQEYRVA8",
      "0EP73HK4C4",
      "0W809K6AW0"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0TZAG60ZYT/",
  "links": [
    {
      "field": "uses",
      "target": "0EWQEYRVA8"
    },
    {
      "field": "uses",
      "target": "0EP73HK4C4"
    },
    {
      "field": "uses",
      "target": "0W809K6AW0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0TZAG60ZYT.md",
  "msc": [
    "15A03"
  ],
  "name": "Row rank equals column rank",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0TZ/0TZAG60ZYT.md",
  "status": null,
  "tag": "0TZAG60ZYT",
  "uri": "m2://0TZAG60ZYT",
  "url": "https://math2.tamnd.com/wiki/row-rank-equals-column-rank/",
  "writing": {
    "tag": "0TZAG60ZYT",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}