{
  "aliases": [
    "invariance-of-dimension"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nAny two [[bases|0KCQ9Y3QKA]] of $V$ have the same cardinality.\n\n## Remarks\n\nFor finite bases, this follows from the [[Steinitz exchange lemma|08ZDRN6H41]].\nFor infinite bases, the proof uses cardinal arithmetic: each vector of one basis is a combination of finitely many vectors of the other basis.\nThe common cardinality is the [[dimension|03GSZ8RTZJ]] of $V$.\nThe theorem holds more generally for free modules over a commutative ring that is not the zero ring.\n\n## Explanations\n\n### L1 human\n\nHowever we choose a basis of a space, it always has the same number of vectors.\nSo \"the number of vectors in a basis\" is a property of the space itself.\nWe call this number the dimension.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "invariance-of-dimension"
    ],
    "kind": "statement",
    "msc": [
      "15A03"
    ],
    "name": "Dimension theorem for vector spaces",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q3527205",
      "wikipedia": "Dimension theorem for vector spaces"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Dimension_theorem_for_vector_spaces"
      }
    ],
    "tag": "0WK0Y8RH1M",
    "uses": [
      "0VTQTPQGAV"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0WK0Y8RH1M/",
  "links": [
    {
      "field": "uses",
      "target": "0VTQTPQGAV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0WK0Y8RH1M.md",
  "msc": [
    "15A03"
  ],
  "name": "Dimension theorem for vector spaces",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0WK/0WK0Y8RH1M.md",
  "status": null,
  "tag": "0WK0Y8RH1M",
  "uri": "m2://0WK0Y8RH1M",
  "url": "https://math2.tamnd.com/wiki/dimension-theorem-for-vector-spaces/",
  "writing": {
    "tag": "0WK0Y8RH1M",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}