{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $m$ be a positive integer. Suppose a set of vectors of $\\mathbb{R}^m$ [[spans|088RYKAWK9]] the whole of [[the space|0XPR1E85W0]] $\\mathbb{R}^m$. Then the set has at least $m$ vectors.\n\n## Remarks\n\nEquivalently, a set with fewer than $m$ vectors in $\\mathbb{R}^m$ never spans $\\mathbb{R}^m$.\n\nIn matrix terms, the matrix whose columns are the vectors has $m$ rows. To span, it needs a pivot position in every row. Each column holds at most one pivot, so there must be at least $m$ columns.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "about": [
      "088RYKAWK9"
    ],
    "kind": "statement",
    "msc": [
      "15A03"
    ],
    "name": "A spanning set of R^m has at least m vectors",
    "role": "proposition",
    "sources": [
      {
        "locator": "src/section2-3.xml:1387",
        "url": "https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-3.xml#L1387"
      }
    ],
    "tag": "D23BB74RYS",
    "uses": [
      "088RYKAWK9",
      "0XPR1E85W0"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/D23BB74RYS/",
  "links": [
    {
      "field": "about",
      "target": "088RYKAWK9"
    },
    {
      "field": "uses",
      "target": "088RYKAWK9"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/D23BB74RYS.md",
  "msc": [
    "15A03"
  ],
  "name": "A spanning set of R^m has at least m vectors",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D23/D23BB74RYS.md",
  "status": null,
  "tag": "D23BB74RYS",
  "uri": "m2://D23BB74RYS",
  "url": "https://math2.tamnd.com/wiki/a-spanning-set-of-r-m-has-at-least-m-vectors/",
  "writing": {
    "tag": "D23BB74RYS",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}