{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7Z0QZP0FC0"
    },
    {
      "field": "defines",
      "target": "SYVAV7V0B3"
    }
  ],
  "body": "\n## Summary\n\nThe dual of a [[matroid|PX31ZNYS3Z]] $M$ is the matroid $M^*$ on the same ground set whose [[bases|QTSQR7130G]] are the complements of the bases of $M$.\nDuality is an involution, $M^{**} = M$, and it exchanges deletion and contraction of [[minors|4WCYBSH1P6]].\nFor a planar graph, the dual of its graphic matroid is the graphic matroid of the [[dual graph|1QHTD1DS5J]].\nSo matroid duality extends planar graph duality to all matroids.\n\n## Explanations\n\n### L1 human\n\nIn a matroid, a basis is a largest independent set.\nTaking the complement of every basis gives the bases of a new matroid, the dual.\nFor a connected planar graph, the edges outside a spanning tree correspond to a spanning tree of the dual graph, and this is the same idea.\nFor the uniform matroid in which bases are the $k$-element subsets of an $n$-element set, the dual has the $(n-k)$-element subsets as bases.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "05B35"
    ],
    "name": "Dual matroid",
    "same_as": {
      "wikidata": "Q5310216",
      "wikipedia": "Dual matroid"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dual_matroid"
      }
    ],
    "tag": "6QF83CFK2G"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/6QF83CFK2G/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/6QF83CFK2G.md",
  "msc": [
    "05B35"
  ],
  "name": "Dual matroid",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6QF/6QF83CFK2G.md",
  "status": null,
  "tag": "6QF83CFK2G",
  "uri": "m2://6QF83CFK2G",
  "url": "https://math2.tamnd.com/wiki/dual-matroid/",
  "writing": {
    "tag": "6QF83CFK2G",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}