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  "body": "\n## Definition\n\nLet $M$ be a [[matroid|PX31ZNYS3Z]] on a finite ground set $E$, and let $\\mathcal{B}$ be its set of [[bases|QTSQR7130G]].\nThe *dual matroid* $M^*$ is the matroid on $E$ whose set of bases is\n$$\\mathcal{B}^* = \\set{E \\setminus B : B \\in \\mathcal{B}}.$$\n\n## Remarks\n\nIt is a theorem that $\\mathcal{B}^*$ satisfies the basis exchange property, so $M^*$ is indeed a matroid.\nIts independent sets are the subsets of $E$ disjoint from some basis of $M$.\nIts [[rank function|SYRCTH853H]] is $r^*(A) = \\abs{A} - r(E) + r(E \\setminus A)$, where $r$ is the rank function of $M$, and $(M^*)^* = M$.\nThe bases of $M^*$ are the cobases of $M$, and the circuits of $M^*$ are the cocircuits of $M$.\n\nExample: for the uniform matroid $U_{k,n}$, whose bases are the $k$-element subsets of $\\set{1, \\dots, n}$, the dual has as bases the $(n-k)$-element subsets, so $U_{k,n}^* = U_{n-k,n}$.\n\nNon-example: the complements of the independent sets of $M$ are in general not the independent sets of $M^*$.\nFor $M = U_{1,2}$ on $\\set{1, 2}$, the complement of $\\emptyset$ is $\\set{1, 2}$, which is not independent in $M^* = U_{1,2}$.\n",
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