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  "body": "\n## Definition\n\nLet $E$ be a [[finite set|070R982G4B]], called the ground set.\nA *matroid* on $E$ is a pair $M = (E, \\mathcal{I})$, where $\\mathcal{I}$ is a family of subsets of $E$, called independent sets, such that:\n\n1. $\\emptyset \\in \\mathcal{I}$.\n2. If $A \\in \\mathcal{I}$ and $B \\subseteq A$, then $B \\in \\mathcal{I}$.\n3. If $A, B \\in \\mathcal{I}$ and $\\abs{A} \u003e \\abs{B}$, then there is $x \\in A \\setminus B$ with $B \\cup \\set{x} \\in \\mathcal{I}$.\n\nA subset of $E$ that is not in $\\mathcal{I}$ is *dependent*.\n\n## Remarks\n\nExample: let $E$ be the set of columns of a matrix over a field, and let $\\mathcal{I}$ be the sets of columns that are [[linearly independent|0CRMGPAH83]].\nClauses 1 and 2 are clear.\nFor clause 3, if $\\abs{A} \u003e \\abs{B}$, then $A$ spans a space of larger dimension than $B$.\nSo some $x \\in A$ lies outside the span of $B$, and $B \\cup \\set{x}$ is independent.\nSimilarly, the edge sets of the forests of a graph form a matroid, the graphic matroid.\nThe uniform matroid $U_{k,n}$ has ground set $\\set{1, \\dots, n}$ and independent sets all subsets with at most $k$ elements.\n\nNon-example: on $E = \\set{1, 2, 3}$, let $\\mathcal{I} = \\set{\\emptyset, \\set{1}, \\set{2}, \\set{3}, \\set{1, 2}}$.\nClauses 1 and 2 hold, but clause 3 fails for $A = \\set{1, 2}$ and $B = \\set{3}$: neither $\\set{1, 3}$ nor $\\set{2, 3}$ is independent.\n\nClause 3 is called the augmentation or exchange axiom.\nEquivalent definitions use the [[bases|QTSQR7130G]], the [[rank function|SYRCTH853H]] or the circuits (minimal dependent sets).\nSome authors allow an infinite ground set with extra conditions.\n",
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