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  "body": "\n## Definition\n\nLet $M = (E, \\mathcal{I})$ be a [[matroid|PX31ZNYS3Z]].\nA *basis* of $M$ is a set $B \\in \\mathcal{I}$ such that no set $A \\in \\mathcal{I}$ satisfies $B \\subsetneq A$.\n\n## Remarks\n\nBy the augmentation axiom, all bases of $M$ have the same number of elements.\nThe bases satisfy the exchange property: if $B_1, B_2$ are bases and $x \\in B_1 \\setminus B_2$, then there is $y \\in B_2 \\setminus B_1$ such that $(B_1 \\setminus \\set{x}) \\cup \\set{y}$ is a basis.\nConversely, a nonempty family of subsets of $E$ with this exchange property is the family of bases of exactly one matroid, whose independent sets are the subsets of bases.\n\nExample: let $U_{2,3}$ be the uniform matroid on $\\set{1, 2, 3}$, whose independent sets are the subsets with at most $2$ elements.\nThe set $\\set{1, 2}$ is a basis of $U_{2,3}$.\nIt is independent, and its only proper superset $\\set{1, 2, 3}$ has $3$ elements, so it is dependent.\n\nNon-example: in $U_{2,3}$ the set $\\set{1}$ is independent but not a basis, since $\\set{1, 2}$ is a larger independent set.\n\nIn the graphic matroid of a connected graph, the bases are the [[spanning trees|0S4ZEZM7KE]].\nIn the matroid of the columns of a matrix, the bases are the sets of columns that form a basis of the column space.\n",
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