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  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nLet $u_1, \\dots, u_m \\in V$ be [[linearly independent|0CRMGPAH83]], and let $w_1, \\dots, w_n \\in V$ [[span|088RYKAWK9]] $V$.\nThen $m \\le n$.\nMoreover, after a renumbering of the $w_j$, the family $u_1, \\dots, u_m, w_{m+1}, \\dots, w_n$ spans $V$.\n\n## Remarks\n\nThe lemma follows by induction on $m$: each $u_i$ replaces one of the $w_j$ in a spanning family.\nA consequence is that any two finite bases of $V$ have the same number of elements.\nThe lemma is named after Ernst Steinitz.\n\n## Explanations\n\n### L1 human\n\nAn independent family can never have more vectors than a spanning family.\nWe can trade the vectors of the spanning family, one by one, for the independent vectors, and the family still spans.\nThis lemma is the reason why the dimension of a space is well defined.\n",
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