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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $L \\subseteq V$ be a [[linearly independent|0CRMGPAH83]] set.\nThen there is a [[basis|0KCQ9Y3QKA]] $B$ of $V$ with $L \\subseteq B$.\n\n## Remarks\n\nFor finite-dimensional $V$, add to $L$, one at a time, a vector that is not in the span of the current set.\nThis stops after finitely many steps.\nIn general, the proof uses Zorn's lemma.\nA stronger form: if $L \\subseteq S$, with $L$ independent and $S$ spanning, then a basis $B$ exists with $L \\subseteq B \\subseteq S$.\nFor example, $\\{(1, 1, 0)\\}$ extends to the basis $\\{(1, 1, 0), (0, 1, 0), (0, 0, 1)\\}$ of $\\R^3$.\n\n## Explanations\n\n### L1 human\n\nWe can always complete an independent set of vectors to a basis.\nWe keep adding vectors that point in new directions until the set reaches the whole space.\nThis shows that an independent set can never be larger than a basis.\n",
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