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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $S \\subseteq V$ be a [[spanning set|088RYKAWK9]] of $V$.\nThen there is a subset $B \\subseteq S$ that is a [[basis|0KCQ9Y3QKA]] of $V$.\n\n## Remarks\n\nFor finite $S$, remove from $S$, one at a time, a vector that is a linear combination of the other vectors.\nWhen no such vector remains, the set is linearly independent and still spans $V$.\nIn general, take a maximal linearly independent subset of $S$, by Zorn's lemma.\nFor example, $\\{(1, 0), (0, 1), (1, 1)\\}$ spans $\\R^2$ and contains the basis $\\{(1, 0), (1, 1)\\}$.\n\n## Explanations\n\n### L1 human\n\nIf a set of vectors reaches the whole space, then some part of it is a basis.\nWe throw away vectors that the others can already build.\nWhen nothing redundant remains, the remaining vectors form a basis.\n",
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