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  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nThen $V$ has a [[basis|0KCQ9Y3QKA]].\n\n## Remarks\n\nFor a space with a finite spanning set, the proof is elementary: remove vectors from the spanning set until it is linearly independent.\nIn general, the proof uses Zorn's lemma: a maximal linearly independent subset of $V$ is a basis.\nThe statement for all vector spaces is equivalent to the axiom of choice, over set theory without choice.\nFor example, the axiom of choice gives a basis of $\\R$ as a vector space over $\\Q$, a Hamel basis.\n\n## Explanations\n\n### L1 human\n\nEvery vector space has a set of building blocks that is independent and reaches every vector.\nFor finite-dimensional spaces, we can find one by hand.\nFor very large spaces, we must use the axiom of choice, and we may not be able to write the basis down.\n",
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