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  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nAny two [[bases|0KCQ9Y3QKA]] of $V$ have the same cardinality.\n\n## Remarks\n\nFor finite bases, this follows from the [[Steinitz exchange lemma|08ZDRN6H41]].\nFor infinite bases, the proof uses cardinal arithmetic: each vector of one basis is a combination of finitely many vectors of the other basis.\nThe common cardinality is the [[dimension|03GSZ8RTZJ]] of $V$.\nThe theorem holds more generally for free modules over a commutative ring that is not the zero ring.\n\n## Explanations\n\n### L1 human\n\nHowever we choose a basis of a space, it always has the same number of vectors.\nSo \"the number of vectors in a basis\" is a property of the space itself.\nWe call this number the dimension.\n",
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