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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nA *basis* of $V$ is a family $(b_i)_{i \\in I}$ of vectors of $V$ that is [[linearly independent|0CRMGPAH83]] and [[spans|088RYKAWK9]] $V$.\n\n## Remarks\n\nA family $(b_i)_{i \\in I}$ is a basis if and only if each $v \\in V$ is equal to $\\sum_{i \\in I} a_i b_i$ for exactly one finitely supported family of scalars $(a_i)_{i \\in I}$.\nThe scalars $a_i$ are the [[coordinates|04CYQHN6MC]] of $v$ in the basis.\nThe [[standard basis|0V7ZD6P7RK]] $e_1, \\dots, e_n$ is a basis of $F^n$; the vectors $(1, 1)$ and $(1, -1)$ form another basis of $\\R^2$.\n[[Every vector space has a basis|00MS9VMQ9Z]], and all bases of a space have the same cardinality.\n\n## Explanations\n\n### L1 human\n\nA basis is a set of directions that is just right: enough to reach every vector, and none of them redundant.\nOnce we fix a basis, each vector gets a unique list of coordinates.\nA vector space has many bases, but all of them have the same number of vectors.\n",
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