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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \\dots, b_n)$ be a basis of $V$.\nThen the map\n$$V \\to F^n, \\qquad v \\mapsto [v]_B,$$\nthat sends each vector to its [[coordinate vector|04CYQHN6MC]] in $B$, is a [[linear isomorphism|0B8T10CE7N]].\nIts inverse sends $(a_1, \\dots, a_n)$ to $a_1 b_1 + \\cdots + a_n b_n$.\n\n## Remarks\n\nThe isomorphism depends on the choice of the basis.\nIt sends $b_i$ to the standard basis vector $e_i$.\nThrough this map, each statement about $F^n$ gives a statement about $V$.\n\n## Explanations\n\n### L1 human\n\nChoosing a basis turns every vector into a list of $n$ numbers.\nThis translation respects addition and scaling, and it loses no information.\nSo each $n$-dimensional space is a copy of $F^n$.\n",
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