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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $(b_i)_{i \\in I}$ be a [[basis|0KCQ9Y3QKA]] of $V$.\nIf $S, T \\colon V \\to W$ are [[linear maps|0NRYHJPQG2]] with $S(b_i) = T(b_i)$ for all $i \\in I$, then $S = T$.\nMoreover, for each family $(w_i)_{i \\in I}$ of vectors of $W$, there is exactly one linear map $T \\colon V \\to W$ with $T(b_i) = w_i$ for all $i \\in I$.\n\n## Remarks\n\nThe map is $T(\\sum_i a_i b_i) = \\sum_i a_i w_i$.\nThe values on a basis can be chosen freely; the values on a merely spanning family cannot.\nThis statement is the reason why a linear map between finite-dimensional spaces is described by a matrix.\n\n## Explanations\n\n### L1 human\n\nA linear map is known as soon as we know where it sends the basis vectors.\nEvery other vector is a combination of basis vectors, and linearity tells us where it goes.\nConversely, we may send the basis vectors anywhere we like.\n",
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