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  "body": "\n## Statement\n\nLet $F$ be a field, and let $U$, $V$, $W$ be finite-dimensional vector spaces over $F$ with bases $A$, $B$, $C$.\nLet $S \\colon U \\to V$ and $T \\colon V \\to W$ be linear maps.\nThen\n$$[T \\circ S]_{C,A} = [T]_{C,B}\\, [S]_{B,A},$$\nwhere $[\\,\\cdot\\,]$ denotes the [[matrix of a linear map|0YQSGNYE3K]] and the right side is a [[matrix product|032Q9FB8MT]].\n\n## Remarks\n\nThis identity is the reason for the definition of matrix multiplication.\nIt implies that matrix multiplication is associative, because composition of maps is associative.\nFor $S$ invertible, it gives $[S^{-1}]_{A,B} = [S]_{B,A}^{-1}$.\n\n## Explanations\n\n### L1 human\n\nDoing one linear map after another corresponds to multiplying their matrices.\nThe order is the same as in $T \\circ S$: the matrix of the map that acts first is on the right.\nThis is why matrix multiplication has its unusual row-times-column rule.\n",
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