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  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.\nLet $B, B'$ be bases of $V$, and let $C, C'$ be bases of $W$.\nLet $P$ be the [[change of basis matrix|0MKRHZB3WS]] from $B$ to $B'$, and $Q$ the change of basis matrix from $C$ to $C'$.\nThen, for each linear map $T \\colon V \\to W$,\n$$[T]_{C',B'} = Q^{-1} [T]_{C,B} P,$$\nwhere $[T]$ denotes the [[matrix of $T$|0YQSGNYE3K]] in the given bases.\nIn particular, for an endomorphism $T$ of $V$ and $C = B$, $C' = B'$, one gets $[T]_{B'} = P^{-1} [T]_B P$.\n\n## Remarks\n\nThe formula follows from [[the matrix of a composition|05Q1B7MH0M]], applied to $T = \\id_W \\circ T \\circ \\id_V$.\nFor endomorphisms, the two matrices are [[similar|0E89HQATM8]].\nFor linear maps between two spaces, the two matrices are [[equivalent|0CMHSGTWXB]].\n\n## Explanations\n\n### L1 human\n\nWhen we change bases, the matrix of a linear map changes in a fixed way.\nWe translate the input from the new to the old coordinates, apply the old matrix, and translate the result back.\nThis gives the product $Q^{-1} A P$.\n",
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