For ordered bases $B=(b_j)$ and $C$ of the same space, $P_{C\leftarrow B}$ is the matrix whose column $j$ is $[b_j]_C$. It satisfies $[v]_C=P_{C\leftarrow B}[v]_B$. In a coordinate ambient space, if $S_B,S_C$ hold basis vectors as columns, then $P_{C\leftarrow B}=S_C^{-1}S_B$.
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