For ordered bases $B=(b_j)$ of $V$ and $C$ of $W$, define $[T]_{C\leftarrow B}$ by taking column $j$ to be $[T(b_j)]_C$. Then $[T(v)]_C=[T]_{C\leftarrow B}[v]_B$. Both bases and the conversion direction are part of the representation.
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