For bases $B=(b_1,\ldots,b_n)$ of $V$ and $C=(c_1,\ldots,c_m)$ of $W$, define $a_{ij}$ by $T(b_j)=\sum_i a_{ij}c_i$. Then $[T]_{C\leftarrow B}=(a_{ij})$ has shape $m\times n$. Its $j$th column is the coordinate vector of $T(b_j)$.
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