Let $k$ be a field, let $V$ and $W$ be finite-dimensional vector spaces over $k$, and let $T: V \to W$ be a $k$-linear map. Let $B=(b_1,\ldots,b_n)$ be an ordered basis of $V$, and let $C=(c_1,\ldots,c_m)$ be an ordered basis of $W$. If $[v]_B \in k^n$ denotes the column coordinate vector of $v$ with respect to $B$, if $[w]_C \in k^m$ denotes the column coordinate vector of $w$ with respect to $C$, and if $[T]_{C \leftarrow B} \in k^{m \times n}$ is the matrix whose $j$th column is $[T(b_j)]_C$, then for every $v \in V$,

\begin{align*} [T(v)]_C = [T]_{C \leftarrow B}[v]_B. \end{align*}