Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation, $B=\set{\vectorlist{u}{n}}$ is a basis for $U$ of size $n$, and $C$ is a basis for $V$ of size $m$. Then the matrix representation of $T$ relative to $B$ and $C$ is the $m\times n$ matrix, \begin{equation*} \matrixrep{T}{B}{C}=\left[ \left.\vectrep{C}{\lt{T}{\vect{u}_1}}\right| \left.\vectrep{C}{\lt{T}{\vect{u}_2}}\right| \left.\vectrep{C}{\lt{T}{\vect{u}_3}}\right| \ldots \left|\vectrep{C}{\lt{T}{\vect{u}_n}}\right. \right] \end{equation*}