Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{U}{V}$ are linear transformations, $B$ is a basis of $U$ and $C$ is a basis of $V$. Then \begin{equation*} \matrixrep{T+S}{B}{C}=\matrixrep{T}{B}{C}+\matrixrep{S}{B}{C} \end{equation*}