Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation, $B$ is a basis for $U$, $C$ is a basis for $V$ and $\matrixrep{T}{B}{C}$ is the matrix representation of $T$ relative to $B$ and $C$. Then, for any $\vect{u}\in U$, \begin{equation*} \vectrep{C}{\lt{T}{\vect{u}}}=\matrixrep{T}{B}{C}\left(\vectrep{B}{\vect{u}}\right) \end{equation*} or equivalently \begin{equation*} \lt{T}{\vect{u}}=\vectrepinv{C}{\matrixrep{T}{B}{C}\left(\vectrep{B}{\vect{u}}\right)} \end{equation*}