Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation, $B$ is a basis for $U$ and $C$ is a basis for $V$. Then $T$ is an invertible linear transformation if and only if the matrix representation of $T$ relative to $B$ and $C$, $\matrixrep{T}{B}{C}$ is an invertible matrix. When $T$ is invertible, \begin{equation*} \matrixrep{\ltinverse{T}}{C}{B}=\inverse{\left(\matrixrep{T}{B}{C}\right)} \end{equation*}