The following computations replicate the computations given in Example CM32, only using the basis $C$.
\begin{align*}
\vectrep{C}{\begin{bmatrix}3 & 7 \\ -2 & 4 \\ 0 & -3\end{bmatrix}}&=
\colvector{-9\\12\\-6\\7\\-2\\-1}
&
\vectrep{C}{\begin{bmatrix}-1 & 3 \\ 4 & 8 \\ -2 & 5\end{bmatrix}}&=
\colvector{-11\\34\\-4\\-1\\16\\5}\\
6\colvector{-9\\12\\-6\\7\\-2\\-1} + 2\colvector{-11\\34\\-4\\-1\\16\\5}
&=\colvector{-76\\140\\-44\\40\\20\\4}
&
\vectrepinv{C}{\colvector{-76\\140\\-44\\40\\20\\4}}&=
\begin{bmatrix}16 & 48 \\ -4 & 40 \\ -4 & -8\end{bmatrix}
\end{align*}