Answer
The coordinate vector is;
$\begin{bmatrix}{c_1}\\{c_2}\end{bmatrix} = \begin{bmatrix}8\\{ - 5}\end{bmatrix}$
Work Step by Step
To find the coordinate vector $[x]_{\beta}$ of $x$ relative to the given basis $\beta$
Basis matrix
$\beta = \begin{bmatrix}1&2\\{ - 3}&{ - 5}\end{bmatrix}$ and ${\bf{x}} = \begin{bmatrix}{ - 2}\\1\end{bmatrix}$
finding $[x]_{\beta}$
$\beta {\left[ x \right]_\beta } = x$
$\begin{bmatrix}1&2\\{ -3}&{-5}\end{bmatrix}\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}=\begin{bmatrix}-2\\1\end{bmatrix}$
forming augmented matrix and row reducing;
$\begin{bmatrix}1&2&-2\\{ -3}&{-5}&1\end{bmatrix}\sim\begin{bmatrix}1&0&8\\0&1&-5\end{bmatrix}$
The coordinate vector is;
$\begin{bmatrix}{c_1}\\{c_2}\end{bmatrix} = \begin{bmatrix}8\\{ - 5}\end{bmatrix}$