Answer
The coordinate vector is;
$[x]_{\beta}=\begin{bmatrix}{c_1}\\{c_2}\\c_{3}\end{bmatrix} = \begin{bmatrix}-1\\-1\\3\end{bmatrix}$
Work Step by Step
To find the coordinate vector $[x]_{\beta}$ of $x$ relative to the given basis $\beta$
Basis matrix;
$\beta = \begin{bmatrix}1&-3&2\\{ -1}&{ 4}&-2\\3&9&4\end{bmatrix}$ and ${\bf{x}} = \begin{bmatrix}{ 8\\-9}\\6\end{bmatrix}$
Finding $[x]_{\beta}$
$\beta {\left[ x \right]_\beta } = x$
$\begin{bmatrix}1&-3&2\\-1&{ 4}&-2\\3&9&4\end{bmatrix}\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\end{bmatrix}=\begin{bmatrix}8\\-9\\6\end{bmatrix}$
Forming augmented matrix and row reducing;
$\begin{bmatrix}1&-3&2&8\\-1&{4}&-2&-9\\3&9&4&6\end{bmatrix}\sim\begin{bmatrix}1&0&0&-1\\0&1&0&-1\\0&0&1&3\end{bmatrix}$
The coordinate vector is;
$[x]_{\beta}=\begin{bmatrix}{c_1}\\{c_2}\\c_{3}\end{bmatrix} = \begin{bmatrix}-1\\-1\\3\end{bmatrix}$
