Answer
The coordinate vector is;
$[x]_{\beta}=\begin{bmatrix}{c_1}\\{c_2}\\c_{3}\end{bmatrix} = \begin{bmatrix}-2\\0\\5\end{bmatrix}$
Work Step by Step
Find the coordinate vector $[x]_{\beta}$ of $x$ relative to the given basis $\beta$
Basis matrix;
$\beta = \begin{bmatrix}1&2&1\\{ 0}&{1}&-1\\3&8&2\end{bmatrix}$ and ${\bf{x}} = \begin{bmatrix}{ 3\\-5}\\4\end{bmatrix}$
Finding $[x]_{\beta}$
$\beta {\left[ x \right]_\beta } = x$
$\begin{bmatrix}1&2&1\\{ 0}&{1}&-1\\3&8&2\end{bmatrix}\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\end{bmatrix}=\begin{bmatrix}3\\-5\\4\end{bmatrix}$
Forming augmented matrix and row reducing;
$\begin{bmatrix}1&2&1&3\\{ 0}&{1}&-1&-5\\3&8&2&4\end{bmatrix}\sim\begin{bmatrix}1&0&0&-2\\0&1&0&0\\0&0&1&5\end{bmatrix}$
The coordinate vector is;
$[x]_{\beta}=\begin{bmatrix}{c_1}\\{c_2}\\c_{3}\end{bmatrix} = \begin{bmatrix}-2\\0\\5\end{bmatrix}$
