Express each of the following in terms of the standard basis $\varepsilon$, $[z_1]_{\beta}=\begin{bmatrix}7 \\ -1 \end{bmatrix}$ $[z_2]_{\beta}=\begin{bmatrix}-6 \\ 3 \end{bmatrix}$
The basis $\beta$ is $\begin{bmatrix}3 & 1 \\ 1 & 2 \end{bmatrix}$
Finding what transformation maps $\beta$ to $\varepsilon$ I got $$\begin{bmatrix}\frac{2}{5} & -\frac{1}{5} \\ -\frac{1}{5} & \frac{3}{5}\end{bmatrix}$$
Then finding what the coordinates are
$$\begin{bmatrix}\frac{2}{5} & -\frac{1}{5} \\ -\frac{1}{5} & \frac{3}{5}\end{bmatrix} \begin{bmatrix}x \\ y \end{bmatrix} = \begin{bmatrix}7 \\ -1 \end{bmatrix} $$ $$x=20 \space y= 5$$
and for the other set of coords, $\begin{bmatrix}-6 \\ 3 \end{bmatrix}$, I got $$x=-15 \space y =0$$
I just wanted to make sure I answered the question properly because it can get confusing.