Hallo Martin
$$ \begin{pmatrix}-7 &-9 \\5 &-6 \end{pmatrix} \begin{pmatrix}x \\y \end{pmatrix} =\begin{pmatrix}-7x-9y \\ 5x-6y\end{pmatrix}\\f\begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix}-7x-9y\\5x-6y\end{pmatrix}\\f(\vec{e_1}) = \\ f\begin{pmatrix}\frac{-2}{\sqrt{8}} \\\frac{2}{\sqrt{8}} \end{pmatrix}=\begin{pmatrix}-7\cdot \frac{-2}{\sqrt{8}} - 9\cdot \frac{2}{\sqrt{8}} \\ 5\cdot \frac{-2}{\sqrt{8}}-6\cdot \frac{2}{\sqrt{8}}\end{pmatrix}=\begin{pmatrix}-\sqrt{2} \\ -\frac{11}{\sqrt{2}}\end{pmatrix} = a'_{11} \begin{pmatrix}\frac{-2}{\sqrt{8}} \\\frac{2}{\sqrt{8}} \end{pmatrix} +a'_{21} \begin{pmatrix}\frac{-2}{\sqrt{8}} \\\frac{-2}{\sqrt{8}} \end{pmatrix} \\f(\vec{e_2}) = \\ f\begin{pmatrix}\frac{-2}{\sqrt{8}} \\\frac{-2}{\sqrt{8}} \end{pmatrix}=\begin{pmatrix}-7\cdot \frac{-2}{\sqrt{8}} - 9\cdot \frac{-2}{\sqrt{8}} \\ 5\cdot \frac{-2}{\sqrt{8}}-6\cdot \frac{-2}{\sqrt{8}}\end{pmatrix}=\begin{pmatrix} 8\sqrt{2}\\\frac{1}{\sqrt{2}} \end{pmatrix} = a'_{1,2}\begin{pmatrix}\frac{-2}{\sqrt{8}} \\\frac{2}{\sqrt{8}} \end{pmatrix} +a'_{22} \begin{pmatrix}\frac{-2}{\sqrt{8}} \\\frac{-2}{\sqrt{8}} \end{pmatrix} \\\Rightarrow a_{1,1}=-\frac{9}{2}, \ a_{1,2}=-\frac{15}{2}, \ a_{2,1}=\frac{13}{2}, \ a_{2,2}=-\frac{17}{2}, \ $$
Grüße