I am currently working on task 28 from the following sheet: https://agag-lassueur.math.rptu.de/~lassueur/en/teaching/LIESS16/LASS16/Blatt8.pdf
I have already completed parts 1 and 3. For part 2, I suspect that isomorphism is given if and only if $\lambda=\mu$ holds, but I am stuck in the proof. Do you have any tips for me? Any help is greatly appreciated!
If $\phi_\lambda$ and $\phi_\mu$ are isomorphic $\phi_\lambda(y)=\begin{pmatrix}\lambda\\&\lambda+1\end{pmatrix}$ and $\phi_\mu(y)=\begin{pmatrix}\mu\\&\mu+1\end{pmatrix}$ are conjugate matrices, so their traces are equal. Thus $2\lambda+1=2\mu+1$, i.e., $\lambda=\mu$.