Hint and solution correct - but possibly not suitable for first years

Hello,

The matrix in Hint 2 is correct (my bad :-( ). However, it appears that it would be hard for a student to figure out how to derive the matrix in Hint 2. Perhaps an easier way (assuming they have covered change of basis matrices) is to simply state that under the change of bases (conjugation by an appropriate rotation), the two matrices become

${\begin{pmatrix}1&0\\0&0\end{pmatrix}}$

and

${\begin{pmatrix}0&0\\0&1\end{pmatrix}}$

Best, Brian