The exam question was:
$f$ is a linear map where $f^3=f,$ prove that it is diagonizable.
It was a linear algebra exam.
I asked the tutor if $f$ is a matrix since we only defined that a matrix can be diagonizable not any linear map, but he answered that we should prove it for linear maps in general. (Maybe iI missed something in the lectures?)
So $I$ wrote down that $f^3-f=0$ and that we can factor everything out like so: $$f(f+1)(f-1)=0 $$ And argued that this is the minimal polynomial for the function and since its linear factors are different it means that f is diagonizable.
However as far as I know this holds only for matrices? Is this approach correct/applicable for linear maps ? Was my tutor perhaps wrong? Is there something that I am missing?