I have a proof ,written by someone, of : $A\in M_n$ is diagonalizable $\iff$ the minimal polynomial has distinct roots. The proof says: $A$ is diagonalizable $\iff$ A has n linearly independent eigenvectors $\iff$ A has n distinct eigenvalues...etc
I doubt the last $\iff$ , it is true for "$\Leftarrow$", but I don't see the forward direction! Was it ture? If not, could someone continue the proof?