Q:

If $T:V \to V$ and $T$ is surjective, can i say that $T$ is diagonaizable?

If $T:V \to V$ and $T$ is surjective, can i say that $T$ is diagonaizable?
I can surley say that $T$ is invertible, but what is the relation to diagonaizability, if there is one?

A:

No, you cannot, e.g. the map $\mathbb{R}^2 \to \mathbb{R}^2$ given by
$$
\begin{pmatrix} 1& 1 \\ 0 & 1 \end{pmatrix}
$$
is surjective but not diagonalizable.

