I have a matrix:

$$
A=\begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}
$$

which satisfies: $A\left( e\_{1}\right)=e\_{1},A\left( e\_{2}\right)=ae\_{1}+e\_{2}, a\neq 0$

I need compute all powers $A^{n}, n\in \mathbb{Z}$ and find their matrices.

$$
A^n=
\begin{bmatrix}1 & na \\ 0 & 1\end{bmatrix}
$$

Is it true?

Is there a basis of $V$ (if $V$ is a 2-dimensional real vector space with basis $\left\{e\_{1},e\_{2}\right\}$) so that the matrix of $A$ is diagonal?