If $T^{2} = 0$, what can you say about the relation of the range of $T$ to the null space of $T$?

Question

Let $V$ be a vector space over the field $\textbf{F}$ and $T$ a linear operator on $V$. If $T^{2} = 0$, what can you say about the relation of the range of $T$ to the null space of $T$? Give an example of a linear operator on $\textbf{R}^{2}$ such that $T^{2} = 0$, but $T\neq 0$.

MY ATTEMPT (EDIT)

Let us suppose that $\alpha\in V$. According to the proposed relation, if $T\alpha \in R$, then $T^{2}\alpha = T(T\alpha) = 0$, thus $T\alpha\in N$, that is to say, $R\subset N$.

As an example, we can consider the following operator: $T(1,0) = (0,1)$ and $T(0,1) = (0,0)$. Consequently, $T^{2}(1,0) = T(0,1) = (0,0)$ and $T^{2}(0,1) = T(0,0) = (0,0)$, which means that $T(x,y)\neq(0,0)$, but $T^{2}(x,y)=(0,0)$.

Answer 1

$T^2=0\implies \operatorname{Range}T\subset\operatorname{ker}T$.

For the second part, consider $\begin{pmatrix}0&1\\0&0\end{pmatrix}$.

Answer 2

Take an operator $T$ such that it maps the first basis to zero and the second basis to the first one, $T(e_1)=0, T(e_2)=e_1$. It is easy to check that $T\neq 0$ but $T^2=0$.