Let $V$ be a vector space of dimension $n$ over a field $F$ and let $T:V\rightarrow V$ be a non-zero linear operator s.t. $T\cdot T=0$. Assume $\dim \mathcal{R}(T)=r$ and $W$ is a subspace of $V$ s.t. $V=\mathcal{N}(T)\oplus W$. Here $\mathcal{R}(T)$ and $\mathcal{N}(T)$ denote the range (or image) and the nullspace (or kernel) of $T$, respectively.

I want to show the following 

* $2r\leq n$
* If $\{\mathbf{w}\_1,\dots,\mathbf{w}\_r\}$ is a basis of $W$ then
$\{T(\mathbf{w}\_1),\dots,T(\mathbf{w}\_r)\}$ is a linearly independent
subset of $\mathcal{N}(T)$

Any ideas?