TITLE: Problem related with a linear transformation and its rank,nullity...
QUESTION [5 upvotes]: I am stuck on the following problem:

Suppose $W$ be  a vector space of dimension $m \geq 2.$ Let $F \colon W \to W$ be a linear transformation such that $F^{n+1}=0$ and $F^n \ne 0$ for some $n \ge 1.$ Then I have to check which of  the following statement(s) is/are true?
1.$\text{trace}(F) \ne 0 $
2.$\text{Rank}(F^n) \le \text{Nullity}(F^n)$
3.$n=m$
4.$F$ is diagonalizable.

I do not know how to progress with the problem. Any kind of hints with some explanation will be helpful.

REPLY [2 votes]: $2:True:$ $FF^n=0\\\implies\text{Im}(F^n)\subset\text{Kernel}(F)\subset\text{Kernel}(F^n)\\\implies\text{Rank}(F^n)\le\text{Nullity}(F^n)$
$1,3,4: False:$ $F:\mathbb R^3\to\mathbb R^3$ given by $F=\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}$