Nullity of a composite linear transformation This is the problem. I know that I am going to use the rank-nullity theorem, but I am still figuring how to start and prove this:
Suppose $V$ and $W$ are finite-dimensional vector spaces over a field $F$. Let $f:V \rightarrow W$ and $g:W \rightarrow U$ be linear transformations over $F$. Prove that 
nullity ($g\circ f$) $\leq$ nullity $(g)$ + nullity $(f)$
Thank you

Hint:
Using Null $\,f=\dim\ker f\;$ , rank$\,f=\dim\text{Im}\,f\;$ , we have:
$$g\circ f:V\to U\implies \dim\ker (g\circ f)+\dim\text{Im}(g\circ f)=\dim V=\dim\ker f+\dim\text{Im}\,f\le$$
$$\le\dim\ker f+\dim W=\dim\ker f+\dim\ker g+\dim\text{Im}\,g$$
Take it from here.
