While studying for my linear algebra test I came across the following problem:
Let $f: \mathbb{V} \to \mathbb{W}$ be a linear transformation and let $S$ and $T$ subspaces of $\mathbb{V}$ such that $S \cap T = \text{Ker}(f)$. Show that
$ \dim(S+T) = \dim(\text{Ker}(f)) + \dim(f(S)) + \dim(f(T)). $
I don't really know how to approach this. There's a good change I have to use that $\dim(S+T) = \dim(S) + \dim(T) - \dim(S \cap T)$, but all I can do with that is transform this into $\dim(f(S)) + \dim(f(T)) = \dim(S) + \dim(T)$. I think the problem here is that I don't know anything about $f(S)$ or $f(T)$. How would one go about proving this?