Let $V,W$ be two vector spaces over a field $F$ and let $f:V \to W$ be a linear mapping.
I have going over some facts about vector spaces (not necessarily finite dimensional) and I have been trying to apply the rank nullity theorem to derive a simple lemma. I have not been able to complete the proof and was wondering if I need to use something other then $\ker f + \mathrm{Im}\; f = \dim V$.
How do we show that for ever vector subspace $K \subset W$,
$ \dim(f^{-1}(K)) = \dim(K \cap \mathrm{Im}\; f) + \dim( \mathrm{Ker}\; f)$?