Rank-nullity with annihilators

Question

I've just seen this being used:

$\dim(\text{ann}(\text{im}(\phi))=\dim(W)-\dim(\text{im}(\phi))$

and I can't figure out why it works. I've looked through all the relevant definitions and I still don't see why it works. The justification for it being used is 'eqns-solns', so I'm not really sure where that's come from.

In this case, $\phi:V \to W$ is a linear map and $\phi^T :W^* \to V^*$ is its transpose. We also know that $\ker(\phi^T)=\dim(\text{ann}(\text{im}(\phi))$.

Answer 1

Thats rank nullity theorem applied to $\phi^T$, using the fact that $dimW^*=dimW$ and that the $rank(\phi^T)=rank(\phi)$.