Summing up, how can one use linear functionals, transpose matrices, row and column rank equality and annihilators to prove the rank-nullity theorem?
While studying linear algebra, I'm trying to get the precise relation between the following concepts: given two vector spaces $V$ and $W$, I can form the dual spaces $V^*$ and $W^*$ by taking all the linear functionals on $V$ and $W$. The basis of each one of those spaces lifts to the duals (though this demonstration I'm still going to carry on, but I have the idea how to do so). If I have a linear transformation $T: V \to W$, I have a natural way of defining a transformation $T^*: W^* \to V^*$, by composition: given a functional $g \in W^*$, I define $f \in V^*$ by: $$f(\alpha) = g(T\alpha)$$
This way of defining $T^*$ is familiar to me, as it is similar a construction commonly done on modules over a ring $R$. Now, if I represent $T$ as a matrix on a choice of basis for $V$ and $W$, and $T^*$ as the matrix on the basis given by the lifting, then $T^*$ will be the transpose of $T$ (the exchange of rows and columns of $T$). I don't see clearly why this happens. Furthermore, the rank of $T$ is equal to the rank of $T^*$, what proves that the column rank of a matrix equals it's row rank. Given that the rank is the dimension of the image subspace, I think that this can be show by carrying out the image of $V$ on $W$ to the duals.
I also read that there is a relation between the duals and the kernel of a linear transformation (the annihilator (?)), but that isn't very clear to me either. Using that and the facts above, one can prove the rank-nullity theorem. The reason I'm asking this is that I was given a proof of the rank-nullity theorem without using the linear functionals, and that seemed to depend only on $T$ having the same row and column rank (which was proved by smart manipulation of some equations on vectors). That proof didn't gave me a satisfactory intuition on the rank-nullity theorem, specially when $V \neq W$. I believe that the proof through linear functionals will be more enlightening.
EDIT: I was following the treatment given on Hoffman's Linear Algebra, chapters 3.5 through 3.7 (linear functionals, annihilators and transposes), if that's of interest.