Consider an n x m matrix A. Find dim(im(A)) + dim(ker(A^T)), in terms of m and n.
I think this is equal to rank(A) + nullity(A^T). If so, I'm not sure what nullity(A^T) is equal to and how to the answer to this equation in terms of m and n.
Imagine reducing A to "row echelon" form by row reduction. You may have,say, k rows all 0 where k can be anything from 0 to n. In any case, the nullity of A is k. There will be, then n-k rows that are not all 0. The rank of A will be n-k. What do those add to?