Transpose of Matrix as Linear Map

[chingkui]

What are the relations between a matrix H and its transpose H^T? I am not asking about the relations between the coefficients, I am asking the relations as linear maps (H: F^m->F^n; H^T: F^n->F^m). I am not sure exactly how I should pose the question actually, but I am thinking there is some deeper relations than between their coefficients, like for example, can we say something about the kernel and image of H^T if we know something about the kernel and image of H? What can we say about HH^T: F^n->F^n and H^TH: F^m->F^m?
(F is an arbitrary field)

 

[matt grime]

row rank=column rank, that pretty much tells you everything, and the fact that the images and kernels are related to the rows and cols of H's implies soemthing obvious since the row/cols of the transpose are the cols/rows of the original. (i haven't said which way round since it depends on what side you make your matrices act)

 

[mathwonk]

the transpose represents the pullback map on linear functions.