Well, you know that
and
right? Can you see where to go from here?
Well, you know that
and
right? Can you see where to go from here?
Any linear transformation from U to V maps a basis for U onto a basis for T(U).
If T is "one to one", then it maps the set of basis vectors of U onto an independent set and so maps U onto a subspace of V having exactly the same dimension as U. If not, then it maps U onto a subset of V having smaller dimension. "rank T" is the dimension of T(U).
"rank T" is the dimension of T(U) as above and, of course, T(U) is a subspace of V.rank (T) dim (V)