Hi there!!!

Let b1,...,bm be a basis for V and let c1,...,cn be a basis for W.

One way is this:

define m times n linear maps f_{i,j}: V to W like this:

f_{i,j} (b_k) = cj if i=k and 0 otherwise.

Then one can show that the f_{i,j} are linearly independent in HOM(V,W),

and that each linear map in HOM(V,W) can be written as linear combination

of the f_{i,j}. That's pretty much it (you need to work out the details a bit).

The other two statements follow as a corollary

(recall that F is a one dimensional

vector space over itself).

Best,

ZD