linear algebra?

Question

if V and W are finite dimensional vector spaces, prove that L(V,W) is isomorphic to L(W,V)

dont look for a mapping L(V,W) -->L(W,V)

Answer 1

Suppose V and W are n and m-dimensional, respectively. Choose bases for V and W -- then each linear transformation may be uniquely specified by the images of the n basis vectors of V, and the image of each vector may be uniquely specified by its m components in the basis of W. So it follows that each linear transformation in L(V, W) may be uniquely specified by nm components, thus L(V, W) is of dimension nm. However, by the same argument, L(W, V) is of dimension mn = nm, so L(V, W) and L(W, V) have the same dimension, and are thus isomorphic. Q.E.D.