I haven't been having problems finding explicit isomorphisms and proving them for practical vector spaces, I am however baffled by this theoretical isomorphism question.

QUESTION:

Let V and W be vector spaces with dim V = n, dim W = m

Let L : V -> W be a linear mapping

Let A be the matrix L with respect to bases B for V, and C for W

Define an explicit isomorphism from Range(L) to Col(A). Prove that your map is an isomorphism.

ATTEMPT:

Since A is the matrix L with respect to bases B and C, can I deduce that B and C are of the same size, and therefore, that V and W have the same dimension?

This would mean that V and W are one-to-one iff they are onto, and if I can use this to show that V and W are onto, then Range(L) = W.

But how can I show that V and W are one-to-one in order to do this, and what can I do with Col(A)?

Any help would be greatly appreciated, thanks.