Prove that an isomorphism produces a basis of W

I think I may have the answer for this one, but I could use a second opinion.

Suppose and are vector spaces over a field .

Now suppose is an isomorphism and is a basis of . Prove that is a basis of .

Here is what I have so far:

Let be a basis for where

.

Then we consider the map

where .

Since the bases are inverse to each other, we can conclude that is a basis of .

I don't know if I have any major holes in my proof, so if you see any, please let me know.