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
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.