Suppose and are vector spaces over a field .
Now suppose is an isomorphism. The inverse of this is defined as . Prove that is an isomorphism, by verifying that it is linear and bijective.
So far, this is what I have available.
such that . This is for the surjectivity of .
Additionally, the aforementioned is unique, which is for the injectivity of .
As such, I can define .
This implies that and
My guess is that I should do the same things for , but just in reverse of .
What I mean is:
such that (surjectivity of ).
Since the aforementioned will be unique, this implies that is injective.
Both of these together would imply that is bijective.
I haven't gotten to the linearity part yet, as I'd like to check to make sure I'm on the right track, or if I'm assuming too much.