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.