The last part in particular sounds strange.
What does it mean that the bases are inverse to each other, and why does it imply that is a basis for ?
I would do like this: We need to prove that is a basis for . So we need to check two things, namely that the vectors in the proposed basis are linearly independent and that they span all of .
Let for some scalars . Apply linearity of to get
Remember that is an isomorphism, so what do you know about vectors that map to 0?
To see that the basis spans all of , take some vector , and choose a vector , such that (why does this vector exist?).
You know that is a basis for , so you can write as a linear combination of these basis vectors. Try applying to this relation and see what happens.