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.