Finding the number of bases for $ \mathbb{F}_3^2 $ and isomorphisms

Question

Find the number of bases for $ \mathbb{F}_3^2 $ and also the amount of Isomorphisms $ \mathbb{F}_3^2 \rightarrow \mathbb{F}_3^2 $

Here if $(v1,v2)$ is a basis, then $(v2, v1)$ is a different basis.

I have no idea how to approach this question. Could someone help me out?

Answer 1

it may be useful to think of the non-singular 2 x 2 matrices. the first column can be any of $3^2-1$ vectors. the second column can then be any of $3^2-3$ vectors. this result may obviously be generalized giving the result that $GL(2,F_{p^n})$ consists of: $$ \prod_{j=0}^{n-1} (p^n-p^j) $$ non-singular transformations