How many projections $\pi: \mathbb{F}_3^2 \to \mathbb{F}_3^2 $ there?

Question

How many projections $\pi: \mathbb{F}_3^2 \to \mathbb{F}_3^2 $ there? Justify your answer!

If someone can give me a tipps on this question.D

Answer 1

For clarity here's a definition of projection. Given a vectorspace $V$, a projection is a linear map $\pi : V \rightarrow V$ such that $\pi \circ \pi = \pi$. In particular $\pi$ acts as the identity on its image $Im(\pi) \subseteq V$ so we let's take cases on the rank of $\pi$.

I'll do one of the cases and leave the rest for you. Suppose that $\pi$ has rank one. First we count the number of possible images for each projection, i.e. the number of $1$-dimensional subspaces of $\mathbb F_3^2$. This is equal to the number of non-zero vectors up to scalar which is $(9 - 1)/2 = 4$ i.e. $9-1$ non-zero vectors and $2$ non-zero scalars in $\mathbb F_3$.

Now we need to count the number of possible projections for each possible image. Take a basis for the image of a projection and extend it to a basis for the whole space. In this case our projection has rank one so our basis has one extra non-zero vector. The projection must send this basis vector inside the $1$-dimensional subspace and any vector is possible. Since the subspace has dimension one, there are $3$ possible choices for the image of this vector.

So there are $4 \cdot 3 = 12$ projections of rank one. How many projections are there of rank zero and two?