Problem (comments after):
Let $\mathbb{F_q}$ be a finite field of cardinal $q$ and $\mathcal{E}$ an affine espace of dimension $n$ directed by the vector space $E$.
Show that:
$\#\left\{\text{vector subspaces of dimension } k \text{ of } E\right\} = \#\{\text{vector subspaces of dimension }k \text{ of }(\mathbb{F_q})^n\}$
$\#\{\text{affine subspaces of dimension }k \text{ of }\mathcal{E}\} = \#\{\text{affine subspaces of dimension }k \text{ of }(\mathbb{F_q})^n\}$
It is not the point to calculate those cardinals. I tried to create a bijection between the sets but nothing. I know, this must be really easy but I do not see how to solve it. Can someone help me here, please?