Exercise 9, § 9

Let $K$ be a finite field with $q$ elements and $V$ an $n$-dimensional vector space over $K$.

(a) Show that the set of sequences $(x_1, x_2, \ldots, x_m)$ of $m \leq n$ vectors of $V$ forming a free system has cardinal equal to

(argue by induction on $n$).

(b) Deduce from (a) that the cardinal of the set of $m$-dimensional linear varieties in an $n$-dimensional projective space over $K$ is equal to