Exercise 2.22 (Probabilistic independence and linear independence)

Let $V$ be a finite-dimensional vector space over a finite field $F$, and let $X$ be a random variable drawn uniformly at random from $V$. Let $⟨,⟩:V\times V\to F$ be a non-degenerate bilinear form on $V$, and let $v_1,\dots ,v_n$ be non-zero vectors in $V$. Show that the random variables $⟨X,v_1⟩,\dots ,⟨X,v_n⟩$ are jointly independent if and only if the vectors $v_1,\dots ,v_n$ are linearly independent.