I am trying to solve the following exercise yet I do not know from which angle to attack it. Therefore, I need a hint or two to get me started on both "if" and "only if" implications.
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 $\langle, \rangle: V \times V \rightarrow F$ 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 $\langle X, v_1 \rangle, \dots, \langle X, v_n \rangle$ are jointly independent if and only if the vectors $v_1,\dots,v_n$ are linearly independent.
