  Given $k$ sets $\mathcal{A}_i \subseteq \mathbb{F}_q^d$ and a non-degenerate
bilinear form $B$ in $\mathbb{F}_q^d$. We consider the system of $l \leq
\binom{k}{2}$ bilinear equations \[ B (\tmmathbf{a}_i, \tmmathbf{a}_j) =
\lambda_{i j}, \tmmathbf{a}_i \in \mathcal{A}_i, i = 1, ..., k. \] We show that
the system is solvable for any $\lambda_{i j} \in \mathbb{F}_q^{*}$, $1 \leq
i,j \leq k$, given that the restricted sets $\mathcal{A}_i$'s are sufficiently
large.
