Given a symmetric bilinear map $B:\mathbb{R}^8 \times \mathbb{R}^8 \to \mathbb{R}$ and a subspace $U,\ \dim{U}=7$ with the signature $(5,3)$ then $rk(B|_U)\geq6$?
Meaning if we look at $B|_{U \times U}$ is it possible to find such subspace on which the rank will be less then 6 (i.e a counter example)? Or is this statement true?
I've tried constructing all kind of "counter examples" (for instance choosing 3 vectors that span a 3 dimensional sub-space $W$ then $\forall u,v B(u,v) =0 $ but then when I've tried completing it to a base of a $7$ dimensional space I've always ended up with rank 6 or seven). Is there some property I'm missing? If there is, I'd really like a direction...