My Algebra books gives the following exercises:
Let f be a reflexive $(f(u,w)=0 $\Leftrightarrow$ f(w,u)=0)$ bilinear form on a vector space $V$ $(\dim V=n)$.
Let $U \subset V$ and
(1) $f$ is symmetric and non-degenerate on $U$
(2) $f$ is alternating on $U^\perp$ .
Show the existence of $U$.
I don't really see how I can prove this exercise. I tried to define a random subspace $U'$, on which $f$ is symmetric and non-degenerate. Then there should exist (I guess?) a subspace $W$ of the orthogonal complement of $U'$ with $U'\oplus W=X$ (direct sum). But is this even helpful?
Any help is appreciated :)