For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ be a vector subspace of $E$. We denote by $\varphi_F$ the restriction of $\varphi$ to $F^2$. We will say that $F$ is singular if and only if $\varphi_F$ is degenerate.
Show that $F$ is non-singular if and only if one of the following properties is verified:
\begin{itemize}
\item $F \cap F^\perp = \{0\}$;
\item $E = F \oplus F^\perp$;
\item $F^\perp$ is non-singular.
\end{itemize}