If $q$ is a quadratic form on a finite-dimensional real vector space $V$, what is the associated symmetric bilinear form $\varphi(\cdot, \cdot)$ ? Prove that there is a basis for $V$ with respect to which the matrix for $\varphi$ is diagonal. What is the signature of $q$ ?

If $R \leqslant V$ is a subspace such that $\varphi(r, v)=0$ for all $r \in R$ and all $v \in V$, show that $q^{\prime}(v+R)=q(v)$ defines a quadratic form on the quotient vector space $V / R$. Show that the signature of $q^{\prime}$ is the same as that of $q$.

If $e, f \in V$ are vectors such that $\varphi(e, e)=0$ and $\varphi(e, f)=1$, show that there is a direct sum decomposition $V=\operatorname{span}(e, f) \oplus U$ such that the signature of $\left.q\right|_{U}$ is the same as that of $q$.