Let $q$ be a nonsingular quadratic form on a finite-dimensional real vector space $V$. Prove that we may write $V=P \bigoplus N$, where the restriction of $q$ to $P$ is positive definite, the restriction of $q$ to $N$ is negative definite, and $q(x+y)=q(x)+q(y)$ for all $x \in P$ and $y \in N$. [No result on diagonalisability may be assumed.]

Show that the dimensions of $P$ and $N$ are independent of the choice of $P$ and $N$. Give an example to show that $P$ and $N$ are not themselves uniquely defined.

Find such a decomposition $V=P \bigoplus N$ when $V=\mathbb{R}^{3}$ and $q$ is the quadratic form $q((x, y, z))=x^{2}+2 y^{2}-2 x y-2 x z$