Let $f$ be a quadratic form on a finite-dimensional real vector space $V$. Prove that there exists a diagonal basis for $f$, meaning a basis with respect to which the matrix of $f$ is diagonal.

Define the rank $r$ and signature $s$ of $f$ in terms of this matrix. Prove that $r$ and $s$ are independent of the choice of diagonal basis.

In terms of $r, s$, and the dimension $n$ of $V$, what is the greatest dimension of a subspace on which $f$ is zero?

Now let $f$ be the quadratic form on $\mathbb{R}^{3}$ given by $f(x, y, z)=x^{2}-y^{2}$. For which points $v$ in $\mathbb{R}^{3}$ is it the case that there is some diagonal basis for $f$ containing $v$ ?