Let $Q$ be a quadratic form on a real vector space $V$ of dimension $n$. Prove that there is a basis $\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}$ with respect to which $Q$ is given by the formula

$$Q\left(\sum_{i=1}^{n} x_{i} \mathbf{e}_{i}\right)=x_{1}^{2}+\ldots+x_{p}^{2}-x_{p+1}^{2}-\ldots-x_{p+q}^{2}$$

Prove that the numbers $p$ and $q$ are uniquely determined by the form $Q$. By means of an example, show that the subspaces $\left\langle\mathbf{e}_{1}, \ldots, \mathbf{e}_{p}\right\rangle$ and $\left\langle\mathbf{e}_{p+1}, \ldots, \mathbf{e}_{p+q}\right\rangle$ need not be uniquely determined by $Q$.