Assume $f(x_1,x_2,\cdots,x_n)$ is a quadratic form on $\mathbb{R}^n$. The positive and negative inertia index is $p$ and $q$. It is evident that the vector $\alpha$ that satisfies $f(\alpha)=0$ form a subspace, how to prove that the dimension is no more than $\min\{n-p,n-q\}$?
My idea is that find a bases $\{\alpha_i\}$ so that the representation matrix is congruence normal form,and we can say that $\alpha_1+\alpha_{p+1},\ldots,\alpha{p}+\alpha_{2p},\alpha_{p+q+1},\ldots,\alpha_n$ are the base that we want. However, how to prove it is the maximum?