What does a quadratic form need to satisfy for its isotropic set (Not sure about the word :) ) to be a subspace?

Question

Suppose I have a quadratic form $q : \textbf{R}^n \rightarrow R$, and let $U = \{\textbf{x} \in \textbf{R}^n\ |\ q(\textbf{x}) = 0\}$. Is $U$ always a subspace of $\textbf{R}^n$? If not, what does $q$ need to satisfy for $U$ to be a subspace of $\textbf{R}^n$? (In terms of sign, rank, and stuff like that). Thanks!

Answer 1

The nullspace of a quadratic form (the set of vectors $v$ such that for every $w$ we have $q(v,w)=0$) is always a subspace.

If $q$ is (positive or negative) semidefinite, then you can use Lagrange's diagonalisation theorem to conclude that the nullspace consists of exactly the vectors satisfying $q(v)=0$ (so the set of those vectors is a subspace).

On the other hand, if it is not semidefinite, then (also by Lagrange's theorem) we can find a two-dimensional subspace $\Pi$ such that the restriction of $q$ to it has signature $(1,1,0)$, and in this case, the set of vectors satisfying $q(v)=0$ is not a subspace (its intersection with $\Pi$ is the union of two lines through the origin).