This is a seemingly simple question but I'm having some problems with it:

Suppose $q:V \to R$ is a positive semidefinite quadratic form.

How can I show that $$L_0=\{v\in V | q(v)=0\}$$ is a subspace of dimension $n-\rho$ where $\rho$ is $q$ s rank?

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Let $\varphi$ be the symmetric bilinear form such that $q(v)=\varphi(v,v)$. If $q(v_1)=q(v_2)=0$, then $q(v_1+v_2) = 2 \varphi(v_1,v_2)$ and $q(v_1-v_2) = -2 \varphi(v_1,v_2)$ and both have to be nonnegative, so $\varphi(v_1,v_2)=0$ and so $v_1 + v_2 \in L_0$. It is obvious that $0 \in L_0$ and that $L_0$ is stable under scalar multiplication, so $L_0$ is indeed a vector space.