$V=\mathbb{R^n}$
$q_1,q_2:\mathbb{R}^n\to \mathbb{R}$ are quadratic forms.
$L=\left\{v \in \mathbb{R^n}:q_1\left(v\right) \ge q_2\left(v\right)\right\}$ is a subspace of $\mathbb{R^n}$
Show that $\forall v \in \mathbb{R^n}, q_1(v) \ge q_2(v) \:or\: q_2(v) \ge q_1(v)$
My approach:
What I actually need to show is that if we mark $q=q_1 - q_2$, then $q$ is positive definite or $q$ is negative definite. I assume by contradiction that $q$ is neither of the above options, I somehow wants to get to a contradiction to the fact that $L$ is a subspace of $\mathbb{R^n}$
How should I proceed ?