forgive my math-english.
Let $L=\{v∈\Bbb R^n|q(v)≥0\}$
My question is , why $L$ is a vector space if $q(x)$ is definite positive/negative or semi positive/negative ? Can you prove it?
And why $L$ is not a vector space if $q(x)$ is none of the above?
Thank you!
If the form is definite or semi-definite , $L$ is either $V$ or $\{0\}$. (You can confirm this yourself.)
If $q$ is not semidefinite in any way, move to the bilinear form for $q$, and find vectors $x,y$ that are orthogonal relative to the bilinear form and having opposite nonzero signs in the form, and scale such that $q(x)=2$ and $q(y)=-1$. The guarantees that $q(y-x)=1$, and then $x+(y-x)\notin L$ despite the fact $x$ and $y-x$ are. Good luck with the details.