From the definition of the quadratic form follows that for every $ x \in \mathbb{R}^n,\; c \in \mathbb{R}$ holds $ q(cx) = c^2q(x) $ ( I understand this)
From this you can to prove that (I don't know how), for example, $q$ is positively definite if and only if $ q(x)>0 $ for each $ x \in \mathbb{R}^n $ satisfying $ \| x \|=1 $.
Does anyone please have idea how to prove that or at least how to start?