Let $V$ be a finite dimensional vector space over $\mathbb{R}$, and let $f,g$ be two non-zero linear functionals on $V$ such that whenever $f(x)\geq 0$, we also have $g(x)\geq 0$. Then, what can be say about the relation between $\ker (f)$ and $\ker (g)$? Is $f = \alpha g$ for some $\alpha >0$?
I could do it in the case, when $f(x)=0$ implies $g(x)=0$, but I don't know how proceed in the case inequalities. Any help would be higly appreciated.
Suppose $f(x) = 0$ did not imply $g(x) = 0$. Thus there is some $v\in V$ with $g(v) \ne 0$ while $f(v) = 0$. Replacing $v$ by $-v$ if necessary, we can assume $g(v) < 0$. Thus it is not true that whenever $f(x) \ge 0$, $g(x) \ge 0$.