I am self-studying Hoffman and Kunze's book Linear Algebra. This is the exercise 13 from page 106.
Let $\mathbb{F}$ be a subfield of the field of complex numbers and let $V$ be any vector space over $\mathbb{F}.$ Suppose that $f$ and $g$ are linear functionals on $V$ such that the function $h$ defined by $h(v)=f(v)g(v)$ is also a linear functional on $V$. Prove that either $f=0$ or $g=0.$
I was able to show that $h=0$. Therefore $V=\operatorname {Ker} (f)\cup \operatorname{Ker}(g)$. I am assuming that $f\neq 0$ and I would like to show that $\operatorname {Ker} (f)\subset \operatorname {Ker} (g)$, but I wasn't able to acomplish that.
I would appreciate your help.