Let V be a vector space over F field and $f: V \rightarrow \mathbb{F}$ is nonzero linear functional. So $f(\alpha u+\beta v) = \alpha f(u) + \beta f(v)$ for every $u,v \in V$ and $\alpha , \beta \in \mathbb{F}$. Show that for every $\lambda \in \mathbb{F}$ $H_\lambda =\{ x \in V | f(x) = \lambda \}$ is a hyperplane.
It's fairly straight forward isn't it? I would first look at $H_0= \{x\in V| f(x)= 0\}$. It's easy to show that $H_0$ is a subspace. Now, given any other $\lambda$ Let y be a single vector such that $f(y)= \lambda$ and show that for any x in $H_\lambda$, x- y is in $H_0$.