Question about hyperplane

Let V be a vector space over F field and $\displaystyle f: V \rightarrow \mathbb{F}$ is nonzero linear functional. So $\displaystyle f(\alpha u+\beta v) = \alpha f(u) + \beta f(v)$ for every $\displaystyle u,v \in V$ and $\displaystyle \alpha , \beta \in \mathbb{F}$. Show that for every $\displaystyle \lambda \in \mathbb{F}$ $\displaystyle H_\lambda =\{ x \in V | f(x) = \lambda \}$ is a hyperplane.

Thank you!

Re: Question about hyperplane

It's fairly straight forward isn't it? I would first look at $\displaystyle H_0= \{x\in V| f(x)= 0\}$. It's easy to show that $\displaystyle H_0$ is a subspace. Now, given any other $\displaystyle \lambda$ Let y be a single vector such that $\displaystyle f(y)= \lambda$ and show that for any x in $\displaystyle H_\lambda$, x- y is in $\displaystyle H_0$.