Exercise 6

Let $U$ be a subspace of $\mathbb{R}^{3}$. Prove that the set $U^{\perp}=\text{$\{\hskip 1.00006pt\vec{v}\mid\text{$\vec{v}\mathbin{\smash{\bullet}}\vec{u}=0$ for all $\vec{u}\in U$}\hskip 1.00006pt\}$}$ of vectors that are perpendicular to each vector in $U$ is also subspace of $\mathbb{R}^{3}$. Does this hold if $U$ is not a subspace?