Affine Sets are Translates of Subspaces

Let $X$ be a vector space and let $\Omega\subseteq X$ be nonempty. Then $\Omega$ is affine if and only if, for any $\omega\in\Omega$,

is a linear subspace of $X$.

Equivalently, $\Omega$ is affine if and only if $\Omega=\omega+L$ for some $\omega\in X$ and linear subspace $L\subseteq X$.