I've seen the notion of an affine subspace defined differently as follows:
$S \subset \mathbb R^3$, non-empty, is an affine subspace if $(1-t)u + tv \in S$ whenever $u,v \in S$.
$S$ is an affine subspace if $S=V+x_0$, where $V\subset \mathbb R^3$ is a subspace and $x_0 \in \mathbb R^3$.
Are these two definitions equivalent ?
I see that if $S$ is an affine subspace with respect to the second definition, then it is also with respect to the first definition. However, I cannot prove the other way around ?