Exercise 5, § 9

For a non-empty subset V of an affine space E over a field K with at least three elements to be a linear variety, it is necessary and sufficient that for every ordered pair $(x, y)$ of distinct points of V the line $D_{xy}$ passing through $x$ and $y$ be entirely contained in V. If K has two elements, for V to be a linear variety, it is necessary and sufficient that the barycentre of any three points of V belong to V.