Definition 2

Given an affine space $E$, a subset $V$ of $E$ is called an affine linear variety (or simply a linear variety or an affine subset of $E$) if, for every family $(x_i)_{i \in I}$ of points of $V$ and every family $(\lambda_i)_{i \in I}$ of elements of $K$ of finite support such that $\sum_{i \in I} \lambda_i = 1$, the barycentre $\sum_{i \in I} \lambda_i x_i$ belongs to $V$.

It amounts to the same to say that the condition of Definition 2 holds for every finite family of points of $V$.

The empty set is a linear variety; every intersection of linear varieties is a linear variety.

Let $V$ be a non-empty subset of $E$ and $a$ a point of $V$; the relation

means that $x$ is a barycentre $\sum_{i=1}^n \lambda_i x_i + \left(1 - \sum_{i=1}^n \lambda_i\right)a$ of the family consisting of the $x_i$ and $a$. Therefore: