Proposition 3

Given a family $(a_i)_{i \in I}$ of points of an affine space E, the set V of barycentres $\sum_{i \in I} \lambda_i a_i$ ((\lambdai) of finite support, $\sum_{i \in I} \lambda_i = 1$) is a linear variety of* E.

If the family $(a_i)$ is empty, then $V = \varnothing$ because of the condition $\sum_i \lambda_i = 1$. It may therefore be assumed that the family $(a_i)$ is non-empty and in this case the proposition is obvious, taking one of the $a_i$ as origin in E.

The variety V is obviously the smallest linear variety containing the $a_i$; it is said to be generated by the family $(a_i)$ and this family is called a generating system of V.

In the notation of Proposition 3, assuming the family $(a_i)$ is non-empty, for the expression for every point $x \in V$ in the form $x = \sum_i \lambda_i a_i$ to be unique, it is necessary and sufficient that, denoting an arbitrary index of I by $\kappa$, the family of vectors $a_i - a_\kappa$, where $i$ runs through the set of indices $\neq \kappa$, be free in T. Then the family $(a_v)_{v \in I}$ of points of E is said to be affinely free (or that its elements form an affinely free system, or are affinely independent) and that $\lambda_i$ is the barycentric coordinate of $x$ of index $i$ with respect to the affinely free family $(a_i)$.

A family $(a_i)_{i \in I}$ of points of E which is not affinely free is said to be affinely related.