\begin{definition}[Definition:Affinely Dependent]
Let $\R^n$ be the $n$-dimensional real Euclidean space.
Let $S = \set {x_1, \dots, x_r}$ be a finite subset of $\R^n$.
An element $x \in \R^n$ is '''affinely dependent''' on $S$ if there exist real numbers $\set {\lambda_i: 1 \le i \le r}$ such that:
:$(1): \quad x = \ds \sum_{i \mathop = 1}^r \lambda_i x_i$ 
:$(2): \quad \ds \sum_{i \mathop = 1}^r \lambda_i = 1$
\end{definition}