Definition:Affinely Dependent

Definition

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 and only if there exist real numbers $\set {\lambda_i: 1 \le i \le r}$ such that:

$(1): \quad x = \ds \sum_{i \mathop = 1}^r \lambda_i x_i$

$(2): \quad \ds \sum_{i \mathop = 1}^r \lambda_i = 1$

Affinely Independent

Let $\R^n$ be the $n$-dimensional real Euclidean space.

Let $X = \set {x_1, \dots, x_r}$ be a finite subset of $\R^n$.

The subset $X$ is affinely independent if and only if no element $x \in X$ is affinely dependent on $X \setminus \set x$.

Also see

• Definition:Affinely Independent

Sources

• 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 3.$ Examples of Matroids