Definition:Affinely Dependent

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

$(1) \quad x = \displaystyle \sum_{i = 1}^r \lambda_i x_i$
$(2) \quad \displaystyle \sum_{i = 1}^r \lambda_i = 1$

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 no element $x \in X$ is affinely dependent on $X \setminus \set x$


Also see

Sources