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