# Definition:Linearly Dependent/Set/Real Vector Space

## Definition

Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space.

Let $S \subseteq \R^n$.

Then $S$ is a linearly dependent set if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence.

That is, such that:

$\ds \exists \set {\lambda_k: 1 \le k \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$

where $\set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n} \subseteq S$, and such that at least one of $\lambda_k$ is not equal to $0$.