Definition:Linearly Independent/Set/Real Vector Space

Definition
Let $\left({\R^n,+,\cdot}\right)_{\R}$ be a real vector space.

Let $S \subseteq \R^n$.

Then $S$ is a linearly independent set if every sequence of distinct terms in $S$ is a linearly independent sequence.

That is, such that:
 * $\displaystyle \forall \left\{{\lambda_n}\right\} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$

where $S = \left\{{\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_k}\right\}$

Otherwise $S$ is a linearly dependent set.