# Definition:Linearly Dependent/Set/Real Vector Space

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## Definition

Let $\left({\R^n,+,\cdot}\right)_{\R}$ 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:

- $\displaystyle \exists \left\{{\lambda_k: 1 \le k \le n}\right\} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$

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

## Also see

- Linearly Independent Set: a subset of $\R^n$ which is not
**linearly dependent**.