# 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 of real vectors** if every finite sequence of distinct terms in $S$ is a linearly independent sequence.

That is, such that:

- $\displaystyle \forall \left\{{\lambda_k: 1 \le k \le n}\right\} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$

where $\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n$ are distinct elements of $S$.

## Also see

- Definition:Linearly Dependent Set of Real Vectors: a subset of $\R^n$ which is not
**linearly independent**.

## Sources

- For a video presentation of the contents of this page, visit the Khan Academy.