# Definition:Linearly Independent/Set/Real Vector Space

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## Definition

Let $\tuple {\R^n, +, \cdot}_\R$ 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:

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

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

- 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE

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