# Definition:Linearly Independent/Set

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## Definition

Let $G$ be an abelian group whose identity is $e$.

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.

Let $S \subseteq G$.

Then $S$ is a **linearly independent set (over $R$)** if and only if every finite sequence of distinct terms in $S$ is a linearly independent sequence.

That is, such that:

- $\ds \forall \sequence {\lambda_n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$

where $a_1, a_2, \ldots, a_k$ are distinct elements of $S$.

### Linearly Independent Set on a Real Vector Space

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

Let $S \subseteq \R^n$.

Then $S$ is a **linearly independent set of real vectors** if and only if every finite sequence of distinct terms in $S$ is a linearly independent sequence.

That is, such that:

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

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

### Linearly Independent Set on a Complex Vector Space

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

Let $S \subseteq \C^n$.

Then $S$ is a **linearly independent set of complex vectors** if and only if every finite sequence of distinct terms in $S$ is a linearly independent sequence.

That is, such that:

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

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

## Also see

- Definition:Linearly Dependent Set: a subset of $G$ which is not
**linearly independent**.

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 4$. Vector Spaces - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 33$. Definition of a Basis - 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): $\text{A}.2$: Linear algebra and determinants: Definition $\text{A}.4$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**linearly dependent**and**independent**