# Definition:Linearly Dependent/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 dependent set** if and only if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence.

That is, such that:

- $\ds \exists \set {\lambda_k: 1 \le k \le n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$

where $a_1, a_2, \ldots, a_n$ are distinct elements of $S$, and where at least one of $\lambda_k$ is not equal to $0_R$.

### Linearly Dependent 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 dependent set** if and only if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence.

That is, such that:

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

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

### Linearly Dependent 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 dependent set** if and only if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence.

That is, such that:

- $\ds \exists \set {\lambda_k: 1 \le k \le n} \subseteq \C: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$

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

## Also see

- Definition:Linearly Independent Set: A subset $S \subseteq G$ which is not a
**linearly dependent set**.

## 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 - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**linearly dependent**and**independent**