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


Sources

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