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 $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module.

Let $S \subseteq G$.

Then $S$ is a linearly dependent set if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence.

That is, such that:

$\displaystyle \exists \left\{{\lambda_k: 1 \le k \le n}\right\} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$

where $\left\{{a_1, a_2, \ldots, a_n}\right\} \subseteq S$, and such that at least one of $\lambda_k$ is not equal to $0_R$.

Let $\left({\R^n,+,\cdot}\right)_{\R}$ be a real vector space.

Let $S \subseteq \R^n$.

Then $S$ is a linearly dependent set if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence.

That is, such that:

$\displaystyle \exists \left\{{\lambda_k: 1 \le k \le n}\right\} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$

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

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