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:

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

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


Linearly Independent Set on a Real Vector Space

Let $\left({\R^n, +, \cdot}\right)_{\R}$ 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:

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

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


Also see


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