Definition:Linearly Independent/Set

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 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 $S$.

Also see

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