Definition:Linearly Independent/Sequence

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 $\sequence {a_n}$ be a sequence of elements of $G$ such that:
 * $\ds \forall \sequence {\lambda_n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$

That is, the only way to make $e$ with a linear combination of $\sequence {a_n}$ is by making all the terms of $\sequence {\lambda_n}$ equal to $0_R$.

Such a sequence is linearly independent.

Also see

 * Linearly Dependent Sequence: a sequence $\sequence {a_n} \subseteq G$ which is not linearly independent.


 * Linearly Dependent Sequence of Vector Space