Definition:Linearly Independent

Definition
Let $$G$$ be a 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.

Sequence
Let $$\left \langle {a_n} \right \rangle$$ be a sequence of elements of $$G$$ such that $$\forall \left \langle {\lambda_n} \right \rangle \subseteq R: \sum_{k=1}^n \lambda_k \circ a_k = e \Longrightarrow \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$$.

That is, the only way to make $$e$$ with a linear combination of $$\left \langle {a_n} \right \rangle$$ is by making all the elements of $$\left \langle {\lambda_n} \right \rangle$$ equal to $$0_R$$.

Such a sequence is linearly independent.

A sequence $$\left \langle {a_n} \right \rangle \subseteq G$$ which is not linearly independent is linearly dependent.

Set
Let $$S \subseteq G$$.

Then $$S$$ is a linearly independent set if every sequence of distinct terms in $$S$$ is a linearly independent sequence.

Otherwise $$S$$ is a linearly dependent set.