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

Let $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $G$ such that:
 * $\displaystyle \exists \left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$

where not all of $\lambda_k$ are equal to $0_R$.

That is, it is possible to find a linear combination of $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ which equals $e$.

Such a sequence is linearly dependent.

Also see

 * Linearly Independent Sequence: A sequence $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n} \subseteq G$ which is not linearly dependent.


 * Linearly Dependent Sequence of Vector Space