# 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 $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.

Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $G$ such that:

$\ds \exists \sequence {\lambda_k}_{1 \mathop \le k \mathop \le n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$

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

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

Such a sequence is linearly dependent.

### Linearly Dependent Sequence on a Real Vector Space

Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space.

Let $\mathbf 0 \in \R^n$ be the zero vector.

Let $\sequence {\mathbf v_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of vectors in $\R^n$.

Then $\sequence {\mathbf v_k}_{1 \mathop \le k \mathop \le n}$ is linearly dependent if and only if:

$\ds \exists \sequence {\lambda_k}_{1 \mathop \le k \mathop \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$

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

That is, it is possible to find a linear combination of $\sequence {\mathbf v_k}_{1 \mathop \le k \mathop \le n}$ which equals $\mathbf 0$.