Definition:Linearly Dependent

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.

Sequence

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.

Linearly Dependent Sequence on a Real Vector Space

Let $\left({\R^n,+,\cdot}\right)_{\R}$ be a real vector space.

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

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

Then $\left \langle {\mathbf v_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ is linearly dependent iff:

$\displaystyle \exists \left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$

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

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

Set

Let $S \subseteq G$.

Then $S$ is a linearly dependent set if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence.

That is, such that:

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

where $a_1, a_2, \ldots, a_n$ are distinct elements of $S$, and where at least one of $\lambda_k$ is not equal to $0_R$.