Definition:Linearly Dependent

From ProofWiki
Jump to navigation Jump to search


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.


Let $S \subseteq G$.

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

That is, such that:

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

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 $0_R$.

Also see

  • Results about linear dependence can be found here.