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 $\mathbf 0$.
Also see
- Linearly Independent Sequence: A sequence $\sequence {a_k}_{1 \mathop \le k \mathop \le n} \subseteq G$ which is not linearly dependent.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases