Definition:Linearly Independent/Sequence
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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 $\sequence {a_n}$ be a sequence of elements of $G$ such that:
- $\ds \forall \sequence {\lambda_n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$
That is, the only way to make $e$ with a linear combination of $\sequence {a_n}$ is by making all the terms of $\sequence {\lambda_n}$ equal to $0_R$.
Such a sequence is linearly independent.
Linearly Independent Sequence on a Real Vector Space
Let $\struct {\R^n, +, \cdot}_\R$ be a real vector space.
Let $\sequence {\mathbf v_n}$ be a sequence of vectors in $\R^n$.
Then $\sequence {\mathbf v_n}$ is linearly independent if and only if:
- $\ds \forall \sequence {\lambda_n} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$
where $\mathbf 0 \in \R^n$ is the zero vector and $0 \in \R$ is the zero scalar.
Also see
- Linearly Dependent Sequence: a sequence $\sequence {a_n} \subseteq G$ which is not linearly independent.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases