Definition:Linearly Independent/Set

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.

Let $S \subseteq G$.

Then $S$ is a linearly independent set if every sequence of distinct terms in $S$ is a linearly independent sequence.

That is, such that:

$\displaystyle \forall \left\{{\lambda_n}\right\} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$

where $S = \left\{{a_1, a_2, \ldots, a_k}\right\}$

Linearly Independent Set on a Real Vector Space

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

Let $S \subseteq \R^n$.

Then $S$ is a linearly independent set if every sequence of distinct terms in $S$ is a linearly independent sequence.

That is, such that:

$\displaystyle \forall \left\{{\lambda_k: 1 \le k \le n}\right\} \subseteq \R: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$

where $\left\{{\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n}\right\} = S$

Also see

• Linearly Dependent Set: a subset of $G$ which is not linearly independent.

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 27$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 7.33$
• John F. Humphreys: A Course in Group Theory (1996): $\text{A}.2$: Definition $\text{A}.4$