Definition:Linear Combination/Sequence

Definition

Let $M$ be an $R$-module.

Let $\sequence {a_n} := \sequence {a_j}_{1 \mathop \le j \mathop \le n}$ be a sequence of elements of $M$ of length $n$.

An element $b \in M$ is a linear combination of $\sequence {a_n}$ if and only if:

$\ds \exists \sequence {\lambda_n} \subseteq R: b = \sum_{k \mathop = 1}^n \lambda_k a_k$

Also see

• Definition:Linear Combination
• Definition:Linear Combination of Subset
• Definition:Linear Combination of Empty Set

• Definition:Linear Span
• Definition:Linear Transformation

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases