Definition:Linear Combination

Definition
Let $$G$$ be an $R$-module.

Let $$\left \langle {a_n} \right \rangle$$ be a sequence of elements of $$G$$.

An element $$b \in G$$ is a linear combination of $$\left \langle {a_n} \right \rangle$$ if $$\exists \left \langle {\lambda_n} \right \rangle \subseteq R: b = \sum_{k=1}^n \lambda_k a_k$$.

Linear Combination of a Subset
Let $$\varnothing \subset S \subseteq G$$.

Let $$b \in G$$ be a linear combination of some sequence $$\left \langle {a_n} \right \rangle$$ of elements of $$S$$.

Then $$b$$ is a linear combination of $$S$$.

Linear Combination of Null
$$b$$ is a linear combination of $$\varnothing$$ if $$b = e_G$$.

Also see
An integer combination is also called a linear combination. The definition is compatible with the one on this page.