Definition:Generator of a Module

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Definition

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

Let $S \subseteq G$.

The submodule generated by $S$ is the smallest submodule $H$ of $G$ containing $S$.

In this context, we say that:

Let $G$ be a vector space over a field $F$.

Let $S \subseteq G$.

Then $S$ is a spanning set (for $G$, or of $G$) iff every vector of $G$ can be expressed as a linear combination of elements of $S$.

That is, iff $S$ is a generator for $G$.

If this is the case, then $S$ spans $G$.