Definition:Generated Submodule/Linear Span

Definition
Let $V$ be a vector space over $K$, and let $A \subseteq V$ be a subset.

Then the linear span of $A$ is the set


 * $\displaystyle \left\{{\sum_{k=1}^n \alpha_k f_k: n \in \N_{\ge 1}, \alpha_i \in \Bbb F, f_i \in A}\right\}$

It is a linear subspace of $V$, as proved in Linear Span is Linear Subspace.