# Definition Given a set of [[vector|vectors]] $S = \{\pmb{x}_1, \pmb{x}_2, \ldots \pmb{x}_m\} \in V$, where $V$ is a [[vector space]] over the [[field]] $\mathbb{F}$, we define its span: $ \text{span}(S) = \left\{\sum_{i=1}^{m} a^i\pmb{x}_i\Bigg|a^i \in\mathbb{F }\right\}, $ i.e., the span is the set of all linear combinations of the vectors in $S$ If $S$ has is infinitely many elements, then the span still needs to be a *finite* linear combination by definition. We exclude infinite linear combinations since they raise questions about convergence of the sum.