In Example VFSAL we saw the solution set of a homogeneous system described as all possible linear combinations of two particular vectors. This is a useful way to construct or describe infinite sets of vectors, so we encapsulate the idea in a definition.
Given a set of vectors $S=\{\vectorlist{u}{p}\}$, their span, $\spn{S}$, is the set of all possible linear combinations of $\vectorlist{u}{p}$. Symbolically, \begin{align*} \spn{S}&=\setparts{\lincombo{\alpha}{u}{p}}{\alpha_i\in\complex{\null},\,1\leq i\leq p}\\ &=\setparts{\sum_{i=1}^{p}\alpha_i\vect{u}_i}{\alpha_i\in\complex{\null},\,1\leq i\leq p} \end{align*}
The span is just a set of vectors, though in all but one situation it is an infinite set. (Just when is it not infinite?) So we start with a finite collection of vectors $S$ ($p$ of them to be precise), and use this finite set to describe an infinite set of vectors, $\spn{S}$. Confusing the finite set $S$ with the infinite set $\spn{S}$ is one of the most persistent problems in understanding introductory linear algebra. We will see this construction repeatedly, so let us work through some examples to get comfortable with it. The most obvious question about a set is if a particular item of the correct type is in the set, or not in the set.
Having analyzed Archetype A in Example SCAA, we will of course subject Archetype B to a similar investigation.