#uw/class/math208 #uw/notes 
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if $\{\mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m}\}$  is a set of vectors in $\mathbf{R}^n$ then the span of the set is given as $\text{span}\{\mathbf{u_{1}}, \mathbf{u_{2}}, \dots, \mathbf{u_{m}}\}$ and represents the set of all linear combinations
$$x_{1}\mathbf{u_{1}}+x_{2}\mathbf{u_{2}}+\dots+x_{m}\mathbf{u_{m}}$$
where $x_{1}, x_{2},\dots, x_{m}$ can be any real numbers

Span represents all points in n-dimensional space that a set of vectors could reach when combined in a certain linear combination.

if $\text{span}\{\mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m}\} = \mathbf{R^n}$ then the vectors $\{\mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m}\}$ spans $\mathbf{R}^n$


In three dimensions the span of two vectors can be visualized by the plane containing both of them. Any point on this plane can be reached by a linear combination of the vectors but any other point that does not lie on the plane cannot.