984 B
#uw/class/math208 #uw/notes
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.