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#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.
[!abstract] Definition 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.
An augmented matrix can be used to find if a point (a,b,c) can be reached by a set of vectors
\mathbf{u}_{1}=\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}, \mathbf{u}_{2}=\begin{bmatrix}4 \\ 5 \\ 6\end{bmatrix}, \mathbf{u}_{3}=\begin{bmatrix}7 \\ 8 \\ 9\end{bmatrix}, \mathbf{v}=\begin{bmatrix}a \\ b \\ c\end{bmatrix} \implies \begin{bmatrix} 1 & 4 & 7 & a \\ 2 & 5 & 8 & b \\ 3 & 6 & 9 & c\end{bmatrix} = \begin{bmatrix}\mathbf{u}_{1} & \mathbf{u}_{2} & \mathbf{u}_{3} & \mathbf{v}\end{bmatrix}
Based on this the point \mathbf{v} can only be reached by \mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m} if the augmented matrix
$$\begin{bmatrix}
\mathbf{u}{1} & \mathbf{u}{2} & \dots & \mathbf{u}_{m} & \mathbf{v}
\end{bmatrix}$$
has a solution
for the set of vectors \mathbf{u}_{1}, \mathbf{u}_{2}, \dots, \mathbf{u}_{m} in \mathbf{R}^n if m<n the set does not span \mathbf{R}^n. If m \ge n the set may span \mathbf{R}^n.