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College/AU 26/MATH 208 (Matrix Algebra)/Chapter 2.2 - Span.md

@@ -7,8 +7,19 @@ 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 $\mathbf{R}^n$
+> [!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 $\mathbf{R}^n$.
+