vault backup: 2026-10-07 10:23:16

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

@@ -1,14 +1,14 @@
 #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
+> [!abstract] Definition
+> 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 $\mathbf{R}^n$
+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.
 
@@ -23,3 +23,7 @@ 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 $\mathbf{R}^n$.
 
+Let $\mathbf{a}_{1}, \mathbf{a}_{2}, \dots, \mathbf{a}_{m}$ be vectors in $\mathbf{R}^n$. If 
+$$A = \begin{bmatrix}
+\mathbf{a}_{1} & \mathbf{a}_{2} \dots
+\end{bmatrix}$$