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

@@ -23,7 +23,16 @@ 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 
-$$A = \begin{bmatrix}
-\mathbf{a}_{1} & \mathbf{a}_{2} \dots
-\end{bmatrix}$$
+> [!abstract] Definition
+> Let $\mathbf{a}_{1}, \mathbf{a}_{2}, \dots, \mathbf{a}_{m}$ be vectors in $\mathbf{R}^n$. If 
+> $$A = \begin{bmatrix}
+> \mathbf{a}_{1} & \mathbf{a}_{2} & \dots & \mathbf{a}_{m}
+> \end{bmatrix} \quad \text{and} \quad \mathbf{x} = \begin{bmatrix}
+> x_{1} \\ x_{2} \\ \vdots \\ x_{m}
+> \end{bmatrix}$$
+> then $A\mathbf{x} = x_{1}\mathbf{a}_{1} + x_{2}\mathbf{a}_{2} + \dots + x_{m}\mathbf{a}_{m}$ (only if the number of columns of $A$ equals the number of components of $\mathbf{x}$)
+
+This means that $A\mathbf{x}$ is the linear combination of the columns of $A$ where the scalars are the components of the vector $\mathbf{x}$.
+
+This is useful to represent vector equations in a more compact way in the form $A\mathbf{x} = \mathbf{b}$ 
+$$A\mathbf{x}=\mathbf{b} \sim x_{1}\mathbf{a}_{1} + x_{2}\mathbf{a}_{2}+\dots+x_{m}\mathbf{a}_{m}=\mathbf{b} \sim \begin{gather} a_{11}x_{1} + a_{1} \end{gather}$$