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College/AU 26/MATH 208 (Matrix Algebra)/Chapter 2.1 Notes.md

@@ -6,7 +6,7 @@ u_{1} \\ u_{2} \\ \vdots \\ u_{n}
 \end{bmatrix}$$
 
 
-> [!NOTE] Basic Properties
+> [!INFO] Basic Properties
 > if $\mathbf{u}$ and $\mathbf{v}$ are vectors defined as 
 > $$\mathbf{u} =\begin{bmatrix}
 > u_{1} \\ u_{2} \\ \vdots \\ u_{n}
@@ -34,11 +34,22 @@ u_{1} \\ u_{2} \\ \vdots \\ u_{n}
 > c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
 > \end{bmatrix}$$
 
+
 A linear combination of vectors is when vectors multiplied by scalar coefficients are added or subtracted
 $$c_{1}\mathbf{u_{1}}+c_{2}\mathbf{u_{2}}+\dots+c_{m}\mathbf{u_{m}}$$
 is a linear combination if all $c$ values are scalars and $\mathbf{u}$ are vectors
 
+
 The general form of a solution to a linear system of equations can be represented as a linear combination of vectors
 
-if the solution of a linear system of equations is 
-$$$$
+> [!EXAMPLE] Example
+> if the solution of a linear system of equations is 
+> $$\begin{gather} x_{1} = 2 - 11s_{1} \\ x_{2} = 2 - 4s_{1} \\ x_{3} = 0 + 1s_{1} \end{gather}$$ then the vector form of the general solution is 
+> $$\mathbf{x} = \begin{bmatrix}
+> x_{1} \\ x_{2} \\ x_{3}
+> \end{bmatrix} = \begin{bmatrix}
+> 2 \\ 2 \\ 0
+> \end{bmatrix} + s_{1}\begin{bmatrix}
+> -11 \\ -4 \\ 1
+> \end{bmatrix}$$
+