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

@@ -7,28 +7,38 @@ u_{1} \\ u_{2} \\ \vdots \\ u_{n}
 
 
 > [!NOTE] Basic Properties
-> if 
+> if $\mathbf{u}$ and $\mathbf{v}$ are vectors defined as 
+> $$\mathbf{u} =\begin{bmatrix}
+> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
+> \end{bmatrix} \quad \text{and} \quad \mathbf{v} =\begin{bmatrix}
+> v_{1} \\ v_{2} \\ \vdots \\ v_{n}
+> \end{bmatrix}$$
+> 
+> **Equality:**
+> 
+> $\mathbf{u}=\mathbf{v}$ if and only if $u_{1}=v_{1}, u_{2}=v_{2},\dots,u_{n}=v_{n}$
+> 
+> **Addition:**
+> $$\mathbf{u}+\mathbf{v} = \begin{bmatrix}
+> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
+> \end{bmatrix} + \begin{bmatrix}
+> v_{1} \\ v_{2} \\ \vdots \\ v_{n}
+> \end{bmatrix} = \begin{bmatrix}
+> u_{1}+v_{1} \\ u_{2}+v_{2} \\ \vdots \\ u_{n}+v_{n}
+> \end{bmatrix}$$
+> 
+> **Scalar Multiplication:**
+> $$c \mathbf{u} = c \begin{bmatrix}
+> u_{1} \\ u_{2} \\ \vdots \\ u_{n}
+> \end{bmatrix} = \begin{bmatrix}
+> c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
+> \end{bmatrix}$$
 
-if $\mathbf{u}$ and $\mathbf{v}$ are vectors defined as 
+A linear combination of vectors is when vectors multiplied by scalar coefficients are added or subtracted
-$$\mathbf{u} =\begin{bmatrix}
+$$c_{1}\mathbf{u_{1}}+c_{2}\mathbf{u_{2}}+\dots+c_{m}\mathbf{u_{m}}$$
-u_{1} \\ u_{2} \\ \vdots \\ u_{n}
+is a linear combination if all $c$ values are scalars and $\mathbf{u}$ are vectors
-\end{bmatrix} \quad \text{and} \quad \mathbf{v} =\begin{bmatrix}
-v_{1} \\ v_{2} \\ \vdots \\ v_{n}
-\end{bmatrix}$$
 
-**Equality:**
+The general form of a solution to a linear system of equations can be represented as a linear combination of vectors
-$\mathbf{u}=\mathbf{v}$ if and only if $u_{1}=v_{1}, u_{2}=v_{2},\dots,u_{n}=v_{n}$
+
-**Addition:**
+if the solution of a linear system of equations is 
-$$\mathbf{u}+\mathbf{v} = \begin{bmatrix}
+$$$$
-u_{1} \\ u_{2} \\ \vdots \\ u_{n}
-\end{bmatrix} + \begin{bmatrix}
-v_{1} \\ v_{2} \\ \vdots \\ v_{n}
-\end{bmatrix} = \begin{bmatrix}
-u_{1}+v_{1} \\ u_{2}+v_{2} \\ \vdots \\ u_{n}+v_{n}
-\end{bmatrix}$$
-**Scalar Multiplication:**
-$$c \mathbf{u} = c \begin{bmatrix}
-u_{1} \\ u_{2} \\ \vdots \\ u_{n}
-\end{bmatrix} = \begin{bmatrix}
-c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
-\end{bmatrix}$$