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

@@ -1,7 +1,34 @@
 #uw/notes #uw/class/math208 
 - - -
-Vectors can be represented vertically in a matrix:
-$$u = \begin{bmatrix}
+Vectors can be represented vertically in a matrix (column vector):
+$$\mathbf{u} = \begin{bmatrix}
 u_{1} \\ u_{2} \\ \vdots \\ u_{n}
 \end{bmatrix}$$
 
+
+> [!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:**
+$$c \mathbf{u} = c \begin{bmatrix}
+u_{1} \\ u_{2} \\ \vdots \\ u_{n}
+\end{bmatrix} = \begin{bmatrix}
+c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
+\end{bmatrix}$$