#uw/notes #uw/class/math208 
- - -
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:**
$$c \mathbf{u} = c \begin{bmatrix}
u_{1} \\ u_{2} \\ \vdots \\ u_{n}
\end{bmatrix} = \begin{bmatrix}
c \cdot u_{1} \\ c \cdot u_{2} \\ \vdots \\ c \cdot u_{n}
\end{bmatrix}$$