1012 B
#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:
$$c \mathbf{u} = c \begin{bmatrix}
u_{1} \ u_{2} \ \vdots \ u_{n}
\end{bmatrix} = \begin{bmatrix}
c \cdot u_{1} \ c \cdot u_{2} \ \vdots \ c \cdot u_{n}
\end{bmatrix}$$