05 Vectors
A
$$ \vec{v} = \mathbf{v}=\begin{bmatrix} v_{1} \\ v_{2} \\ \vdots \\ v_n \end{bmatrix} $$
Column Vector Operations
Vector Addition
If $\vec{u}$ and $\vec{v} \in \mathbb{R}^n,$ then
$$ \vec{u}+\vec{v}=\begin{bmatrix} \vec{u_{1}+\vec{v_{1}}} \\ \vec{u_{2}+\vec{v}_{2}} \\ \vdots \\ \vec{u}_n + \vec{v}_n \end{bmatrix}. $$
$\vec{u}$ and $\vec{v}$ must be the same size. ($\mathbb{R}^{2}$ is not in $\mathbb{R}^{3}$)
Example: Given $\vec{v}=\begin{bmatrix}2 \\2 \end{bmatrix}$ and $\vec{u}=\begin{bmatrix} 3 \\ 1 \end{bmatrix}$,
$$ \mathbb{u+v}= \begin{bmatrix} 2+3 \\ 2+1 \end{bmatrix} = \begin{bmatrix} 5 \\ 5 \end{bmatrix}. $$
Scalar Multiplication
if $\vec{u} \in \mathbb{R}^n$ and $c \in \mathbb{R}$, then
$$ c\vec{u}=\begin{bmatrix} c\vec{u_{1}} \\ c\vec{u_{2}} \\ \vdots \\ c\vec{u_n} \end{bmatrix}. $$
Example:
$$ \begin{aligned} \text{Given } A=\begin{bmatrix} 4 \\ 7 \\ 9 \end{bmatrix} \text{ and } b&=2, \\ Ab&=\begin{bmatrix} 8 \\ 14 \\ 18 \end{bmatrix}. \end{aligned} $$
We will discuss the cross and dot products at a later time.
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