Section 3 Definitions

Vector

A line that has both direction and magnitude (oh yeah!) represented as a vertical matrix of dimensions nx1 where $\vec{u} = \begin{bmatrix} u_1 \\ u_2 \\ ... \\ u_n \end{bmatrix}$

Properties
$\vec{u} + \vec{v} = \vec{v} + \vec{u}$ Commutative Property
$(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$ Associative Property
$\vec{u} + \vec{0} = \vec{u}$ Identity Property
$\vec{u} + (-\vec{u}) = \vec{0}$ Inverse Property
$c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}$ Distributive Property

Linear Combinations of any two vectors are in the span of those vectors, therefore,

(1)

\begin{align} \text{if } \vec{u} = \begin{bmatrix} 2\\1\\0 \end{bmatrix} \text{and } \vec{v} = \begin{bmatrix} 0\\1\\1 \end{bmatrix} \text{is} \begin{bmatrix} 4\\3\\1 \end{bmatrix} \text{an element of the span of } \vec{u},\vec{v} \text{?} \end{align}

(2)

\begin{align} \text{yes, because } 2 (\vec{u}) + \vec{v} = \begin{bmatrix} 4\\3\\1 \end{bmatrix} \end{align}