Module 1: Vector Arithmetic

There are two essential operations associated to a vector space. They are vector addition and scalar multiplication.

Vector addition in $\mathbb{R}^n$ is done componentwise. Let

$\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix} \mbox{ and } \vec{w} = \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{pmatrix}.$

Then we can define the sum of $v$ and $w$ to be

$\vec{v} + \vec{w} := \begin{pmatrix} v_1 + w_1 \\ v_2 + w_2 \\ \vdots \\ v_n + w_n \end{pmatrix}.$

For example, 

$\begin{pmatrix} -1 \\ 3 \\ 8 \end{pmatrix} + \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} = \begin{pmatrix} -1+2 \\ 3-1 \\ 8+3 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 11 \end{pmatrix}.$

Note: this definition requires that the vectors have the same number of components. For our purposes we will not add together vectors of different dimensions.

The second operation is called scalar multiplication: Let $r \in \mathbb{R}$ be any real number and

$\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}.$

Then the product of $r$ and $\vec{v}$ is the vector

$r \vec{v} := \begin{pmatrix} rv_1 \\ rv_2 \\ \vdots \\ rv_n \end{pmatrix}$

For example:

$2 \begin{pmatrix} -1 \\ 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \cdot -1 \\ 2 \cdot 3 \\ 2 \cdot 2 \end{pmatrix} = \begin{pmatrix} -2 \\ 6 \\ 4 \end{pmatrix}$