Module 1: Vectors

In our course a vector will be a list of LaTeX: n $n$ elements from a scalar field (i.e., real numbers) where $n$ is a positive natural number. It turns out that the best way to write these lists is as a column:

$LaTeX: \begin{pmatrix} 1 \end{pmatrix}, \quad \begin{pmatrix} -1 \\ .5 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}, \quad \begin{pmatrix} \pi \\ e \\ \sqrt{2} \\ -1000 \end{pmatrix}$ $\begin{pmatrix} 1 \end{pmatrix}, \quad \begin{pmatrix} -1 \\ .5 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}, \quad \begin{pmatrix} \pi \\ e \\ \sqrt{2} \\ -1000 \end{pmatrix}$

are vectors in $LaTeX: \mathbb{R} ^{1} , \mathbb{R} ^{2},\mathbb{R} ^{3},$ $\mathbb{R} ^{1} , \mathbb{R} ^{2},\mathbb{R} ^{3},$ and $LaTeX: \mathbb{R} ^{4}$ $\mathbb{R} ^{4}$ respectively. The entries inside of a vector are called its components:

For example, consider the vector

$LaTeX: \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}.$ $\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}.$

The "name'' of the vector is LaTeX: x $x$ . It has first component $x$ , second component LaTeX: y $y$ , and third component LaTeX: z $z$ . Note that it is common to use similar variable names for vectors and its components. In some texts it can be hard to distinguish between them and it is up to the reader to understand the difference from context. In our class we will distinguish vectors by using bold print LaTeX: v $v$ or drawing a little arrow on top ( $LaTeX: \vec{v}$ $\vec{v}$ ).

In summary: A vector is a list of LaTeX: n $n$ scalar values arranged in a column. The scalar values are called the components of the vector.