Vectors & Matrices

Introduction

Vectors are a way of notating directed line segments, that is, a line with direction between two points in some space. We will be demonstrating vectors in R2\reals^2 (Introduction to Number Sets).

fig 1.1

fig 1.1\text{fig 1.1}

Notation

The vectors in the diagram above can be notated in several ways. Traditionally a vector is named with a single letter and is usually bold and underlined, although one or the other will usually suffice. For example:

a‾=OA→\underline { \mathbf{ a } } = \overrightarrow { OA }

The value of a‾\underline { \mathbf{ a } } is usually displayed as a vector such as [2,3][2,3], this is its row vector form. Alternatively, a‾\underline { \mathbf{ a } } has a column vector form [23]\begin{bmatrix} 2 \\\\ 3 \end{bmatrix}. The first number denotes how far the vector travels in the xx axis and the second, how far it travels in the yy axis.

Addition

Adding vectors together is as simple as adding their components:

[x1y1]+[x2y2]=[x1+x2y1+y2]\begin{bmatrix} x_1 \\\\ y_1 \end{bmatrix} + \begin{bmatrix} x_2 \\\\ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + x_2 \\\\ y_1 + y_2 \end{bmatrix}

Example:

[52]+[−14]=[46]\begin{bmatrix} 5 \\\\ 2 \end{bmatrix} + \begin{bmatrix} -1 \\\\ 4 \end{bmatrix} = \begin{bmatrix} 4 \\\\ 6 \end{bmatrix}

From figure 1.1\text{figure 1.1} we can see that:

b‾=[20], c‾=[−4−3]\underline { \mathbf{ b } } = \begin{bmatrix} 2 \\\\ 0 \end{bmatrix},\space \underline { \mathbf{ c } } = \begin{bmatrix} -4 \\\\ -3 \end{bmatrix}

and it follows that:

b‾+c‾=[−2−3]\underline { \mathbf{ b } } + \underline { \mathbf{ c } } = \begin{bmatrix} -2 \\\\ -3 \end{bmatrix}

fig 1.2

fig 1.2\text{fig 1.2}

Notice that new vector b‾+c‾ \space\underline { \mathbf{ b } } + \underline { \mathbf{ c } } \space takes the same space as a‾\underline{\mathbf{a}} but the arrow is pointing the other direction. This is because  b‾+c‾=−a‾\space\underline { \mathbf{ b } } + \underline { \mathbf{ c } } = - \underline { \mathbf{ a } }.

A negative vector −u‾- \underline { \mathbf{ u } } points the opposite direction from it’s positive counterpart u‾\underline { \mathbf{ u } }.

fig 1.3

fig 1.3\text{fig 1.3}

In fact −u‾- \underline { \mathbf{ u } } is just −1(u‾)-1 (\underline { \mathbf{ u } }).

Scalar multiples

A scalar is just a number in 1 dimension, part of R\reals, or Z\Z or other number sets. Most importantly, it is not a vector.

A vector cannot be directly multiplied by another vector, but it can be multiplied by a scalar.

Say variable p\space p is a scalar,

p[31]=[31]p=[3pp]p \begin{bmatrix} 3 \\\\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\\\ 1 \end{bmatrix}p = \begin{bmatrix} 3p \\\\ p \end{bmatrix}