0Introduction

IA Vectors and Matrices

0 Introduction

Vectors and matrices is the language in which a lot of mathematics is written

in. In physics, many variables such as position and momentum are expressed as

vectors. Heisenberg also formulated quantum mechanics in terms of vectors and

matrices. In statistics, one might pack all the results of all experiments into a

single vector, and work with a large vector instead of many small quantities. In

group theory, matrices are used to represent the symmetries of space (as well as

many other groups).

So what is a vector? Vectors are very general objects, and can in theory

represent very complex objects. However, in this course, our focus is on vectors

in

R

n

or

C

n

. We can think of each of these as an array of

n

real or complex

numbers. For example, (1

,

6

,

4) is a vector in

R

3

. These vectors are added in the

obvious way. For example, (1

,

6

,

4) + (3

,

5

,

2) = (4

,

11

,

6). We can also multiply

vectors by numbers, say 2(1

,

6

,

4) = (2

,

12

,

8). Often, these vectors represent

points in an n-dimensional space.

Matrices, on the other hand, represent functions between vectors, i.e. a

function that takes in a vector and outputs another vector. These, however, are

not arbitrary functions. Instead matrices represent linear functions. These are

functions that satisfy the equality

f

(

λx

+

µy

) =

λf

(

x

) +

µf

(

y

) for arbitrary

numbers

λ, µ

and vectors

x, y

. It is important to note that the function

x 7→ x

+

c

for some constant vector

c

is not linear according to this definition, even though

it might look linear.

It turns out that for each linear function from

R

n

to

R

m

, we can represent

the function uniquely by an

m × n

array of numbers, which is what we call the

matrix. Expressing a linear function as a matrix allows us to conveniently study

many of its properties, which is why we usually talk about matrices instead of

the function itself.