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.