MATH1231 Mathematics 1B – Algebra
Lecture 1: Introduction to Vector Spaces
Lecturer: Sean Gardiner – [email protected]

Vectors (MATH1131 revision)
In MATH1131, we saw a
vector
is a mathematical object that can be
thought of geometrically as something with direction and magnitude, or
algebraically as an element of
R
n
(an ordered vertical list of
n
real numbers).
We can add any two vectors to produce another vector, written
v
+
w
, and
this
vector addition
is associative and commutative. We can also multiply
any vector by a scalar (real number) to produce another vector, written
λ
*
v
or just
λ
v
, and this
scalar multiplication
is associative. Furthermore, vector
addition and scalar multiplication together are distributive. There is also the
concept of a
zero vector
0
which has the property that any vector added to
it goes unchanged.
The above properties are not unique to elements of
R
n
. Another set of
mathematical objects that satisfy all the above properties is the set of
matrices
of a particular size, for example all
2
×
3
matrices. This makes
sense, since we can consider any element of
R
n
as an
n
×
1
matrix. Similarly,
another set of mathematical objects that satisfies all the above properties is
the set of
complex numbers
C
. This makes sense, since we know numbers
plotted in the complex plane correspond with position vectors in
R
2
.
Mathematics 1B – Algebra
Lecture 1: Introduction to Vector Spaces
1/20

Vector spaces
We would like to be able to describe all these systems under a single
unifying concept. The term given to any such sort of system is a
vector
space
, and its elements are called vectors. In this more abstract sense of a
vector space, the term “vector” no longer needs to be so restrictive and will
therefore not always refer to elements of
R
n
.
For a vector space to be well-defined, we must provide four pieces of
information:
•
The set of vectors
V
. (This may be
R
n
or something else entirely.)
•
The set of scalars
F
. (This is typically either
R
or
C
.)
•
The vector addition operation
+
.
•
The scalar multiplication operation
*
. (This symbol is often omitted in
practice, so
λ
*
v
is usually written as
λ
v
.)
To properly reference a vector space, we should provide the list
(
V
, +,
*
,
F
)
.
In practice, once the operations and set of scalars has been established, we
refer to the vector space just as
V
.
Mathematics 1B – Algebra
Lecture 1: Introduction to Vector Spaces
2/20

Field axioms (MATH1131 revision)
Before we proceed with investigating the properties of a vector space, we
should establish the properties of a set of scalars, which is also known as a
field
. Recall from MATH1131 that
Q
,
R
and
C
are fields, and they can each
be described as a set
F
satisfying the following axioms:
•
Closure under addition:
x
+
y
∈
F
for all
x
,
y
∈
F
.
•
Additive associativity:
(
x
+
y
) +
z
=
x
+ (
y
+
z
)
for all
x
,
y
,
z
∈
F
.