Linear Algebra
Lecture Notes in progress
L´
aszl´
o Babai
Version: November 11, 2007
These notes are based on the “apprentice course” and the “discrete mathematics” course
given by the author at the Summer 2007 REU of the Department of Mathematics, University
of Chicago. The author is grateful to the scribes Sundeep Balaji and Shawn Drenning (other
scribes will be acknowledged as their parts get incorporated in the text).
1
Basic structures
1.1
Groups
A group is a set
G
endowed with a binary operation, usually called addition or multiplication,
satisfying the following axioms (written in multiplicative notation):
(a) (
∀
a, b
∈
G
)(
∃
!
ab
∈
G
) (operation uniquely defined)
(b) (
∀
a, b, c
∈
G
)((
ab
)
c
=
a
(
bc
)) (associativity)
(c) (
∃
e
∈
G
)(
∀
a
∈
G
)(
ea
=
ae
=
a
) (identity element)
(d) (
∀
a
∈
G
)(
∃
b
∈
G
)(
ab
=
ba
=
e
) (inverses)
In additive notation, we postulate
(a) (
∀
a, b
∈
G
)(
∃
!
a
+
b
∈
G
) (operation uniquely defined)
(b) (
∀
a, b, c
∈
G
)((
a
+
b
) +
c
=
a
+ (
b
+
c
)) (associativity)
(c) (
∃
e
∈
G
)(
∀
a
∈
G
)(
e
+
a
=
a
+
e
=
a
) (identity element)
(d) (
∀
a
∈
G
)(
∃
b
∈
G
)(
a
+
b
=
b
+
a
=
e
) (inverses)
1
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The multiplicative identity is usually denoted by “1,” the additive identity by “0.” The
multiplicative inverse of
a
is denoted by
a

1
; the additive inverse by (

a
).
The group is
commutative
or
abelian
if it satisfies (
∀
a, b
∈
G
)(
ab
=
ba
) (or (
∀
a, b
∈
G
)(
a
+
b
=
b
+
a
) in the additive notation). The additive notation is customarily reserved
for abelian groups.
Example 1.1.1.
(
Z
,
+) (the additive group of integers), (
Z
n
,
+) (the additive group of mod
ulo
n
residue classes), the general linear group GL
2
(
p
) (2
×
2 matrices over
Z
p
with nonzero
determinant (nonzero mod
p
where
p
is prime)) under matrix multiplication, the special
linear group SL
2
(
p
) (the subgroup of GL
2
(
p
) consisting of those matrices with determinant
= 1 (mod
p
))
Exercise 1.1.2.
If
p
is a prime (
Z
×
p
,
·
) is a group. Here
Z
×
p
is the set of nonzero residue
classes modulo
p
.
The
order
of a group is the number of elements of the group. For instance, the order of
(
Z
n
,
+) is
n
; the order of (
Z
×
p
,
·
) is
p

1; the order of (
Z
,
+) is infinite. Note that the order
of a group is at least 1 since it has an identity element. The idenity element alone is a group.
Exercise 1.1.3.
Calculate the order of the special linear group SL
2
(
p
). (Give a very simple
exact formula.)
1.2
Fields
Informally, a
field
is a set
F
together with two binary operations, addition and multiplication,
so that all the usual identities and rules of inversion hold (all nonzero elements have a
multiplicative inverse). Here is the formal definition.
(
F
,
+
,
·
) is a field if
(a) (
F
,
+) is an abelian group (the additive group of the field);
(b) (
F
×
,
·
) is an abelian group (the multiplicative group of the field) (where
F
×
=
F
\ {
0
}
);
(c) (
∀
a, b, c
∈
F
)(
a
(
b
+
c
) =
ab
+
ac
) (distributivity)
Examples: the real numbers (
R
), the complex numbers (
C
), the rational numbers (
Q
), the
modulo
p
residue classes (
F
p
) (where
p
is a prime). (This latter is the same as
Z
p
; we write
F
p
to emphasize that it is a field.) There are many other examples but only these will matter for
us so you do not need to know the formal definition of a field, just think of these examples.
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 Linear Algebra, Group Theory, Vector Space, Abelian group

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