Page 33 - Advanced Linear Algebra

Preliminaries 17

Theorem 0.18
) .
1 The following sets have cardinality L
)
a The rational numbers .
r
)
b The set of all finite subsets of .
o
)
c The union of a countable number of countable sets.
d The set { of all ordered -tuples of integers.
)

)
2 The following sets have cardinality L .
a The set of all points in s .
)
b The set of all infinite sequences of natural numbers.
)
c The set of all infinite sequences of real numbers.
)
)
d The set of all finite subsets of .
s
e The set of all irrational numbers.
)
Part 2 Algebraic Structures
We now turn to a discussion of some of the many algebraic structures that play a
role in the study of linear algebra.
Groups
Definition A group is a nonempty set . , together with a binary operation
denoted by *, that satisfies the following properties:
1 )(Associativity ) For all Á Á  . ,
² i ³i ~ i² i ³
2 )(Identity ) There exists an element  . for which
i ~ i ~
for all  . .
3 )(Inverses ) For each  . , there is an element c  . for which

c
i c ~ i ~
Definition A group is abelian , or commutative , if
.
i ~ i

for all Á  . . When a group is abelian, it is customary to denote the
operation by +, thus writing i as b . It is also customary to refer to the
i
identity as the zero element and to denote the inverse c by c , referred to as
the negative of .

Example 0.7 The set of all bijective functions from a set to is a group
:
:
<
under composition of functions. However, in general, it is not abelian.
Example 0.8 The set C Á ²-³ is an abelian group under addition of matrices.
The identity is the zero matrix 0 Á of size d . The set C ²-³ is not a
group under multiplication of matrices, since not all matrices have multiplicative