Page 17 - Advanced Linear Algebra

Preliminaries

In this chapter, we briefly discuss some topics that are needed for the sequel.
This chapter should be skimmed quickly and used primarily as a reference.
Part 1 Preliminaries

Multisets

The following simple concept is much more useful than its infrequent
appearance would indicate.

Definition Let be a nonempty set. A multiset 4 with underlying set is a
:
:
set of ordered pairs
4~ ¸² Á ³ “  :Á  { b Á £ for £ ¹

b
where { ~ ¸ Á Á Ã ¹ . The number is referred to as the multiplicity of the

. If the underlying set of a multiset is finite, we say that the
elements in 4
multiset is finite . The size of a finite multiset 4 is the sum of the multiplicities
of all of its elements.
For example, 4 ~ ¸² Á ³Á ² Á ³Á ² Á ³¹ is a multiset with underlying set
: ~ ¸ Á Á ¹. The element has multiplicity . One often writes out the
elements of a multiset according to multiplicities, as in 4 ~ ¸ Á Á Á Á Á ¹ .
Of course, two mutlisets are equal if their underlying sets are equal and if the
multiplicity of each element in the common underlying set is the same in both
multisets.

Matrices
-
The set of d matrices with entries in a field is denoted by C Á ²-³ or
by C when the field does not require mention. The set C Á Á ²³ is denoted
<
by C or C À If (  C ²-³ , the ² Á ³ th entry of will be denoted by ( Á .
(
The identity matrix of size d is denoted by 0 . The elements of the base