Page 467 - Advanced Linear Algebra
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Chapter 18
An Introduction to Algebras
Motivation
We have spent considerable time studying the structure of a linear operator
B ² = - ³ on a finite-dimensional vector space = over a field - . In our
studies, we defined the -´%µ -module = and used the decomposition theorems
for modules over a principal ideal domain to dissect this module. We
concentrated on an individual operator , rather than the entire vector space
B B - ²= ³. In fact, we have made relatively little use of the fact that - ²= ³ is an
algebra under composition. In this chapter, we give a brief introduction to the
theory of algebras, of which B - ²= ³ is the most general, in the sense of Theorem
18.2 below.
Associative Algebras
An algebra is a combination of a ring and a vector space, with an axiom that
links the ring product with scalar multiplication.
Definition An ( associative algebra over a field , or an - -algebra , is a
)
-
(
nonempty set , together with three operations, called addition denoted by
(
(
b ), multiplication ( denoted by juxtaposition ) and scalar multiplication ( also
)
denoted by juxtaposition , for which the following properties hold:
-
1 ( ) is a vector space over under addition and scalar multiplication.
)
2 ( is a ring with identity under addition and multiplication.
)
3 If - and Á ( , then
² ³ ~ ² ³ ~ ² ³
An algebra is finite-dimensional if it is finite-dimensional as a vector space. An
algebra is commutative if ( is a commutative ring. An element ( is
invertible if there is ( for which ~ ~ .
Our definition requires that have a multiplicative identity. Such algebras are
(
called unital algebras . Algebras without unit are also of great importance, but
