Page 470 - Advanced Linear Algebra

454 Advanced Linear Algebra

% ~ bÄb

where  - and  . . If

& ~ bÄb

then we can include additional terms with coefficients and reindex if

necessary so that we may assume that ~ and ~ for all . Then the sum

in -´.µ is given by

b ~ ² b ³
8 9 8 9
~ ~ ~
Also, the product is given by

~

8 9 8 9
~ ~ Á
and the scalar product is

8 9 ~
~ ~
The Usual Suspects
Algebras have substructures and structure-preserving maps, as do groups, rings
and other algebraic structures.
Subalgebras
Definition Let be an -algebra. A subalgebra of is a subset of that is
(
(
-
)
(
a subring of with the same identity as ) and a subspace of .
(
(
(
(
The intersection of subalgebras is a subalgebra and so the family of all
subalgebras of is a complete lattice, where meet is intersection and the join of
(
a family of subalgebras is the intersection of all subalgebras of that contain
<
(
the members of .
<
The subalgebra generated by a nonempty subset ? of an algebra is the
(
smallest subalgebra of that contains and is easily seen to be the set of all
(
?
linear combinations of finite products of elements of ? , that is, the subspace
spanned by the products of finite subsets of elements of :
?
º?» alg ~ º% Ä% “ %  ?»

Alternatively, º?» alg is the set of all polynomials in the variables in ? . In
particular, the algebra generated by a single element %( is the set of all
polynomials in over .
%
-