Page 228 - Advanced Linear Algebra
P. 228
212 Advanced Linear Algebra
Definition Let be an inner product space.
=
)
1 Two vectors "Á # = are orthogonal , written " # , if
º"Á #» ~
)
2 Two subsets ?Á @ = are orthogonal , written ? @ , if º?Á @ » ~ ¸ ¹ ,
that is, if %& for all % ? and & @ . We write # ? in place of
¸#¹ ?.
3 The orthogonal complement of a subset ? = is the set
)
?~ ¸# = # ?¹
The following result is easily proved.
Theorem 9.7 Let be an inner product space.
=
1 The orthogonal complement ? of any subset ? = is a subspace of .
)
=
)
2 For any subspace of ,
=
:
: q : ~ ¸ ¹
Definition An inner product space = is the orthogonal direct sum of
subspaces and if
:
;
=~ : l ;Á : ;
In this case, we write
:p ;
More generally, is the orthogonal direct sum of the subspaces : ÁÃÁ: ,
=
written
:~ : p Ä p :
if
and for £
=~ : l Ä l : : :
Theorem 9.8 Let be an inner product space. The following are equivalent.
=
1) =~ : p ;
2) =~ : l ; and ; ~ :
Proof. If =~ : p ; , then by definition, ; : . However, if # : , then
#~ b ! where : and !; . Then is orthogonal to both ! and # and so
is orthogonal to itself, which implies that ~ and so # ; . Hence, ; ~ : .
The converse is clear.
Orthogonal and Orthonormal Sets
Definition A nonempty set E ~¸" 2¹ of vectors in an inner product
space is said to be an orthogonal set if " " for all £ 2 . If, in
addition, each vector is a unit vector, then is an orthonormal set . Thus, a
E
"
