Page 235 - Advanced Linear Algebra
P. 235
Real and Complex Inner Product Spaces 219
%~ º#Á » ~
for all and so # ~ . Hence, no nonzero vector # ¤ 4 is orthogonal to 4 .
This shows that 4 is a Hilbert basis for the inner product space .
M
On the other hand, the vector space span of 4 is the subspace of all
:
sequences in that have finite support, that is, have only a finite number of
M
nonzero terms and since span²4³ ~ : £ M , we see that 4 is not a Hamel
M
basis for the vector space .
The Projection Theorem and Best Approximations
Orthonormal bases have a great practical advantage over arbitrary bases. From a
=
computational point of view, if 8 ~¸# Á Ã Á # ¹ is a basis for , then each
#= has the form
# ~ # bÄb #
In general, determining the coordinates requires solving a system of linear
equations of size d .
On the other hand, if E ~¸" Á Ã Á " ¹ is an orthonormal basis for and
=
# ~ " bÄb "
then the coefficients are quite easily computed:
º#Á" » ~ º " b Ä b " Á" » ~ º" Á" » ~
Even if E ~¸" Á Ã Á " ¹ is not a basis (but just an orthonormal set), we can
still consider the expansion
V
# ~ º#Á " »" b Ä b º#Á " »"
Theorem 9.14 Let E ~¸" Á Ã Á " ¹ be an orthonormal subset of an inner
product space and let : ~ º E » . The Fourier expansion with respect to of
=
E
a vector #= is
V
# ~ º#Á " »" bÄbº#Á " »"
is called a Fourier coefficient of with respect to .
E
#
Each coefficient º#Á " »
The vector can be characterized as follows:
# V
1 # ) V is the unique vector : for which # ² c ³ . :
#
2 # ) V is the best approximation to from within , that is, is the unique
#
:
V
#
vector : that is closest to , in the sense that
) )#c# ) V )#c
for all : ± ¸#¹ .
V
