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1
Vector Spaces and Eigenvalues
Lecture XXVII
I.
Orthonormal Bases and Projections
A.
Suppose that a set of vectors
1
,
r
xx
for a basis for some space
S
in
m
R
space such that
rm
. For mathematical simplicity, we may want to
form an orthogonal basis for this space. One way to form such a basis is
the GramSchmit orthonormalization. In this procedure, we want to
generate a new set of vectors
1
,
r
yy
that are orthonormal.
B.
The GramSchmit process is:
11
21
2
2
1
3
1
3
2
33
1
1
2
2
'
'
''
yx
xy
y
x
y
x y
x y
y y
y y
which produces a set of orthogonal vectors, and then
1
2
'
i
i
ii
y
z
that produces a orthonormal vectors (vectors of length one).
C.
Example, from last lecture, we know that
16
7
9
,
4
3
1
2
1
x
x
spans a plane in three dimensional space. Setting
,
2
y
is derived as:
2
1
70
9
7 16
3
13
91
4
50
73
13
1
16
4
20
1 3
4
3
13
4
y
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View Full Document AEB 6933 – Mathematical Statistics for Food and Resource Economics
Lecture XXVII
Professor Charles Moss
Fall 2005
2
The vectors can then
be normalized to one. However, to test
for
orthogonality:
70
13
50
1 3 4
0
13
20
13
D.
Theorem
2.13
Every
r
dimensional
vector
space,
except
the
zero
dimensional space 0 , has an orthonormal basis.
E.
Theorem 2.14 Let
1
,
r
zz
be an orthornomal basis for some vector
space
S
, of
m
R
. Then each
m
xR
can be expressed uniquely as
x
u v
where
uS
and
v
is a vector that is orthogonal to every vector in
S
.
F.
Definition 2.10 Let
S
be a vector subspace of
m
R
. The orthogonal
complement of
S
, denoted
S
, is the collection of all vectors in
m
R
that
are
orthogonal
to
every
vector
in
S
:
That
is,
:
0
m
S
x x
R and x y
y
S
.
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This note was uploaded on 07/18/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.
 Fall '09
 CARRIKER

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