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Vector Spaces and Eigenvalues
Charles B. Moss
August 3, 2010
I. Orthonormal Bases and Projections
A. Suppose that a set of vectors
{
x
1
,
···
x
r
}
for a basis for some
space
S
in
R
m
space such that
r
≤
m
. For mathematical sim
plicity, we may want to form an orthogonal basis for this space.
One way to form such a basis is the GramSchmit orthonormaliza
tion. In this procedure, we want to generate a new set of vectors
{
y
1
,
y
r
}
that are orthonormal.
B. The GramSchmit process is
y
1
=
x
1
y
2
=
x
2
−
x
0
2
y
1
y
0
1
y
1
y
1
y
3
=
x
3
−
x
0
3
y
1
y
0
1
y
1
y
1
−
x
0
3
y
2
y
0
2
y
2
y
2
(1)
which produces a set of orthogonal vectors, and then
z
i
=
y
i
q
y
0
i
y
i
(2)
C. Example, the vectors
x
1
=
⎛
⎜
⎝
1
3
4
⎞
⎟
⎠
,x
2
=
⎛
⎜
⎝
9
7
16
⎞
⎟
⎠
(3)
1
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View Full DocumentAEB 6571 Econometric Methods I
Professor Charles B. Moss
Lecture XXVII
Fall 2010
span a plane in three dimension space. Setting
y
1
=
x
1
,
y
2
is
derived as
y
2
=
⎛
⎜
⎝
9
7
16
⎞
⎟
⎠
−
⎡
971
6
±
⎛
⎜
⎝
1
3
4
⎞
⎟
⎠
⎡
134
±
⎛
⎜
⎝
1
3
4
⎞
⎟
⎠
⎛
⎜
⎝
1
3
4
⎞
⎟
⎠
=
⎛
⎜
⎝
70
/
13
−
50
/
13
20
/
13
⎞
⎟
⎠
(4)
The vectors can then be normalized to one. However, to test for
orthogonality
⎡
±
⎛
⎜
⎝
70
/
13
−
50
/
13
20
/
13
⎞
⎟
⎠
=0
(5)
D.
Theorem 2.13
Every
r
dimensional vector space, except the zero
dimensional space
{
0
}
, has an orthonormal basis.
E.
Theorem 2.14
Let
{
z
1
,
···
z
r
}
be an orthornomal basis for some
vector space
S
,o
f
R
m
.T
h
e
ne
a
c
h
x
∈
R
m
can be expressed
uniquely as
x
=
u
+
v
(6)
where
u
∈
S
and
v
is a vector that is orthogonal to every vector
in
S
.
F.
Defnition 2.10
Let
S
be a vector subspace of
R
m
h
eo
r

thogonal complement of
S
, denoted
S
⊥
, is the collection of all
vectors in
R
m
that are orthogonal to every vector in
S
:Th
a
ti
s
,
S
⊥
=(
x
:
x
∈
R
m
and
x
0
y
,
∀
y
∈
S
}
.
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