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Vector Spaces and Eigenvalues: Lecture XXVII
Charles B. Moss
November 12, 2010
Charles B. Moss ()
Vector Spaces and Eigenvalues
November 12, 2010
1 / 22
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View Full Document 1
Orthonormal Bases and Projections
2
Projection Matrices
Idempotent Matrices
3
Eigenvalues and Eigenvectors
4
Kronecker Products
Charles B. Moss ()
Vector Spaces and Eigenvalues
November 12, 2010
2 / 22
Orthonormal Bases and Projections
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 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
{
y
1
,
y
r
}
that
are orthonormal.
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)
Charles B. Moss ()
Vector Spaces and Eigenvalues
November 12, 2010
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View Full Document which produces a set of orthogonal vectors, and then
z
i
=
y
i
q
y
0
i
y
i
(2)
Example, the vectors
x
1
=
1
3
4
,
x
2
=
9
7
16
(3)
span a plane in three dimension space. Setting
y
1
=
x
1
,
y
2
is derived
as
y
2
=
9
7
16

(
9 7 16
)
1
3
4
(
1 3 4
)
1
3
4
1
3
4
=
70
/
13

50
/
13
20
/
13
(4)
Charles B. Moss ()
Vector Spaces and Eigenvalues
November 12, 2010
4 / 22
The vectors can then be normalized to one. However, to test for
orthogonality
(
1 3 4
)
70
/
13

50
/
13
20
/
13
= 0
(5)
Theorem 2.13
Every
r
dimensional vector space, except the
zerodimensional space
{
0
}
, has an orthonormal basis.
Theorem 2.14
Let
{
z
1
,
···
z
r
}
be an orthornomal basis for some
vector space
S
, of
R
m
. Then each
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
.
Charles B. Moss ()
Vector Spaces and Eigenvalues
November 12, 2010
5 / 22
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View Full Document Deﬁnition 2.10
Let
S
be a vector subspace of
R
m
. The orthogonal
complement of
S
, denoted
S
⊥
, is the collection of all vectors in
R
m
that are orthogonal to every vector in
S
: That is,
S
⊥
= (
x
:
x
∈
R
m
and
x
0
y
= 0
,
∀
y
∈
S
}
.
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.
 Spring '10
 Staff

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