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1
1
MAC 2103
Module 10
lnner Product Spaces I
2
Rev.F09
Learning Objectives
Upon completing this module, you should be able to:
1.
Define and find the inner product, norm and distance in a
given inner product space.
2.
Find the cosine of the angle between two vectors and
determine if the vectors are orthogonal in an inner product
space.
3.
Find the orthogonal complement of a subspace of an
inner product space.
4.
Find a basis for the orthogonal complement of a
subspace of
ℜⁿ
spanned by a set of row vectors.
5.
Identify the four fundamental matrix spaces of an m x n
matrix A, and know the column rank of A, the row rank of
A, and the rank of A.
6.
Know the equivalent statements of an n x n matrix A.
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3
Rev.09
General Vector Spaces II
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Inner Product Spaces,
Inner Product Spaces,
Inner Products,
Inner Products,
Norm, Distance, Fundamental Matrix Spaces,
Norm, Distance, Fundamental Matrix Spaces,
Rank, and
Orthogonality
Orthogonality
The major topics in this module:
4
Rev.F09
Definition of an Inner Product and Orthogonal Vectors
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A.
Inner Product:
Let
be any function from
into
that satisfies the
following conditions for any
u
,
v, z
in V:
and
iff
Then
defines an inner product for all
u
,
v
in V.
u
and
v
in V are
Orthogonal Vectors
iff
!"
,
"#
:
V
$
V
%
!
V
!
V
!
c
)
!
k
!
u
,
!
v
"
=
k
!
!
u
,
!
v
"
,
a
)
!
!
u
,
!
v
"
=
!
!
v
,
!
u
"
,
b
)
!
!
u
+
!
v
,
!
z
"
=
!
!
u
,
!
z
"
+
!
!
v
,
!
z
"
,
d
)
!
!
v
,
!
v
" #
0,
!
!
v
,
!
v
"
=
0
!
v
=
0.
!
!
u
,
!
v
"
!
!
u
,
!
v
"
=
0.
3
5
Rev.F09
Definition of a Norm of a Vector
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Click link to download other modules.
B.
Norm:
Let
be any function from
into
that satisfies the
following conditions for any
u
,
v
in V:
Then
defines a norm for all
u
in V.
The
special norm that is induced by an inner product
is
!
:
V
"
!
+
=
[0,
#
)
V
!
+
!
u
a
)
!
u
!
0,
and
!
u
=
0
iff
!
u
=
!
0,
b
)
s
!
u
=
s
!
u
,
and
c
)
!
u
+
!
v
"
!
u
+
!
v
Triangle
inequality
.
!
u
=
!
!
u
,
!
u
"
.
6
Rev.F09
Definition of the Distance Between Two Vectors
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Click link to download other modules.
C.
Distance:
Let
be any function from
into
that satisfies the
following conditions for any
u
,
v, w
in V:
Then
defines the distance for all
u
,
v
in V.
The
special distance that is induced by a norm
is
d
(
!
,
!
) :
V
"
V
#
!
+
=
[0,
$
)
V
!
V
!
+
d
(
!
u
,
!
v
)
d
(
!
u
,
!
v
)
=
!
u
!
!
v
=
!
u
!
!
v
,
!
u
!
!
v
#
.
a
)
d
(
!
u
,
!
v
)
!
0,
and d
(
!
u
,
!
v
)
=
0
iff
!
u
=
!
v
,
b
)
d
(
!
u
,
!
v
)
=
d
(
!
v
,
!
u
),
and
c
)
d
(
!
u
,
!
v
)
"
d
(
!
u
,
!
w
)
+
d
(
!
w
,
!
v
)
Triangle
inequality
.
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7
Rev.F09
How to Find an Inner Product, Norm, and Distance for
Vectors in
ℜⁿ
?
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules.
For
u
and
v
in
ℜⁿ
, we define
and
Note:
These are our usual norm and distance in
ℜⁿ
.
ℜⁿ
is an
inner product space with the dot product as its inner product.
!
u
=
!
!
u
,
!
u
"
1
2
=
!
u
#
!
u
)
=
1
2
+
u
2
2
+
...
+
u
n
2
,
d
(
!
u
,
!
v
)
=
!
u
!
!
v
=
"
!
u
!
!
v
,
!
u
!
!
v
#
1
2
=
(
!
u
!
!
v
)
$
(
!
u
!
!
v
)
=
u
1
!
v
1
)
2
+
(
u
2
!
v
2
)
2
+
...
+
(
u
n
!
v
n
)
2
.
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This note was uploaded on 10/18/2011 for the course MAS 2103 taught by Professor Shaw during the Summer '11 term at Valencia.
 Summer '11
 Shaw

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