LINEAR ALGEBRA
W W L CHEN
c
W W L Chen, 1997, 2005.
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Chapter 9
REAL INNER PRODUCT SPACES
9.1. Euclidean Inner Products
In this section, we consider vectors of the form
u
= (
u
1
, . . . , u
n
) in the euclidean space
R
n
. In particular,
we shall generalize the concept of dot product, norm and distance, first developed for
R
2
and
R
3
in
Chapter 4.
Definition.
Suppose that
u
= (
u
1
, . . . , u
n
) and
v
= (
v
1
, . . . , v
n
) are vectors in
R
n
. The euclidean dot
product of
u
and
v
is defined by
u
·
v
=
u
1
v
1
+
. . .
+
u
n
v
n
,
the euclidean norm of
u
is defined by
u
= (
u
·
u
)
1
/
2
= (
u
2
1
+
. . .
+
u
2
n
)
1
/
2
,
and the euclidean distance between
u
and
v
is defined by
d
(
u
,
v
) =
u
−
v
= ((
u
1
−
v
1
)
2
+
. . .
+ (
u
n
−
v
n
)
2
)
1
/
2
.
PROPOSITION 9A.
Suppose that
u
,
v
,
w
∈
R
n
and
c
∈
R
. Then
(a)
u
·
v
=
v
·
u
;
(b)
u
·
(
v
+
w
) = (
u
·
v
) + (
u
·
w
)
;
(c)
c
(
u
·
v
) = (
c
u
)
·
v
; and
(d)
u
·
u
≥
0
, and
u
·
u
= 0
if and only if
u
=
0
.
Chapter 9 : Real Inner Product Spaces
page 1 of 15
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Linear Algebra
c
W W L Chen, 1997, 2005
PROPOSITION 9B.
(CAUCHYSCHWARZ INEQUALITY)
Suppose that
u
,
v
∈
R
n
. Then

u
·
v
 ≤
u
v
.
In other words,

u
1
v
1
+
. . .
+
u
n
v
n
 ≤
(
u
2
1
+
. . .
+
u
2
n
)
1
/
2
(
v
2
1
+
. . .
+
v
2
n
)
1
/
2
.
PROPOSITION 9C.
Suppose that
u
,
v
∈
R
n
and
c
∈
R
. Then
(a)
u
≥
0
;
(b)
u
= 0
if and only if
u
=
0
;
(c)
c
u
=

c

u
; and
(d)
u
+
v
≤
u
+
v
.
PROPOSITION 9D.
Suppose that
u
,
v
,
w
∈
R
n
. Then
(a)
d
(
u
,
v
)
≥
0
;
(b)
d
(
u
,
v
) = 0
if and only if
u
=
v
;
(c)
d
(
u
,
v
) =
d
(
v
,
u
)
; and
(d)
d
(
u
,
v
)
≤
d
(
u
,
w
) +
d
(
w
,
v
)
.
Remark.
Parts (d) of Propositions 9C and 9D are known as the Triangle inequality.
In
R
2
and
R
3
, we say that two nonzero vectors are perpendicular if their dot product is zero. We
now generalize this idea to vectors in
R
n
.
Definition.
Two vectors
u
,
v
∈
R
n
are said to be orthogonal if
u
·
v
= 0.
Example 9.1.1.
Suppose that
u
,
v
∈
R
n
are orthogonal. Then
u
+
v
2
= (
u
+
v
)
·
(
u
+
v
) =
u
·
u
+ 2
u
·
v
+
v
·
v
=
u
2
+
v
2
.
This is an extension of Pythagoras’s theorem.
Remarks.
(1) Suppose that we write
u
,
v
∈
R
n
as column matrices. Then
u
·
v
=
v
t
u
,
where we use matrix multiplication on the right hand side.
(2) Matrix multiplication can be described in terms of dot product. Suppose that
A
is an
m
×
n
matrix and
B
is an
n
×
p
matrix. If we let
r
1
, . . . ,
r
m
denote the vectors formed from the rows of
A
, and
let
c
1
, . . . ,
c
p
denote the vectors formed from the columns of
B
, then
AB
=
r
1
·
c
1
. . .
r
1
·
c
p
.
.
.
.
.
.
r
m
·
c
1
. . .
r
m
·
c
p
.
9.2. Real Inner Products
The purpose of this section and the next is to extend our discussion to define inner products in real
vector spaces. We begin by giving a reminder of the basics of real vector spaces or vector spaces over
R
.
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 Fall '08
 PETRINA
 Linear Algebra, Vector Space, Hilbert space, inner product, real inner product, W W L Chen

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