Definition 11.
An
inner product
on a vector space
V
is a way of assigning a scalar value
x
,
y
to any pair
x
,
y
of vectors such that for all vectors
x
,
y
,
z
1. Positivity:
x
,
x
is a nonnegative real number, which can be zero only if
x
= 0
;
2. Symmetry:
x
,
y
=
y
,
x
*
(
=
y
,
x
for real v.sp);
3. Linearity:
x
,
αy
+
βz
=
α x
,
y
+
β x
,
z
.
Consequences:
1.
x
, 0
= 0;
2.
αx
,
y
=
y
,
αx
*
= (
α y
,
x
)
*
=
α
*
x
,
y
which can be called “antilinearity in the
first component”.
Definition 12.
Length
or
Norm
or
Magnitude
of
x
:

x

=
x
,
x
, which is always
≥
0
.
x
is called a
unit vector
or
normalized vector
if

x

= 1
.
Examples 8.
1.
Dot product
in
R
n
:
x
,
y
=
x
·
y
=
x
1
y
1
+
· · ·
+
x
n
y
n
=
n
i
=1
x
i
y
i
=
x
T
y
More generally, one has weighted dot products: given positive real numbers called
weights
w
1
,
. . .
,
w
n
,
x
,
y
w
=
n
i
=1
w
i
x
i
y
i
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22
CHAPTER 1.
LINEAR ALGEBRA (APPROX. 9 LECTURES)
2. in
C
n
:
x
,
y
=
x
*
1
y
1
+
· · ·
+
x
*
n
y
n
=
x
†
y
or the same with positive weights
w
i
.
3.
V
=
C
[0, 1]
:
f
,
g
=
1
0
f
*
(
x
)
g
(
x
)
dx
and for any positive weight function
w
(
x
)
f
,
g
w
=
1
0
f
*
(
x
)
g
(
x
)
w
(
x
)
dx
Definition 13.
x
,
y
∈
V
are
orthogonal
if
x
,
y
= 0
.
Example 5.
The Bessel functions
J
n
,
Y
n
satisfy:
∞
0
J
n
(
x
)
Y
n
(
x
)
x dx
=
J
n
,
Y
n
x
= 0
and so are orthogonal with respect to a weighted inner product on
C
(0,
∞
)
.
There is a
reason for this: we will find out later when we study Bessel’s equation.
Definition 14.
1. A set of nonzero vectors
{
u
1
,
u
2
,
. . .
,
u
n
} ∈
V
is called an
orthog
onal set
if they are “mutually orthogonal”:
u
i
,
u
j
= 0
for all pairs
i
=
j
;
2. An orthogonal set which is “normalized”, that is
u
i
,
u
i
= 1
for all
i
, is called an
orthonormal set
. In summary,
u
i
,
u
j
=
δ
ij
for all
i
,
j
.
3.
Kronecker delta
:
δ
ij
= 1
if
i
=
j
and
δ
ij
= 0
if
i
=
j
.
Examples 9.
1. It is easy to check that
e
†
i
e
j
=
δ
ij
for the standard basis
{
e
1
,
. . .
,
e
N
}
for
C
N
and so this is an orthonormal basis with respect to the standard dot product.
2. The Fourier exponential functions
v
n
(
x
) =
e
inx
for
n
∈
Z
are an orthonormal set in
the inner product space
C
[0, 2
π
]
:
v
m
,
v
n
=
2
π
0
e

imx
e
inx
dx
=
δ
mn
,
for all integers
m
,
n
Question:
Why are orthogonal sets useful?
Answer:
First, orthogonal sets are guaranteed to be linearly independent. If
∑
i
α
i
u
i
= 0,
one can take the inner product with
u
j
to show that
α
j
= 0. If the set in fact forms a basis
for
V
, then they are really useful because of the following: any
x
∈
V
has
v
–coordinates
α
which can be calculated directly
α
j
=
u
j
,
x
u
j
,
u
j
,
j
= 1,
. . .
,
n
as shown in Assignment 2. For a general basis, remember that finding the coordinates
requires a major exercise in gaussian elimination.
1.8.
INNER PRODUCT SPACES
23
Example 6.
This can be made more concrete in the case of
C
n
with the standard dot
product. Suppose
{
v
i
}
n
i
=1
is an orthogonal set (hence a basis, since there are
n
of them).
Finding
v
–coordinates of a vector
x
means finding
a
= (
α
j
)
j
=1:
n
such that
x
=
Va
where
V
= [
v
1
,
v
2
,
. . .
,
v
n
]
is the matrix constructed from the basis.
The solution is
a
=
V

1
x
, which requires one to calculate the inverse of
V
.
Normally, this requires
Gaussian elimination, but for an orthogonal basis it doesn’t.
One can check that
V

1
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