Definition 11 an inner product on a vector space v is

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 non-negative 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

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 non-zero 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