Inner Products
MATH 322, Linear Algebra I
J. Robert Buchanan
Department of Mathematics
Spring 2007
J. Robert Buchanan
Inner Products
Introduction
We have discussed the inner (dot) product in vector
spaces where V = Rn .
Today we extend the concept of the inner product to
abstract vector spaces.
J. Robert Buchanan
Inner Products
General Inner Products
Definition
An inner product on a vector space V is a function which
associates a real number, denoted hu, vi, with each pair of
vectors u, v ∈ V such that the following axioms hold for all u, v,
and w in V and all scalars k.
1
2
3
4
hu, vi = hv, ui
(Symmetry Axiom)
hu + v, wi = hu, wi + hv, wi
hku, vi = khu, vi
(Additivity Axiom)
(Homogeneity Axiom)
hu, ui ≥ 0 and hu, ui = 0 if and only if u = 0.
Axiom)
(Positivity
A real vector space with an inner product is called a real inner
product space.
J. Robert Buchanan
Inner Products
Example
Example
If u, v ∈ Rn , then the dot product u · v = hu, vi is an inner
product.
J. Robert Buchanan
Inner Products
Weighted Euclidean Inner Product
Definition
Let w1 , w2 , . . . , wn be positive real numbers and let u, v ∈ Rn
then
hu, vi = u1 v1 w1 + u2 v2 w2 + · · · + un vn wn
is called the weighted Euclidean inner product with weights
w1 , w2 , . . . , wn .
J. Robert Buchanan
Inner Products
Example
Example
The center of mass of a system of points masses m1 , m2 , . . . ,
mk located along the x-axis at positions x1 , x2 , . . . , xk is
x
m1 x1 + m2 x2 + · · · + mk xk
m1 + m2 + · · · + mk
1
(m1 x1 + m2 x2 + · · · + mk xk )
=
m
= hm, xi
=
a weighted Euclidean inner product with weights
w1 = w2 = · · · = wk = m1 where m is the total mass of the
system.
J. Robert Buchanan
Inner Products
Length and Distance
Definition
If V is an inner product space, the norm of a vector u ∈ V is
defined as
kuk = hu, ui1/2 .
The distance between two vectors u, v ∈ V is defined as
d (u, v) = ku − vk.
Definition
If V is an inner product space, the subset of V satisfying
kuk = 1 is called the unit sphere of V .
J. Robert Buchanan
Inner Products
Length and Distance
Definition
If V is an inner product space, the norm of a vector u ∈ V is
defined as
kuk = hu, ui1/2 .
The distance between two vectors u, v ∈ V is defined as
d (u, v) = ku − vk.
Definition
If V is an inner product space, the subset of V satisfying
kuk = 1 is called the unit sphere of V .
J. Robert Buchanan
Inner Products
Example
Example
Let V be the vector space of functions which are continuous on
the interval [−π/2, π/2].
1
Show that the function defined as
Z π/2
u(x)v(x) dx
hu, vi =
−π/2
is an inner product.
2
Use this inner product to define the norm of u and the
distance between u and v.
J. Robert Buchanan
Inner Products
Unit Circles and Spheres
Let V = R2 and consider the graph of the unit circle kuk for two
different inner products.
hu, vi = u1 v1 + u2 v2
u12 + u22 = 1
hu, vi = u1 v1 + 41 u2 v2
u12 + 14 u22 = 1
u2
2
1
u2
-1
1
-0.5
0.5
0.5
-1
-1
-0.5
0.5
1
u1
-0.5
-2
-1
J. Robert Buchanan
Inner Products
1
u1
Unit Sphere in an Abstract Vector Space
Example
Let V be the vector space of functions which are continuous on
the interval [−π/2, π/2] with inner product
hu, vi =
Z
π/2
u(x)v(x) dx
−π/2
√
Show that u = 1/ π lies on the unit sphere of V .
J. Robert Buchanan
Inner Products
Inner Products Generated by Matrices
Definition
Suppose u, v ∈ Rn and A is an invertible n × n matrix. The
function
hu, vi = Au · Av
defines and inner product on Rn called the inner product on
Rn generated by A.
Note: hu, vi = Au · Av = (Au)T Av = uT AT Av.
J. Robert Buchanan
Inner Products
Inner Products Generated by Matrices
Definition
Suppose u, v ∈ Rn and A is an invertible n × n matrix. The
function
hu, vi = Au · Av
defines and inner product on Rn called the inner product on
Rn generated by A.
Note: hu, vi = Au · Av = (Au)T Av = uT AT Av.
J. Robert Buchanan
Inner Products
Examples
Example
Suppose V =
R2
and A =
hu, vi.
1 2
, find an expression for
−1 1
Example
Let V = Mnn be the vector space of n × n matrices. Confirm the
function defined as
hA, Bi =
n X
n
X
aij bij
i=1 j=1
is an inner product.
J. Robert Buchanan
Inner Products
Examples
Example
Suppose V =
R2
and A =
hu, vi.
1 2
, find an expression for
−1 1
Example
Let V = Mnn be the vector space of n × n matrices. Confirm the
function defined as
hA, Bi =
n X
n
X
aij bij
i=1 j=1
is an inner product.
J. Robert Buchanan
Inner Products
Properties of Inner Products
Theorem
If u, v, and w are vectors in a real inner product space, and if k
is any scalar then
1
2
3
4
5
h0, ui = hu, 0i = 0
hu, v + wi = hu, vi + hu, wi
hu, kvi = khu, vi
hu − v, wi = hu, wi − hv, wi
hu, v − wi = hu, vi − hu, wi
Proof.
(2)
J. Robert Buchanan
Inner Products
Properties of Inner Products
Theorem
If u, v, and w are vectors in a real inner product space, and if k
is any scalar then
1
2
3
4
5
h0, ui = hu, 0i = 0
hu, v + wi = hu, vi + hu, wi
hu, kvi = khu, vi
hu − v, wi = hu, wi − hv, wi
hu, v − wi = hu, vi − hu, wi
Proof.
(2)
J. Robert Buchanan
Inner Products
Homework
Read Section 6.1 and work exercises 1–4, 6, 8a, 9ab, 10, 11,
16, 20, 21, 27.
J. Robert Buchanan
Inner Products