INNER PRODUCT SPACE
Definition 1
Suppose u, v and w are all vectors in vector space V and c is any scalar. An inner product
space on the vectors space V is a function that associates with each pair of vectors in V, say u
and v , a real number denoted by
that satisfies the following axioms:
1.
[symmetry axiom]
2.
[additivity axiom]
3.
[homogeneity axiom]
4.
A vector space with its inner product is called an inner product space.
Euclidean inner product
If
defines
are vectors in Rn, then the formula
and
to be the Euclidean inner product on Rn.
The weighted Euclidean inner product
If
are positive real numbers, which we shall call weights, and if
are vectors in Rn, then the formula
and
defines an inner product on Rn. It is called the weighted Euclidean inner product with weights
.
Proof:
Suppose that u, v, and a are all vectors in Rn and that c is a scalar. where
positive real numbers,
;
and
are
First axiom we know that real numbers commute with multiplication we have,
The second axiom
The third axiom
The last axiom
because
are positive real
numbers so ≥0. And suppose that
. So the fourth axiom
is also satisfied.
Therefore proved that
defines an inner
product on Rn. It is called the weighted Euclidean inner product with weights (
are positive real number).
Definition 2
Suppose that V is an inner product space. The norm or length of a vector u in V is defined to be,
Definition 3
Suppose that V is an inner product space and that u and v are two vectors in V. The distance
between u and v, denoted by
is defined to be,
Example 1
Let f = f(x) and g = g(x) be two functions on C[a, b], the set of all
continuous functions on [a, b]. Define
Then
defines an inner product on C[a, b].
Proof:
Suppose that f, g, and h are continuous functions in C[a, b] and that c is any scalar.
Here is the work showing the first axiom is satisfied.
The second axiom
Here’s the third axiom
Finally, the fourth axiom.
Now, recall that if you integrate a continuous function that is greater than or equal to zero then
the integral must also be greater than or equal to zero. Hence,
clearly we’ll have
also have
Next, if
. Likewise, if we have
then
then we must
.
Example 2
Let A be an invertible
matrix. The following defines an inner product on Rn
Example 3
in Rn Let
For A and B
then
defines an inner product
in
Rn .
Proof:
Suppose that
of
are two matrices in
. And an inner product space
we can express
This formula is very similar to the Euclidean inner product formula. Therefore it’s proved that A
and B
in Rn an inner product of
matrices defines the formula
Cauchy-Schwarz inequality
Suppose that any vectors u and v in a inner product space V then
or
Property of Length (norm)
If u and v are vectors in an inner product space V, and if k is any scalar, then
a.
b.
c.
d.
(Triangle inequality)
e. The angle θ between u and v is defined by
Triangle Inequality
If V is an inner product space with norm
then
Proof:
We have
because
hence
So the Cauchy-Schwarz inequality implies that
Therefore prove that
.
Property of distance
If u, v, and w are vectors in an inner product space V, and if k is any scalar, then
a.
b.
c.
d.
(Triangle inequality)
Definition Orthogonality
Suppose that u and v are two vectors in an inner product space then u and v are called
orthogonal if
Orthogonal sets and bases
Suppose that V an inner product space and let
V. W is called orthogonal set
if any 2 member mutually orthogonal or
for every i≠j with i,j=1,2,3,….k.
And W is called orthonormal set if orthogonal set W and distance of any member of W is one or we can
express
.
Phythagoras Theorem
Let V be an inner product space. If
are orthogonal then
.
Proof:
Since
hence the v and w are orthogonal so
.
Therefore prove that the phytagoras theorem.
Projection theorem
If W is a finite-dimensional subspace of an inner product space V, then any vector u in V can be
expressed in exactly one way as
where
is in W and
is in W⊥
Orthogonal Projection
Suppose that W is spanned by orthogonal set
The reason
in W and
member of set is one or
or
and orthogonal projection v to W is
orthonormal set it’s means the distance or norm the
. And orthogonal v component to W is
Gram-Schmidt process
Let W be a subspace of an inner product space V. Suppose that
W, not necessarily orthogonal. An orthogonal basis
be a basis of
we can construct from B
as follows:
or we can write
The method of constructing the orthogonal vector
process.
is known as the Gram-Schmidt
Orthogonal matrix
Let V be an n-dimensional inner product space. A linear transformation T : V → V is called an
isometry if for any
,
Theorem
A linear transformation T : V → V is an isometry if only if T preserving inner product, e.g for
,
Proof:
for vectors
identity with
So it’s clear that the length preserving is equivalent to the inner product preserving.
An
matrix Q is called orthogonal if
i.e;
Theorem
Let Q be an
matrix. The following are equivalent
a) Q is orthogonal
b)
is orthogonal
c) The column vectors of Q are orthonormal
d) The row vectors of Q are orthonormal
Diagonalizing real symmetric matrices
Let V be an n-dimensional real inner product space. A linear mapping T : V → V is said to be
symmetric if
for all
Let A be an real symmetric
matrix. Let T: Rn →Rn be defined by T(x)=A(x). Then T is
symmetric for the Euclidean n-space. In fact
An
we have
real symmetric matrix A is called positive definite if for any nonzero vector
;
Complex inner product space
Definition
Suppose that V be an complex inner product space. An inner product space of V is a function
:
satisfying the following properties:
1. Linearity
2. Conjugate symmetric property
3. Positive definite property
For any
For any
, since
,
is a nonnegative real number, the length of u to be real number
Cauchy-Schwarz inequality
Let V be a complex inner product space. Suppose that any vectors u and v in a inner product
space V then
or
Like real inner product space, one can similarly define orthogonality, the angle between 2
vectors, orthogonal set, orthonormal set, orthogonal projection and Gram-Schmidt process, etc.
Complex Euclidean space Cn
For vectors
Let
and
where
defines
be orthogonal basis of an inner product space V. Then for any
Let W be a subspace of V with an orthogonal basis
is given by
;
. Then orthogonal projection
RESUME OF INNER PRODUCT SPACE
Linear Algebra 2
by
Rini Kurniasih
K1310069
Mathematics Education 2010
Training Teacher and Education Faculty
SEBELAS MARET UNIVERSITY (UNS)
2011