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# Lecture 4 - Inner Product spaces, Orthogonal and Orthonormal Vectors Engineering Mathematics Notes | EduRev

## Engineering Mathematics : Lecture 4 - Inner Product spaces, Orthogonal and Orthonormal Vectors Engineering Mathematics Notes | EduRev

``` Page 1

Inner product spaces, Orthogonal and Orthonormal Vectors

Institute of Lifelong Learning, University of Delhi

Lesson: Inner product spaces, Orthogonal and Orthonormal
Vectors

Lesson Developer : Umesh Chand

Department / College: Department of Mathematics, Kirorimal
College, University of Delhi

Page 2

Inner product spaces, Orthogonal and Orthonormal Vectors

Institute of Lifelong Learning, University of Delhi

Lesson: Inner product spaces, Orthogonal and Orthonormal
Vectors

Lesson Developer : Umesh Chand

Department / College: Department of Mathematics, Kirorimal
College, University of Delhi

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 2

Chapter: Inner product spaces, Orthogonal and Orthonormal
Vectors
1. Learning outcomes
2. Introduction
3. Inner product space
4. Norm
5. Orthogonaility
6. Orthogonal complement
? Summary
? Exercise

Page 3

Inner product spaces, Orthogonal and Orthonormal Vectors

Institute of Lifelong Learning, University of Delhi

Lesson: Inner product spaces, Orthogonal and Orthonormal
Vectors

Lesson Developer : Umesh Chand

Department / College: Department of Mathematics, Kirorimal
College, University of Delhi

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 2

Chapter: Inner product spaces, Orthogonal and Orthonormal
Vectors
1. Learning outcomes
2. Introduction
3. Inner product space
4. Norm
5. Orthogonaility
6. Orthogonal complement
? Summary
? Exercise

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 3

1. Learning outcomes
After studying the whole content student will be able to understand
? Inner product space
? Norm
? Orthogonaility
? Orthogonal complement
? Schwarz inequality
? Bessel inequality

2. Introduction
In linear algebra, an inner product space is vector space with
associates each pair of vectors in the space with a scalar quantity
know as the inner product of vectors. Inner products allow the
rigorous introduction of intuitive geometrical notions such as the
length of a vector or angle between two vectors. Inner product
space generalize Euclidean space (in which the inner product is the
det product, also known as scalar product) to vector spaces of any
(possibility infinite) functional analysis.

Page 4

Inner product spaces, Orthogonal and Orthonormal Vectors

Institute of Lifelong Learning, University of Delhi

Lesson: Inner product spaces, Orthogonal and Orthonormal
Vectors

Lesson Developer : Umesh Chand

Department / College: Department of Mathematics, Kirorimal
College, University of Delhi

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 2

Chapter: Inner product spaces, Orthogonal and Orthonormal
Vectors
1. Learning outcomes
2. Introduction
3. Inner product space
4. Norm
5. Orthogonaility
6. Orthogonal complement
? Summary
? Exercise

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 3

1. Learning outcomes
After studying the whole content student will be able to understand
? Inner product space
? Norm
? Orthogonaility
? Orthogonal complement
? Schwarz inequality
? Bessel inequality

2. Introduction
In linear algebra, an inner product space is vector space with
associates each pair of vectors in the space with a scalar quantity
know as the inner product of vectors. Inner products allow the
rigorous introduction of intuitive geometrical notions such as the
length of a vector or angle between two vectors. Inner product
space generalize Euclidean space (in which the inner product is the
det product, also known as scalar product) to vector spaces of any
(possibility infinite) functional analysis.

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 4

3.  Inner Product Space
Let V be a vector space over F. An inner product on V is a function
V ?V to f that assign every ordered pair of vectors x, y in V, to a
scalar in F, denoted by x, y , in such a way that
(i) x y,z x,z y,z x,y,z V ? ? ? ? ?
(ii) cx,y c x,y c F ? ? ?  and x, y ? V
(iii) x,y y,x x,y V ? ? ?
(iv) x,x 0 if x 0 ??
Then the vector space V is said to be inner product space with
respect to the specified inner product defined on it.
1. (i) and (ii) are equivalent to
x y, z x,z y,z , F, x,y,z V ? ? ? ? ? ? ? ? ? ? ? ?
2. (i) and (ii) imply inner product is linear in Ist component,
property (1) and (ii) Jointly is called linearly property, (iii) is
called conjugate symmetry, (iv) is called non-negative.

Example 1: Let V = F
n
, F = R (or C)
Let  x = (a
1
, a
2
, …, a
n
),
? V
y ? (b
1
, b
2
, …,b
n
).
Page 5

Inner product spaces, Orthogonal and Orthonormal Vectors

Institute of Lifelong Learning, University of Delhi

Lesson: Inner product spaces, Orthogonal and Orthonormal
Vectors

Lesson Developer : Umesh Chand

Department / College: Department of Mathematics, Kirorimal
College, University of Delhi

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 2

Chapter: Inner product spaces, Orthogonal and Orthonormal
Vectors
1. Learning outcomes
2. Introduction
3. Inner product space
4. Norm
5. Orthogonaility
6. Orthogonal complement
? Summary
? Exercise

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 3

1. Learning outcomes
After studying the whole content student will be able to understand
? Inner product space
? Norm
? Orthogonaility
? Orthogonal complement
? Schwarz inequality
? Bessel inequality

2. Introduction
In linear algebra, an inner product space is vector space with
associates each pair of vectors in the space with a scalar quantity
know as the inner product of vectors. Inner products allow the
rigorous introduction of intuitive geometrical notions such as the
length of a vector or angle between two vectors. Inner product
space generalize Euclidean space (in which the inner product is the
det product, also known as scalar product) to vector spaces of any
(possibility infinite) functional analysis.

Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 4

3.  Inner Product Space
Let V be a vector space over F. An inner product on V is a function
V ?V to f that assign every ordered pair of vectors x, y in V, to a
scalar in F, denoted by x, y , in such a way that
(i) x y,z x,z y,z x,y,z V ? ? ? ? ?
(ii) cx,y c x,y c F ? ? ?  and x, y ? V
(iii) x,y y,x x,y V ? ? ?
(iv) x,x 0 if x 0 ??
Then the vector space V is said to be inner product space with
respect to the specified inner product defined on it.
1. (i) and (ii) are equivalent to
x y, z x,z y,z , F, x,y,z V ? ? ? ? ? ? ? ? ? ? ? ?
2. (i) and (ii) imply inner product is linear in Ist component,
property (1) and (ii) Jointly is called linearly property, (iii) is
called conjugate symmetry, (iv) is called non-negative.

Example 1: Let V = F
n
, F = R (or C)
Let  x = (a
1
, a
2
, …, a
n
),
? V
y ? (b
1
, b
2
, …,b
n
).
Inner product spaces, Orthogonal and Orthonormal Vectors
Institute of Lifelong Learning, University of Delhi                                       pg. 5

Define
n
1 1 2 2 n n i i
i1
x, y a b a b .... a b a b
?
? ? ? ? ?
?

Then it is an inner product  on V, called standard inner product.
Verification: Let x, y, z ? V, where
x = (a
1
, a
2
, …, a
n
), y = (b
1
, b
2
, ….,.b
n
), z = (c, c
2
, …, c
n
)
(i)
n
i i i
i1
x y,z (a b )c
?
? ? ?
?

nn
i i i i
i 1 i 1
a c b c
??
??
??

x,z y,z ??
(ii)
n
ii
i1
cx, y ca b
?
?
?

n
ii
i1
c a b
?
?
?

c x, y ?
(iii)
n
ii
i1
x, y a b
?
?
?

n
ii
i1
ab
?
?
?

n
ii
i1
ba
?
?
?

y,x ?
(iv)
n
ii
i1
x,x a a 0
?
??
?
if x ? 0
Thus x, y is an inner product on V.
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