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

Linear Algebra

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 
 
Table of Contents: 
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 
 ? References/Bibliography/Further reading  
 
 
 
 
 
 
 
 
 
 
 
 
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 
 
Table of Contents: 
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 
 ? References/Bibliography/Further reading  
 
 
 
 
 
 
 
 
 
 
 
 
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 
additional structure called inner product. This additional structure 
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 
 
Table of Contents: 
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 
 ? References/Bibliography/Further reading  
 
 
 
 
 
 
 
 
 
 
 
 
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 
additional structure called inner product. This additional structure 
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. 
Value Addition : Note 
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 
 
Table of Contents: 
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 
 ? References/Bibliography/Further reading  
 
 
 
 
 
 
 
 
 
 
 
 
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 
additional structure called inner product. This additional structure 
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. 
Value Addition : Note 
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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