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Richa Nandra, Lovely Professional University Unit 24: Orthogonal Complements

Unit 24: Orthogonal Complements Notes

CONTENTS
Objectives
Introduction

24.1 Orthogonal Complement
24.1.1 Orthogonal Vectors
24.1.2 Pythagorean Theorem

24.1.3 Orthogonal Sets
24.1.4 Orthogonal Compliment: Definition
24.1.5 The Orthogonal Decomposition Theorem or Projection Theorem
24.2 Summary
24.3 Keywords

24.4 Review Questions
24.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define Orthogonal complement
 Understand theorems on it
 Understand the Orthogonal decomposition theorem
 Solve problems related to Orthogonal complement.

Introduction

In this unit, we shall start with orthogonality. Then we shall move on to definition of orthogonal
complement. Let M be a closed linear subspace of H. We know that M is also a closed linear
subspace, and that M and M are disjoint in the sense that they have only the zero vector in
common. Our aim in this unit is to prove that H = M M , and each of our theorems is a step in
this direction.

24.1 Orthogonal Complement

24.1.1 Orthogonal Vectors

Let H be a Hilbert space. If x, y H then x is said to be orthogonal to y, written as x y, if
(x, y) = 0.
By definition,

(a) The relation of orthogonality is symmetric, i.e.,
x y y x

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