Page 273 - DMTH502_LINEAR_ALGEBRA
P. 273

Sachin Kaushal, Lovely Professional University   Unit 25: Linear Functional and Adjoints of Inner Product Space





                    Unit 25: Linear Functional and Adjoints of                                  Notes
                                   Inner Product Space


            CONTENTS

            Objectives
            Introduction
            25.1 Linear Functional

            25.2 Adjoint of Linear Operators
            25.3 Summary
            25.4 Keywords
            25.5 Review Questions
            25.6 Further Readings


          Objectives

          After studying this unit, you will be able to:
              Understand that any linear functional  f on a  finite-dimensional inner product space is
               ‘inner product with a  fixed vector in the space’.
              Prove the existence of the ‘adjoint’ of a linear operator T on V, this being a linear operator
               T* such that (T   ) = (  T*  )  for all  and  in V.

              A  linear  operator  T  such that  T =  T* is called  self-adjoint  (or  Hermitian).  If    is an
               orthonormal basis for V, then [T*]  = [T] .
                                          B
          Introduction

          The idea of the linear functional helps in understanding the nature of the inner product.

          The concept of adjoint of a linear transformation with the help of the inner product helps in
          understanding the self-adjoint operators or Hermitian operators.
          This unit also makes a beginning to the understanding of unitary operators and normal operators.
          The normal operator T has the property that T*T = TT*.

          25.1 Linear Functional

          In this section we treat linear functionals on inner product space and their relation to the inner
          product. Basically any linear functional f on a finite dimensional inner product space is ‘inner
          product with a fixed vector in the space’, i.e. that such an f has the form f( ) = (  ) for some fixed
            in V. We use this result to prove the existence of the ‘adjoint’ of a linear operator T on V, this
          being a linear operator T* such that (T  ) = (  T* ) for all  and  in V. Through the use of an
          orthonormal basis, this adjoint operation on linear operators (passing from  T to T*) is identified
          with the operation of forming the conjugate transpose of a matrix.

          We define a function f  from V, any inner product space into the scalar field by
                                           f  ( ) = (  ).




                                           LOVELY PROFESSIONAL UNIVERSITY                                   267
   268   269   270   271   272   273   274   275   276   277   278