Page 281 - DMTH502_LINEAR_ALGEBRA

Unit 25: Linear Functional and Adjoints of Inner Product Space

f ( ) = ( ) Notes
A linear operator T* is an adjoint of T on V, such that (T ) = ( T* ) for all and in V.
Self-adjoint (or Hermitian): A linear operator T such that T = T* is called self-adjoint
(or Hermitian). If is an orthonormal basis for V, then [T*] = [T]* and so a self-adjoint if and only
if its matrix in every orthonormal basis is a self-adjoint matrix.

25.5 Review Questions

2
1. Let T be the linear operator on C defined by T = (1 + i, 2), T = (i, i). Using the standard
1 2
inner product, find the matrix of T* in the standard ordered basis.
2. Let V be a finite-dimensional inner product space and T a linear operator V. Show that the
range of T* is the orthogonal complement of the null space of T.

25.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
I N. Herstein, Topics in Algebra
Michael Artin, Algebra

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