Page 212 - DMTH502_LINEAR_ALGEBRA

Linear Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 19: Cyclic Subspaces and Annihilators

CONTENTS
Objectives
Introduction

19.1 Cyclic Subspaces
19.2 Annihilators
19.3 Summary

19.4 Keywords
19.5 Review Questions
19.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand clearly the meaning of cyclic vector, cyclic-vector subspace and T-annihilator
of .
 See that in the case of a nilpotent linear operator one finds out the basis of the vectors ,
T, T ,... as the basis that spans the space of the linear transformation T.
2
 Know that closely related to the idea of cyclic vector  one understands the T-annihilator
of  i.e., finds a polynomial g in F such that g(T = 0
 See that with the help of these ideas one can understand the rational forms as well as the
Jordan forms.

Introduction

The cyclic subspaces, the cyclic vector  and the T annihilators of  help us in the factoring of a
linear operator T on the finite dimensional space to give a simple and elementary form.
2
In this unit the nilpotent transformation helps us in finding the basis vectors , T, T ,... that
spans the space and this will help us in introducing the rational and the Jordan forms.

19.1 Cyclic Subspaces

We are considering an arbitrary but fixed linear operator on V, a finite-dimension vector space
over the field F. If  is any vector in V, there is a smallest subspace of V which is invariant under
T and contains . This subspace can be defined as the intersection of all, T-invariant subspaces
which contains , if W is any subspace of V which is invariant under T and contains , then W
2
3
must also contain the vector T; hence W must contain T , T , etc. In other words, W must
contain g(T) for every polynomial g over F. The set of all vectors of the form g(T), with g in
F(x), is clearly invariant under T, and is thus the smallest T-invariant subspace which contains .

206 LOVELY PROFESSIONAL UNIVERSITY