Page 216 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 19.3 Summary

 Know what is a cyclic vector, cyclic subspaces of a linear operator T acting on a finite
dimension vector space.

 See that if the cyclic vector is found then the basis vectors of the sub-space of the linear
transformation can be found that span the space of the linear operator.

 Understand how to find the T-annihilator of a vector and also find out that the monic
polynomial which generates it has a degree equal to the dimension of the cyclic subspace.

19.4 Keywords

A Cyclic Vector: If the T-cyclic subspace generated by the vector  spans the whole finite
dimensional space V then  is called a cyclic vector for the linear T.
Cyclic Subspace: If  is a vector in a finite dimensional space V of a linear operator T, then the
invariant subspace W which contains all g(T) for every polynomial g over F is called T-cyclic
subspace generated by .
T-annihilator of a Vector:  consisting of all polynomials g over F such that g(T) = 0 is called
T-annihilator of . The unique monic polynomial which generates this set will also be called the
T-annihilator of .

19.5 Review Questions

1. Let T be a linear operator on the finite dimensional space V . Suppose that T is
n
diagonalizable. If T has a cyclic vector, then show that T has n-distinct characteristic values.
2. If S and T are nilpotent linear transformation which commute, prove that ST and S + T are
nilpotent linear transformations.

19.6 Further Readings

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

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