Page 213 - DMTH502_LINEAR_ALGEBRA

Unit 19: Cyclic Subspaces and Annihilators

T-cyclic Subspace Notes

If  is any vector in V, the T-cyclic subspace generated by  is the subspace Z(; T) of all vectors
of the form g(T), g in F[x]. If Z(; T) = V, then  is called a cyclic vector for T.
K
In other words, the subspace Z(; T) is the subspace spanned by the vectors T , K  0 and thus
 is a cyclic vector for T if and only if these vectors span V.

Example 1: For any T, the T-cyclic subspace generated by the zero vector is the zero
space.
If the vector  is a characteristic vector for T the space Z(; T) is one dimensional.

For the identity operator, every non-zero vector generates a one dimension cyclic subspace,
thus, if dim V > 1, the identity operator has no cyclic vector.

2
Example 2: Consider the linear operator T on F which is represented in the standard
basis by the matrix
 0 0
A   
 1 0 
Here the cyclic vector is  = (1, 0); for
1
A = 
1 2
So that for any vector  given by
(a, b)
We have (a, b) = a + b ,
1 2
so = a + bA
1 1
= (a + bA)
1
2
Thus the polynomial g in F (x) can be taken as
g = a + bx

For the same operator T, the cyclic subspace generated by  = (0, 1) is the one dimensional space
2
spanned by  , because  is a characteristic vector of T.
2 2
3
Example 3: Consider the linear operator T on F , which is represented by A;
 0 1 0
 
A  1 0 1
 
 0 1 0
 
Here A = 0, but A  0. So if  is a vector such that A   0 i.e.,  =  = (1,0,0), then the basic
2
2
3
1
vectors will be (, T, T ) and space generated by  is a cyclic subspace.
2
1
19.2 Annihilators
For any T and , we shall be interested in linear relations
k
c  + c T + ... + c T  = 0
0 1 k

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