cyclic subspace
Let V be a vector space over a field k, and x∈V. Let T:V→V be a linear transformation. The T-cyclic subspace generated by x is the smallest T-invariant subspace
which contains x, and is denoted by Z(x,T).
Since x,T(x),…,Tn(x),…∈Z(x,T), we have that
W:=span{x,T(x),…,Tn(x),…}⊆Z(x,T). |
On the other hand, since W is T-invariant, Z(x,T)⊆W. Hence Z(x,T) is the subspace generated by x,T(x),…,Tn(x),… In other words, Z(x,T)={p(T)(x)∣p∈k[X]}.
Remark. If Z(x,T)=V we say that x is a cyclic vector of T.
Title | cyclic subspace |
---|---|
Canonical name | CyclicSubspace |
Date of creation | 2013-03-22 14:05:03 |
Last modified on | 2013-03-22 14:05:03 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 12 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 15A04 |
Classification | msc 47A16 |
Synonym | cyclic vector subspace |
Related topic | CyclicDecompositionTheorem |
Related topic | CyclicVectorTheorem |
Defines | cyclic vector |