In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector v in a vector space V and a linear transformation T of V is called the T-cyclic subspace generated by v. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.
Let be a linear transformation of a vector space and let be a vector in . The -cyclic subspace of generated by is the subspace of generated by the set of vectors . This subspace is denoted by . In the case when is a topological vector space, is called a cyclic vector for if is dense in . For the particular case of finite-dimensional spaces, this is equivalent to saying that is the whole space .
There is another equivalent definition of cyclic spaces. Let be a linear transformation of a topological vector space over a field and be a vector in . The set of all vectors of the form , where is a polynomial in the ring of all polynomials in over , is the -cyclic subspace generated by .
Let be the two-dimensional vector space and let be the linear operator on represented by the matrix relative to the standard ordered basis of . Let . Then . Therefore and so . Thus is a cyclic vector for .