Cylindrical vector beams are described by the representation formalism of finite electromagnetic beams in free space. This is achieved by factorizing the field vector into a mapping matrix and a Jones-like vector. The vectorial property of a finite beam can be described by a degree of freedom of the mapping matrix that can be determined by the azimuthal angle of a fixed unit vector with respect to the wave vector. In addition, it is a simple and flexible experimental device for converting linearly polarized light beams into cylindrical vector beams. The radially or azimuthally polarized beam is easily converted into either one of these by varying the applied voltage.