We propose a linear mapping relationship between the polarization of the fundamental mode and the cylindrical vector (CV) modes on the first-order Poincaré sphere (FOPS) in fiber. The new method is based on the four-dimensional complex Jones matrices in terms of the linearly polarized mode bases. With our theoretical model, an all-fiber approach to generate arbitrary CV beams on the FOPS is proposed theoretically and verified experimentally. In the experiment, through the combination of a mode converter and a two-segment cascaded few-mode fiber with fixed stresses, it is possible to generate all CV modes on the FOPS by only adjusting the polarization of the fundamental mode. The Stokes parameters of the output light are measured to verify our scheme, which shows good agreement with the theoretical prediction. The method may provide a convenient way to generate CV beams and evolve the polarization states in any path on the FOPS, which is expected to have potential applications in encoding information and quantum computation.