By using multiple-scale methods, we study analytically the propagation properties of the weak linear polarization probe optical field in a cold lifetime-broadened five-level V type atomic system via electromagnetically induced transparency. It is shown that with the strength of the coupling field increasing, the absorption of the probe optical field decreases rapidly and the group velocity of the probe optical field increases quickly under the consideration of linear effect. However, the velocity is several orders of magnitude slower than the light speed in vacuum. Meanwhile, we find that for the same magnetic field the nonlinear Faraday rotation direction is opposite to linear Faraday rotation, and its rotation angle grows bigger than that of linear Faraday rotation. These results mean that the Faraday rotation of the electromagnetically induced transparency medium can be controlled by the nonlinear effect.