Linear flexible polymers of N-g monomers, end-grafted at the density sigma onto a spherical surface of radius R ("hairy sphere"), including the case of a flat impenetrable wall (R --> infinity), are simulated using the cooperative motion algorithm. The simulations have been carried out for a wide range of parameters characterizing the hairy surfaces (N-g, sigma, and R) and concern in more detail the influence of length of chains constituting the surrounding medium on the anchored ones. This is achieved by gradual varying the polymerization degree N of the matrix chains between the two extremes of a dense melt of identical chains (N = N-g) and a simple solvent consisting of nonbounded beads (N = 1). The analysis of the properties of the grafted layer is based on concentration profiles, distributions of polymer centers of mass, and distributions of free ends of chains. Moreover, some of the scaling relations and analytical formulas predicted for grafted chains are tested.