Antenna selection is capable of reducing the hardware complexity of massive multiple-input multiple-output (MIMO) networks at the cost of certain performance degradation. Reconfigurable intelligent surface (RIS) has emerged as a cost-effective technique that can enhance the spectrum-efficiency of wireless networks by reconfiguring the propagation environment. By employing RIS to compensate the performance loss due to antenna selection, in this paper we propose a new network architecture, i.e., RIS-assisted massive MIMO system with antenna selection, to enhance the system performance while enjoying a low hardware cost. This is achieved by maximizing the channel capacity via joint antenna selection and passive beamforming while taking into account the cardinality constraint of active antennas and the unit-modulus constraints of all RIS elements. However, the formulated problem turns out to be highly intractable due to the non-convex constraints and coupled optimization variables, for which an alternating optimization framework is provided, yielding antenna selection and passive beamforming subproblems. The computationally efficient submodular optimization algorithms are developed to solve the antenna selection subproblem under different channel state information assumptions. The iterative algorithms based on block coordinate descent are further proposed for the passive beamforming design by exploiting the unique problem structures. Experimental results will demonstrate the algorithmic advantages and desirable performance of the proposed algorithms for RIS- assisted massive MIMO systems with antenna selection.