A digital pre-distortion (DPD) scheme for concurrent multi-beam transmission in fully digital multiple-input, multiple-output (MIMO) systems, using fast Fourier transform (FFT) beamforming and so-called virtual array processing, is proposed. In a MIMO array with nonlinear power amplifiers (PAs), transmitting multiple beams concurrently yields intermodulation products that end up in both user and non-user directions. In the setting with few users in a large array, the array dimension will typically be much larger than the number of generated intermodulation products. At the same time, linearization per-PA is excessively costly for large arrays. This work shows that is instead possible to linearize the system by producing predistorted user beams, and non-user intermodulation products, through DPD processing in a virtual array, of a much smaller dimension than the physical array. Theoretical derivations and simulation examples show how this approach can lead to manyfold reductions in DPD complexity.