  We apply a hybrid approach which combines loop and Fock quantizations to
fully quantize the linearly polarized Gowdy $T^3$ model in the presence of a
massless scalar field with the same symmetries as the metric. Like in the
absence of matter content, the application of loop techniques leads to a
quantum resolution of the classical cosmological singularity. Most importantly,
thanks to the inclusion of matter, the homogeneous sector of the model contains
flat Friedmann-Robertson-Walker (FRW) solutions, which are not allowed in
vacuo. Therefore, this model provides a simple setting to study at the quantum
level interesting physical phenomena such as the effect of the anisotropies and
inhomogeneities on flat FRW cosmologies.
