Experimental and theoretical piezospectroscopic investigation of A^+ centers in GaAs/AlGaAs quantum wells doped with beryllium is presented. Spectra of linearly polarized photoluminescence are studied experimentally depending on applied uniaxial pressure. A model of the A^+ center in the quantum well in the presence of uniaxial deformation in the plane of the quantum well has been constructed. Analytical expressions for the level energy, optical transition intensities, and polarization ratio have been obtained. In the framework of the proposed theory, the experimentally observed change in the polarization ratio depending on pressure and the shift of the line maximum towards short waves are explained.