High-mobility two-dimensional electron gas (2DEG) which resides at the interface between GaAs and AlGaAs layered semiconductors has been used experimentally and theoretically to study ballistic electron transport. The present paper is motivated by recent experiments in magnetic electron focusing. The proposed device consists of two quantum point contacts (QPCs) serving as electron injector and collector which are placed in the same semiconductor GaAs/AlGaAs heterostructure. Here we focus on a theoretical study of the injection of electrons via a quantum wire/QPC into an open two-dimensional (2D) reservoir. The transport is considered for non-interacting electrons at different transmission regimes using the mode-matching technique. The proposed mode-matching technique has been implemented numerically. Electron flow through the quantum wire with hard-wall rectangular, conical and rounded openings has been studied. We have found for these three cases that the geometry of the opening does not play a crucial role for the electron propagation. When a perpendicular magnetic field is applied the electron paths in the 2D reservoir are curved. We analyse this case both classically and quantum-mechanically. The effect of spin-splitting due to exchange interactions on the electron flow is also considered. The effect is clearly present for realistic choices of device parameters and consistent with observations. The results of this study may be applied in designing magnetic focusing devices and spin separation.