Yes, it works with the same principle as any other motor.  However on a linear motor, the magnets are placed alligned on a a flat surface, making the element that we want to move (i.e. a door leaf), slide along the line of magnets by attraction/repulsion of the same.

Does it work with magnetic levitation? No… but almost.  The attraction force of the magnets is around 40Kg. For a door leaf weighing less than 40 Kg, the magnets compenssate almost its full weight, however we still need the carriers and its wheels to guide the door leaf along the track.  Having heavy door leaves levitating at all times, even in stand-by mode would require bigger magnets and high power consumption that would make the product too expensive to maintain. Linear motor and magnetic levitation technologies share the same principle of operation, where motion is generated by the attraction and repulsion of the magnets placed in-line.