Linear brakes feature on things like train tracks and rollercoasters, where the track itself (or something mounted on it) works as part of the brake.
The simplest linear, eddy-current brakes have two components, one of which is stationary while the other moves past it in a straight line. In a rollercoaster ride, you might have a series of powerful, permanent magnets permanently mounted at the end of the track, which produce eddy currents in pieces of metal mounted on the side of the cars as they whistle past. The cars move freely along the track until they reach the very end of the ride, where the magnets meet the metal and the brakes kick in.
This kind of approach is no use for a conventional train, because the brakes might need to be applied at any point on the track. That means the magnets have to be built into the structure that carries the train's wheels (known as the bogies) and they have to be the kind of magnets you can switch on and off (electromagnets, in other words). Typically, the electromagnets move a little less than 1cm (less than 0.5 in) from the rail and, when activated, slow the train by creating eddy currents (and generating heat) inside the rail itself. It's a basic law of electromagnetism that you can only generate a current when you actually move a conductor through a magnetic field (not when the conductor is stationary); it follows that you can use an eddy current brake to stop a train, but not to hold it stationary once it's stopped (on something like an incline). For that reason, vehicles with eddy current brakes need conventional brakes as well.