When he was a young boy Matthew Ryder was saved from a subway cave in by a time traveling hero named Waverider. An experience that would give Matthew a strong admiration for superheroes of the 20th century. A few days after his recuse the boy and his parents watched as Superman and the Metal Men attempted to move an unstable generator. Realizing that this was the event that would trigger an explosion that would kill the boys' (actually Waveriders' past self) parents Waverider transported the generator out of time. Although he succeeded in saving the Ryder family he found that he could not alter his own past. Thanks to the nature of his change into Waverider his own existence was not disrupted. As a result the version of Matthew Ryder he had saved did not become Waverider in the altered timeline but co-existed with him as the leader of the Linear Men. This version devoted his life to his passion in time travel forming a group in an attempt to protect the timeline from outside forces. They set up their base in an outpost at the end of time called the Vanishing Point.
As the post-crisis means of time travel were severally limited the group had to frequently replace their human parts for machine replacements in order to survive. Ryders' team was comprised of archivist Liri Lee (his wife), a cyborg version of Rip Hunter, Travis O'Connell and later Waverider himself. O'Connells' last mission was to bring hero Booster Gold back to his point of origin but was intercepted by Superman who was mistakenly sent into the timestream during their struggle. Eventually O'Connell sacrificed his life during a confrontation with Superman in the 30 century. When Superman questioned the Linear Mens' methods and O'Connells' mission to return Booster to the 25th century the Linear Men claimed their teammates' actions went against their own. Hunter stating that although Booster didn't know it he was essential to the evolution of man and needed to stay in the past. At some point the Linear Men expanded their organization to become leaders of a force of temporal time enforcers called the Linear Authority.
Different Timelines?: The Linear Men have gone through a few changes over the years. In The Kingdom, a tie in from the main DCU timeline into the popular Kingdom Come series, Rip Hunter rebelled against his teammates to help the heroes and protect an aspect of time he had discovered called Hypertime. In Superman The Man of Steel #118 all the remaining Linear Men save Liri Lee were reduced to brains that communicated telepathically. (Yeah, I don't get it either.) Deemed insane all of them given new lives.
Where are they now?: In the new Earth established after Infinite Crisis it isn't clear how much the Linear Men have been altered. Waverider was apparently murdered in the 52 series while refusing to give any information on Rip Hunter (he claimed he didn't know much anyway.) The cyborg version of Rip Hunter hasn't been seen and according to Black Beetle only Matthew Ryder and Liri Lee remain. They were said to have been locked away in a cell at Vanishing Point by Hunter for disagreements over the existence of people who should not or will not exist. This seems to go against past characterizations of Rip Hunter but given the speaker of such accusations it's likely that this is stretching the truth.
First Appearance: Adventures of Superman #476