Abstract: Event cameras are ideal sensors for line-based motion estimation since they predominantly respond to edges in the scene. Recent work has shown that events generated by a single line satisfy a system of polynomial equations that uniquely describe a manifold in the space-time volume. In solving this system, partial linear velocity and line parameters can be recovered. In this work, we show that, with a suitable line parametrization, this polynomial system is actually linear in the unknowns, which allows us to design a novel linear solver. Unlike existing solvers, our linear solver (i) is fast and numerically stable since it does not rely on expensive root finding, (ii) can solve both minimal and overdetermined systems with more than 5 events (i.e. $N \geq 5$), and (iii) admits the characterization of all degenerate cases and multiple solutions. The found line parameters are singularity-free and, implicitly, scale-free, eliminating the need for auxiliary constraints typically encountered in previous work. To recover the full linear camera velocity we fuse observations from multiple lines with a novel velocity averaging scheme that explicitly minimizes a geometric error, instead of solving an algebraic equation, as done in previous work. Extensive experiments in synthetic and real-world settings demonstrate that our method surpasses the previous work in accuracy, and operates over 600 times faster. We plan on open-sourcing the code upon acceptance.