The study of the equations defining projective varieties together with all higher relations amongst them has been very actively studied since the beginnings of algebraic geometry, yet virtually all natural questions are completely open. In this talk, I will discuss how the very classical technique of "projection", can be made into an operation acting on the linear-part of these higher relations ("linear syzygies"). I will state a theorem which tells you that if you study enough projections, no information is lost under this operation. As a consequence, we prove an earlier conjecture of myself and Farkas on syzygies of curves embedded by line bundles of high degree, which may be thought of as "geometrizing" a result of Ein-Lazarsfeld.