  This paper develops a new class of algorithms for general linear systems and
eigenvalue problems. These algorithms apply fast randomized sketching to
accelerate subspace projection methods, such as GMRES and Rayleigh--Ritz. This
approach offers great flexibility in designing the basis for the approximation
subspace, which can improve scalability in many computational environments. The
resulting algorithms outperform the classic methods with minimal loss of
accuracy. For model problems, numerical experiments show large advantages over
MATLAB's optimized routines, including a $100 \times$ speedup over gmres and a
$10 \times$ speedup over eigs.
