Krylov subspace based typicality methods have found success in the study of the thermodynamics of quantum systems. We will survey two well-known algorithms for equilibrium systems: the finite temperature Lanczos method (FTLM) and the kernel polynomial method (KPM). Our overview will shed light on some of their commonalities and differences. We will then provide an overview to the finite precision behavior of the famous Lanczos algorithm, and address some widespread concerns about its stability. In particular, we will argue that the FTLM works fine in finite precision arithmetic, and that the KPM can (and often should) be implemented using the Lanczos algorithm. We will conclude with a brief introduction to some recent work on randomized algorithms for estimating partial traces. No background in numerical linear algebra is assumed!