This is the first post in a (planned) series on Krylov subspaces, projection processes, and related algorithms. We already discussed projection processes when talking about the Bregman algorithm and we will see that the Krylov (sub-)spaces will be generated by a set of vectors that are not necessarily orthogonal. Getting an orthogonal basis for these spaces can for example be done by applying Gram-Schmidt. Forthcoming posts will discuss further interesting applications such as the Arnoldi iteration or the Lanczos method.