Bases for Krylov subspaces

In this chapter we consider the computation of bases for the Krylov subspace \(\mathcal{K}_k(A,r_0)\). We could have used another notation for the vector on which the Krylov subspace is constructed, but in the next chapters, the Krylov subspaces we have to deal with will be based on the initial residual vector \(r_0=b-Ax_0\).