Past exam of the numerical linear algebra and approximation course of the University of Oxford / 2008 / 4 / a

The Krylov subspace is $\boxed{K_k(A,b)=\mathrm{span}\{b,Ab,\dots ,A^{k-1}b\}}$. If $y\in K_k$, write $y=q(A)b$ with $\mathrm{deg}q\le k-1$. Since $b=Ax$,which has degree at most $k$ and $p(0)=1$. Conversely, if $x-y=p(A)x$ with these properties, then $q(t)=(1-p(t))/t$ is a polynomial of degree at most $k-1$ andThis is the polynomial description of the admissible errors in a Krylov subspace method.