= Solution
The <Krylov subspace> is $\boxed{K_k(A,b)=\operatorname{span}\{b,Ab,\ldots,A^{k-1}b\}}$. If $y\in K_k$, write $y=q(A)b$ with $\deg q\leq k-1$. Since $b=Ax$,
$$
x-y=[I-Aq(A)]x=p(A)x,\qquad p(t)=1-tq(t),
$$
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 $k-1$ and
$$
\boxed{y=Aq(A)x=q(A)b\in K_k(A,b).}
$$
This is the polynomial description of the admissible errors in a <Krylov subspace method>.
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