(a) For $A \in \mathbb{R}^{n \times n}$ and nonzero $\boldsymbol{v} \in \mathbb{R}^{n}$, define the $m$-th Krylov subspace $K_{m}(A, \boldsymbol{v})$ of $\mathbb{R}^{n}$. Prove that if $A$ has $n$ linearly independent eigenvectors with at most $s$ distinct eigenvalues, then

$$\operatorname{dim} K_{m}(A, \boldsymbol{v}) \leqslant s \quad \forall m$$

(b) Define the term residual in the conjugate gradient (CG) method for solving a system $A \boldsymbol{x}=\boldsymbol{b}$ with a symmetric positive definite $A$. State two properties of the method regarding residuals and their connection to certain Krylov subspaces, and hence show that, for any right-hand side $\boldsymbol{b}$, the method finds the exact solution after at most $s$ iterations, where $s$ is the number of distinct eigenvalues of $A$.

(c) The preconditioned CG-method $P A P^{T} \widehat{\boldsymbol{x}}=P \boldsymbol{b}$ is applied for solving $A \boldsymbol{x}=\boldsymbol{b}$, with

Prove that the method finds the exact solution after two iterations at most.

(d) Prove that, for any symmetric positive definite $A$, we can find a preconditioner $P$ such that the preconditioned CG-method for solving $A \boldsymbol{x}=\boldsymbol{b}$ would require only one step. Explain why this preconditioning is of hardly any use.