I am requesting a solution or elaboration for educational purposes to an examination problem that I wasn't able to proceed with on our examination yesterday (Winter 5th Semester - (31/1/2018)).
The following problem was an exams question on Numerical Linear Algebra at the School of Applied Mathematics and Physics - Mathematics sector of National Technical University of Athens.
Problem 4
$(a)$ For the non stable iterative method of Richardson type, with $Q=I$, show that : $$r^{(k)} = \prod_{j=0}^{k-1} (I-a_jA)r^{(0)}$$
$(b)$ Show that the $k-$iteration $x^{(k)}$ exists in the space $W=\{v=x^{(0)}+y,y\in K_k(A,r^{(0)})\}$, $\quad$ $\space$ $\space$ $\space$ where $K_k(A,r^{(0)})=\text{span}\space\{r^{(0)},Ar^{(0)},\dots,A^{k-1}r^{(0)}\}$ is the Krylov subspace of order $k$.
I will also attach the question in Greek just for the mention, since it's the original form of it stated by our acting professor :
Θεμα 4
$(a)$ Για τη μη στάσιμη διαδικασία τύπου Richardson, με $Q=I$, να δείξετε ότι : $$r^{(k)} = \prod_{j=0}^{k-1} (I-a_jA)r^{(0)}$$ $(b)$ Στη συνέχεια να δείξετε ότι η $k-$επανάληψη $x^{(k)}$ ανήκει στο χώρο : $$W=\{v=x^{(0)}+y,y\in K_k(A,r^{(0)})\}$$ $\quad$ όπου $K_k(A,r^{(0)})=\text{span}\space\{r^{(0)},Ar^{(0)},\dots,A^{k-1}r^{(0)}\}$ ο υπόχωρος Krylov τάξης $k$.