Consider the linear system $Ax=b,$ where $A\in\mathbb{R}^n$ is symmetric positive definite. In the conjugate gradient method, show that
a) Every vector $r_i, i=0,\dots,n-1$ can be written as $$r_i=\sum_0^i\gamma_{ik}A^kr_0,$$ where $\gamma_{ik}\in\mathbb{R}$.
b)Every vector $p_i, i=0,...,n-1$ can be written as $$p_i=\sum_0^i\tilde\gamma_{ik}A^kr_0,$$ where $\tilde\gamma_{ik}\in\mathbb{R}.$
My step are as follows:
In the conjugate gradient method, I choose the vector $r_i$ such that $$r_i=b-Ax_i$$ where $$x_i=\sum_{k<i}\alpha_kp_k$$
Via the Gram-Schmidt process I arrived at $$r_{i+1}=r_i-\alpha_iAp_i$$ where $$\alpha_i=\frac{{r_i}^Tr_i}{p^TAp_i}$$ and $$p_{i+1}=r_{i+1}-\frac{{r_{i+1}}^Tr_{i+1}}{{r_i}^Tr_i}p_i.$$
But I am not able to proceed further to the answer.
Help would be greatly appreciated.