If we choose the conjugate vectors $\hat{p}_k$ carefully, then we may not need all of them to obtain a good approximation to the solution $\hat{x}$. We want to regard the conjugate gradient method as an iterative method. This will us to solve systems where $n$ is so large that the direct method would take too much time.

We denote the initial guess for $\hat{x}$ as $\hat{x}_0$. We can assume without loss of generality that $$\begin{equation*} \hat{x}_0=0, \end{equation*}$$ or consider the system $$\begin{equation*} \hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, \end{equation*}$$ instead.