One can show that the solution $\hat{x}$ is also the unique minimizer of the quadratic form

$$\begin{equation*} f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. \end{equation*}$$

This suggests taking the first basis vector $\hat{r}_1$ (see below for definition) to be the gradient of $f$ at $\hat{x}=\hat{x}_0$, which equals

$$\begin{equation*} \hat{A}\hat{x}_0-\hat{b}, \end{equation*}$$

and $\hat{x}_0=0$ it is equal $-\hat{b}$.