The coefficients are given by $$ \begin{equation*} \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. \end{equation*} $$ Multiplying with \( \hat{p}_k^T \) from the left gives $$ \begin{equation*} \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b}, \end{equation*} $$ and we can define the coefficients \( \alpha_k \) as $$ \begin{equation*} \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k} \end{equation*} $$