Integration points and weights with orthogonal polynomials
We can derive this result in an alternative way by defining the vectors
$$
\begin{equation*}
\hat{x}_k=\left(\begin{array} {c} x_0\\
x_1\\
.\\
.\\
x_{N-1}\end{array}\right) \hspace{0.5cm}
\hat{\alpha}=\left(\begin{array} {c} \alpha_0\\
\alpha_1\\
.\\
.\\
\alpha_{N-1}\end{array}\right),
\end{equation*}
$$
and the matrix
$$
\begin{equation*}
\hat{L}=\left(\begin{array} {cccc} L_0(x_0) & L_1(x_0) &\dots &L_{N-1}(x_0)\\
L_0(x_1) & L_1(x_1) &\dots &L_{N-1}(x_1)\\
\dots & \dots &\dots &\dots\\
L_0(x_{N-1}) & L_1(x_{N-1}) &\dots &L_{N-1}(x_{N-1})
\end{array}\right).
\end{equation*}
$$