Obviously, there is no problem in changing the numbering of the matrix elements i,k=0,1,2,\dots,N-1 to
i,k=1,2,\dots,N . We have chosen to start from zero, since we deal with polynomials of degree N-1 .
Summarizing, for Legendre polynomials with N=2 we have
weights
\begin{equation*}
\omega : \left\{1,1\right\},
\end{equation*}
and mesh points
\begin{equation*}
x : \left\{-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right\}.
\end{equation*}