Exercise 20

If we have $n$ data points $(x_{1},y_{1}),(x_{2},y_{2}),\dots\,,(x_{n},y_{n})$ and want to find a polynomial $p(x)=a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_{1}x+a_{0}$ passing through those points then we can plug in the points to get an $n$ equation/$n$ unknown linear system. The matrix of coefficients for that system is the Vandermonde matrix. Prove that the determinant of the transpose of that matrix of coefficients

equals the product, over all indices $i,j\in\text{$\{\hskip 1.00006pt1,\dots,n\hskip 1.00006pt\}$}$ with $i<j$, of terms of the form $x_{j}-x_{i}$. (This shows that the determinant is zero, and the linear system has no solution, if and only if the $x_{i}$’s in the data are not distinct.)