Polynomial Interpolation using Vandermonde Matrix

For a system of $N+1$ points, the order is $N$

The following ansatz is said to be true

$$y_i={\alpha }_0+{\alpha }_1{x}_i\cdots {\alpha }_{N-1}{x}_i^{N-1}+{\alpha }_N{x}_i^N,i=0,\dots ,N$$

Eg. when $N=1$, $y_i={\alpha }_1{x}_i+{\alpha }_0$

In Matrix form

$$\left(\begin{array}{c}{y}_0\\ y_1\\ ⋮\\ y_N\end{array}\right)=\underset{\text{Vandermond Matrix}}{\underbrace{\left(\begin{array}{ccccc}1& x_0& x_0^2& \cdots & x_0^N\\ 1& x_1& x_1^2& \cdots & x_1^N\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& x_N& x_N^2& \cdots & x_N^N\end{array}\right)}}\left(\begin{array}{c}{\alpha }_0\\ {\alpha }_1\\ ⋮\\ {\alpha }_N\end{array}\right)$$

The Interpolation polynomial is then given by

$$p(x)={\alpha }_0+{\alpha }_{1}x+\cdots +{\alpha }_{N-1}{x}^{N-1}+{\alpha }_N{x}^N$$

Note: $i$ represents each point, it’s much simpler to write it as a function (implicitly)