Let us be working over the field of complex numbers. Suppose $f(x)= a_n x^n + \cdots +a_1 x + a_0$ is a degree $n$ polynomial with $n$ distinct roots $z_1,\ldots,z_n$.
Is the following matrix always invertible?
$$\left( \begin{array}{ccc} 1 & 1 & \cdots & 1 \\ z_1 & z_2 & \cdots & z_n \\ \vdots & \vdots & & \vdots \\ z_1^{n-1} & z_2^{n-1} & \cdots & z_n^{n-1} \end{array} \right) $$
Motivation: I was learning how to solve linear recurrence relations with constant coefficients, and the standard method involves writing down the characteristic polynomial and then solving for the roots. After which, we use the initial conditions to solve for the constants ->this is the step which I am wondering about: is it always possible to do that? This quesiton amounts to the same question above.
Edit: I guess I can extend my question since I am concerned with general linear recurrence relations with constant coefficients. If the roots of the characteristic polynomial are repeated, then we use $z^n, nz^n, n^2z^n,...$ for our "guesses", so the general matrix to invert is something like:
$$\left( \begin{array}{ccc} 1 & 0 & \cdots & 1 \\ z_1 & z_1 & \cdots & z_k \\ \vdots & \vdots & & \vdots \\ z_1^{n-1} & (n-1)z_1^{n-1} & \cdots & z_k^{n-1} \end{array} \right) $$
How do we know this matrix is invertible?
