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I want to show that the following statement is false. Consider $n+1$ linearly independent functions on $\mathbf{C}$

$$ \phi_0, \cdots, \phi_n, $$

and the generalized Vandermonde matrix given by

$$ V_n = \phi_k(z_j) $$

for $n+1$ points in $\mathbf{C}$. Then $V_n$ is invertible IFF if $z_0, \cdots, z_n$ are pairwise distinct.

The only thing that came up in my mind would be to show that it is possible to have $det(V_n) = 0$ even if they are distinct, and that it is possible to have $det(V_n) \neq 0$ even if they are not pairwise distinct.

Nevertheless, I do not know if such a matrix is used for not pairwise distinct points, so maybe the second point is useless to prove.

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In fact, given $n+1$ points $z_j$, there is a plenty of linearly independent functions which have equal values in these points (for example, polynomials of degree $d> n+1$). Taking these functions for the given points we get singular matrix $V_n = \phi_k(z_j)$.

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    Hi again! Thanks. Yes the idea was that. The only problem is the following: do I have to prove both? Or I do not *care* about not distinct points? Because now I do not know if we are in the realm of interpolation. – 2017-02-25
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    @Andrei Kulunchakov I am not sure your counterexample is valid : do you think that polynomials with degree $>n+1$ taking the same values on these points could be linearly independent ? – 2017-02-25
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    @JeanMarie, for example, i think that it is sufficient to take polynomials of distinct degrees. Their linear independency could be proven by induction – 2017-02-25
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    @LorenzoFabbri, for each set of points $z_j$ invertibility is not guaranteed. If the set contains equal points, then its is not invertible either, as you will have equal columns in your matrix – 2017-02-25
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    @AndreiKulunchakov I am not completely sure. Fir of all, the matrix: is the last element $\phi_n(z_n^n)$ or $\phi_n(z_n)^n$? So, you are saying that I do not have to bother myself about the case of equal points. But I'm not sure for the case of distinct points: how can I obtain equal columns? – 2017-02-25
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    @AndreiKulunchakov I am still working on the proof. I even have doubts about the structure of the matrix: is it $(\phi _k (z_j))^2$ or $\phi _k (z_j^2)$? Moreover, each column is, for instance, $\phi_0$ or each row is $\phi_0$? – 2017-03-04