Suppose $\alpha_1,\alpha_2,\dots,\alpha_n \in \mathbb{C}$. Consider the set of vectors $S = \{x\in \mathbb{C}^n : x = (1, \alpha_i, \alpha^2_i, \dots,\alpha^{n-1}_i), \ 1\leq i \leq n \ \}$. Under what conditions on $\alpha_1, \alpha_2, \dots , \alpha_n$ is $S$ linearly dependent?

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    $\begingroup$ If and only if $\alpha_i = \alpha_j$ for some $i\ne j$. See en.wikipedia.org/wiki/Vandermonde_matrix. $\endgroup$ – Song Jan 12 at 16:24
  • $\begingroup$ @DavidC.Ullrich Thanks, edited $\endgroup$ – Fortox Jan 12 at 16:37
  • $\begingroup$ If it's not clear why it matters consider a simpler example: (i) If $a\in\Bbb R$ then $\{x\in\Bbb R:x\ge a\}=[a,\infty)$. (ii) Otoh $\{x\in\Bbb R:a\in\Bbb R. x\ge a\}=\Bbb R$. (Because in the second example there's a "such that $a\in\Bbb R$" inside the braces; for any $x$ there is $a$ such that $x\ge a$.) $\endgroup$ – David C. Ullrich Jan 12 at 16:48

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