When is a set of vectors in $\mathbb{C}^n$ where the $n$th component of each vector is the $(n-1)$st power of some scalar linearly dependent?

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?

• If and only if $\alpha_i = \alpha_j$ for some $i\ne j$. See en.wikipedia.org/wiki/Vandermonde_matrix. – Song Jan 12 at 16:24
• @DavidC.Ullrich Thanks, edited – Fortox Jan 12 at 16:37
• 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 $x\ge a$.) – David C. Ullrich Jan 12 at 16:48