Hey, so the idea is to look at what the conditions are for (1,x) and (1,y) to be linearly dependent then for (1,x,x^2), (1,y,y^2), and (1,y,y^2), then to generalize to n such vectors in C^n.

The most obvious thing (which is definitely sufficient) is if we're calling our n vectors v_i=(1,b_i,b_i^2,...,b_i^n-1), if b_i=b_j for i!=j then they're linearly dependent.

In fact this is verified by looking at their determinants. They are (up to sign) of the form (b1-b2) for n=2, (b1-b2)(b2-b3)(b1-b3) for n=3, and (b1-b2)(b1-b3)(b1-b4)(b2-b3)(b2-b4)(b3-b4) for n=4, etc... product of differences of all possible pairings ( I haven't proved this pattern, only tested in for n=2,3,4,5) If the determinant of the concatenation of these column vectors is 0 that's equivalent to their being linearly dependent, so we're good.

The question is how to actually prove that this pattern about the determinants is valid or just to prove that any of the bases being equal is also a necessary condition for these vectors to be linearly dependent.

Thanks