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yaser · #2 04-24-2013, 03:39 PM

Nice argument! (and thank you for the compliment)

Quote:

Originally Posted by Elroch

There is a non-trivial linear relation on these points:

$\sum\limits_{x\in X}a_x x = 0$

Rearrange so all the coefficients are positive (changing labels for convenience)

$\sum\limits_{x\in X_A}a_x x =\sum\limits_{x\in X_B}b_x x$

and

must be non-empty subsets of

because the relation is non-trivial, and the first co-ordinate of all the points is 1.

Just to elaborate, the coefficients are not all zeros so because of the constant 1 coordinate, there has to be at least one positive and one negative coefficient, hence the rest of the argument even if all other coefficients are zeros.