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Prove that every nonzero vector $\mathbf r=(r_1, r_2, \dots, r_n)^\top\in\mathbb Z^n$ such that $r_1 \geq r_2 \geq \cdots \geq r_n$ and $\sum_{i=1}^n r_i =0$ can be written as a linear combination of the vectors $\mathbf x_1,\mathbf x_2,\ldots,\mathbf x_{n-1}$ with integer coefficients, where $\mathbf x_p = (n-p,\,n-p,\,\dots,\,n-p,\,-p,\,-p,\,\dots,\,-p)^\top$ in which $n-p$ is repeated $p$ times and $-p$ is repeated $n-p$ times.

Thoughts: They form a $n \times (n-1)$ matrix, if every minor of size $n-1$ has not $0$ determinant we are done, but this strategy doesn't seem easy.

Thanks!

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  • $\begingroup$ Which are those that form an $n\times(n-1)$ matrix? $\endgroup$ – user26857 Apr 5 '17 at 8:05
  • $\begingroup$ @user26857 They are those rows made setting $p=1, \cdots , n-1$. $\endgroup$ – Maffred Apr 5 '17 at 17:28
  • $\begingroup$ @user1551 Every vector $(r_1, \cdots, r_n)$ with those properties can be generated as a linear combination of those exact $(n-1)$ vectors, one for every $p=1, \cdots , n-1$. $\endgroup$ – Maffred Apr 5 '17 at 17:30

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