Prove affine independence of these points I am having some trouble with this problem.
I want to prove that the following points are affine independent.
Let $k\geq 2$ and $i\in\{0,1,\dots,2k\}$ we define $S_i =(s_i^t)$ where $s_i^{i+2t (\bmod 2k+1)}=1$ for $t\in\{0,1,\dots,k-1\}$ and $s_i^t=0$ in the other cases, $s_i\in \mathbb{R}^{2k+1}$. Let now $X$ be defined as $\left\{ S_{i}\,:\, i\in\{0,1,\dots,2k\}\right\}$, I need to show that the points in $X$ are affine independent. 
Any advice?
Regards,
John
 A: The $S_i$ are actually linearly independent vectors in $\mathbb R^{2k+1}$ so they are certainly affinely independent.
Let $P$ be the $(2k+1)\times(2k+1)$ matrix with general entry $t^j_i=1$ if $j\equiv i+1\hbox{ (mod $2k+1$)}$ and $0$ otherwise. This is the matrix of a linear tranformation that simply shifts the coordinates of every vector one position to the right (and the last coordinate becomes the first one).
Let $S$ be the matrix whose rows are the coordinates of the $S_i.$ Then
$$S=I+P^2+\ldots+P^{2(k-1)}$$
Because $P$ represents a coordinate shift, repeating it $2k+1$ times leaves every vector unchanged:
$$P^{2k+1}=I$$
and therefore all eigenvalues $\omega$ of $P$ are complex numbers that satisfy $\omega^{2k+1}=1.$ Apart from $\omega=1$ these numbers all satisfy $\omega^2\neq1$ and $\omega^{2k}\neq1.$
The eigenvalues $\lambda$ of $S$ all have the form
$$\lambda=1+\omega^2+\ldots+\omega^{2(k-1)}=\frac{1-\omega^{2k}}{1-\omega^2}$$
where the second equality holds only for $\omega\neq1$ but in any case all eigenvalues of $S,$ and thus its determinant, are nonzero. This shows that the rows of $S$ are linearly independent.
