Adjacent sums of a cyclic basis
= Adjacent sums of a cyclic basis
{title2=$\det(I+S)=1-(-1)^n$}
For a <basis> $e_1,\ldots,e_n$ with cyclic indexing, the vectors $e_j+e_{j+1}$ have coefficient matrix $I+S$, where $S$ cyclically shifts coordinates. Its <determinant> is $1-(-1)^n$. They therefore form a <basis> exactly when $n$ is odd over a field of characteristic different from two. For even $n$, alternating coefficients produce a nontrivial <linear dependence>.