Linear independence of a special set of vectors

Question

Let $$ S_1=(e_1,e_2,..,e_n)$$ a set of vectors.And suppose that $$ S_2=(v_1,v_2,..,v_n)$$ is linearly independent , where $$ v_i = e_i+e_{i+1} \quad if \quad 1 \le i \le n-1 \quad and \quad v_n = e_n+e_1$$ How can we show that if $$S_2$$ is linearly indenpendent then $$S_1$$ is linearly indenpendent as well ?

Answer 1

Hint: $e_1=v_1-e_2=v_1-v_2+e_3=\dots\ $ write it out, then you can express $e_1$ by $v_i$. Then $e_2=v_1-e_1$, and so on..

It turns out that $n$ must be odd.
Then we have $\def\s{\mathrm{span}} \s(S_1)\subseteq\s(S_2)\subseteq\s(S_1)$, which has dimension $n$, so both $S_2$ and $S_1$ are bases of this common spanned subspace.