The question I have is: Suppose $x_1, x_2, ... , x_k$ are linearly independent n-dimensional vectors. Show that $x_1+x_2, x_2+x_3, ... ,x_{k-1}+x_k, x_k+x_1$ are also linearly independent. Here k is an odd number.
I thought I had a proof done, but was told that I showed linear dependence instead. What I did was took a set of constants $a_1, a_2, ..., a_k$ and set $a_1(x_1+x_2)+a_2(x_2+x_3)+...+a_k(x_k+x_1)=0$ and proved that each constant was 0. Is that right? If not, how can I prove this?