If we have a set of distinct vectors, say $w_1$ to $w_n$ from a vector space W, and we have that $$c_1w_1+c_2w_2+...+c_nw_n=0$$ and
$$(c_1-c_2)w_1+(c_2-c_3)w_2+...+(c_{n-2}-c_{n-1})w_{n-1}+c_nw_n=0$$, does this gaurantee that c1=c2=...=0? Since I assume $w_1$ to $w_n$ is not fixed, or would they actually be fixed values.
Take $w_1=w \not=0$, $w_2 = 2w$, $w_3 = 3w$, $c_1 = c_3 = 1$ and $c_2 = -2$. Then $$ c_1w_1 + c_2w_2 + c_3w_3 = 0 $$ and $$ (c_1-c_2)w_1 + (c_2-c_3)w_2 + w_3 = 0 $$