Do nontrivial linear combinations stay nontrivial when you re-write the vectors as combinations?

Question

Suppose $v_1,\ldots,v_n$ are vectors such that $v_i \ne c v_j$ for any $i,j$ and there is a nontrivial linear combination $$a_1v_1+\cdots+a_nv_n=0.$$ Suppose further that for each $i$, $v_i=w_{i_1}+w_{i_2}$ (the $w_{i_j}$ are not assumed distinct). Certainly $$a_1(w_{1_1}+w_{1_2})+\cdots+a_n(w_{n_1}+w_{n_2})=0,$$ but is it a nontrivial linear combination?

Answer 1

E.g. if $v_1 = w_1 + w_2$, $v_2 = w_2 + w_3$, $v_3 = w_3 + w_4$, $v_4 = w_4 + w_1$, $v_1 - v_2 + v_3 - v_4 = 0 w_1 + 0 w_2 + 0 w_3 + 0 w_4$. Is that what you meant?