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