Linear independence of nonzero orthogonal vectors ($⟨q_i,q_j⟩=0(i\ne j),q_i\ne 0$)

A finite collection of nonzero pairwise orthogonal vectors is linearly independent. Given ${\sum }_i{a}_i{q}_i=0$, take the inner product with $q_j$ to obtain $a_j\Vert q_j{\Vert }^2=0$. Positive definiteness and $q_j\ne 0$ force $a_j=0$ for every index. The nonzero condition matters: adjoining a zero vector preserves pairwise orthogonality but destroys linear independence.