 Solution

As written, the assertion needs a nonzero-vector assumption: the singleton set containing the zero vector is mutually orthogonal but is linearly dependent. For the linear independence of nonzero orthogonal vectors, suppose ${\sum }_{i=1}^k{\alpha }_i{q}_i=0$. Taking the inner product with $q_j$ givesSince $q_j\ne 0$, positive definiteness gives $\Vert q_j{\Vert }^2>0$, hence ${\alpha }_j=0$. This holds for every $j$, proving linear independence under the necessary assumption.