Lemma 3

Let q be a positive quadratic form on a real vector space V and let B be the associated symmetric bilinear form. Let $a_1, \ldots, a_n$ be elements of V such that $B(a_i, a_j) \leq 0$ for $i \neq j$.

(i) If $c_1, \ldots, c_n$ are real numbers such that $q \left( \sum_i c_i a_i \right) = 0$, then

(ii) If q is non-degenerate and if there exists a linear form f on V such that $f(a_i) > 0$ for all i, the vectors $a_1, \ldots, a_n$ are linearly independent.

The relation $B(a_i, a_j) \leq 0$ for $i \neq j$ immediately implies that

hence (i). If q is non-degenerate, the relation $\sum_i c_i a_i = 0$ thus implies that it follows that, for any linear form f on V, we have $\sum_i |c_i| f(a_i) = 0$, and hence $c_i = 0$ for all i if we also have $f(a_i) > 0$ for all i. This proves (ii).