(i) Let $V$ be a finite-dimensional real vector space with an inner product. Let $e_{1}, \ldots, e_{n}$ be a basis for $V$. Prove by an explicit construction that there is an orthonormal basis $f_{1}, \ldots, f_{n}$ for $V$ such that the span of $e_{1}, \ldots, e_{i}$ is equal to the span of $f_{1}, \ldots, f_{i}$ for every $1 \leqslant i \leqslant n$.

(ii) For any real number $a$, consider the quadratic form

$$q_{a}(x, y, z)=x y+y z+z x+a x^{2}$$

on $\mathbf{R}^{3}$. For which values of $a$ is $q_{a}$ nondegenerate? When $q_{a}$ is nondegenerate, compute its signature in terms of $a$.