= Solution
{c}
Symmetry of the real <inner product> gives $A^T=A$. Moreover, <4/b/i> with $y=x$ shows that
$$
x^TAx=\left\|\sum_{i=1}^n x_iv_i\right\|^2\geq0
\qquad(x\in\mathbb R^n),
$$
so the <Gram matrix> is a <positive semidefinite matrix>. To also check reality of every <eigenvalue>, allow a nonzero <vector> $z=a+ib\in\mathbb C^n$ with $Az=\lambda z$. Since $A$ is a real <symmetric matrix>,
$$
z^*Az=a^TAa+b^TAb+i(a^TAb-b^TAa)=a^TAa+b^TAb\geq0.
$$
Here $z^*$ is the <conjugate transpose>, and $a^TAb=b^TAa$ cancels the imaginary part. Thus
$$
\boxed{\lambda=\frac{z^*Az}{z^*z}\in[0,\infty),}
$$
because $z^*z>0$. This proves both that every <eigenvalue> is real and that it is nonnegative.
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