= Solution
Expand the <quadratic form> of this <Gram matrix>:
$$
\begin{aligned}
Q(\mathbf c)&=\sum_{i,j}c_ic_j\mathbf x_i^T\mathbf x_j\\
&=\left(\sum_i c_i\mathbf x_i\right)^T\left(\sum_jc_j\mathbf x_j\right)
=\left\|\sum_i c_i\mathbf x_i\right\|^2.
\end{aligned}
$$
For $\mathbf c\ne0$, <linear independence> of the real vectors implies $\sum_i c_i\mathbf x_i\ne0$, so its squared <Euclidean norm> is strictly positive. Thus
$$
\boxed{Q(\mathbf c)>0\text{ for every }\mathbf c\ne0,}
$$
proving the <positive definiteness of a Gram matrix> in the independent-vector case.
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