Q:

Link between Gram Matrix and volume of parallelpiped question - Determinant

I'm not sure how to approach this question. I have proved $det(G(v_1,...v_k)≥0$ and have proven the triangle inequality using the Cauchy-Schwarz inequality.

A:

Well, I think the most clean way to proceed here is by induction. Fon $n=1$ we have:
$$det(G(v_1))=Vol_1(P(v_1))^2=<v_1,v_1>$$
Now let us suppose the formula valid for $n-1$ and induce the formula for $n$. First of all we notice that both $det(G(v_1,...,v_N))$ and $Vol_n(P(v_n))$ do not depend from a specific choice of the base (that's because the first one is a determinant and the second one is originated by the norm). 
Now instead of calculating directly $det(G(v_1,...,v_n))$ we calculate the determinant after a change of basis
$$det(G(v_1,...,v_n))=det(S^{-1}G(v_1',...,v_n')S)$$ 
where $v_n'$ is orthogonal to all $v_i'$ with $i<n$.
In this coordinates the calculation should be a little more convenient: 
$$det(G(v_1,...,v_n))=<v_n',v_n'>det(S^{-1}G(v_1',...,v_{n-1}')S)$$
now using the inductive hypotesis
$$<v_n',v_n'>det(G(v_1',...,v_{n-1}'))=<v_n',v_n'>Vol_{n-1}(P(v_1',..,v_{n-1}'))^2$$
And finally using the recursive definition of the Volume and using the fact that the volume is invariant for change of basis we obtain
$$Vol_{n}(P(v_1',..,v_{n}'))^2=Vol_{n}(P(v_1,..,v_{n}))^2$$

