For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$ and $\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = (\det(\operatorname{Gram}(e, e)))^{1/2}$. (a) Show that if $e \in E^p$ is a free family and if $b \in E^p$ is an orthonormal basis of $\operatorname{Vect}(e)$, then $\operatorname{vol}_p(e) = |\det(P_b^e)|$ where $P_b^e$ is the change of basis matrix from $b$ to $e$ i.e. $e_j = \sum_{i=1}^p (P_b^e)_{ij} b_i$ for all $j \in \llbracket 1, p \rrbracket$. (b) Show that for all $e, e^{\prime} \in E^p$, we have $|\Omega_p(e)(e^{\prime})| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.
For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by
$$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$
and $\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = (\det(\operatorname{Gram}(e, e)))^{1/2}$.
(a) Show that if $e \in E^p$ is a free family and if $b \in E^p$ is an orthonormal basis of $\operatorname{Vect}(e)$, then $\operatorname{vol}_p(e) = |\det(P_b^e)|$ where $P_b^e$ is the change of basis matrix from $b$ to $e$ i.e. $e_j = \sum_{i=1}^p (P_b^e)_{ij} b_i$ for all $j \in \llbracket 1, p \rrbracket$.
(b) Show that for all $e, e^{\prime} \in E^p$, we have $|\Omega_p(e)(e^{\prime})| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.