For all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$ (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 $\operatorname{vol}_p(e) = \left|\operatorname{det}\left(P_b^e\right)\right|$ where $P_b^e$ is the change of basis matrix from $b$ to $e$, i.e. $e_j = \sum_{i=1}^p \left(P_b^e\right)_{ij} b_i$ for all $j \in \llbracket 1, p \rrbracket$. (b) Show that for all $e, e^{\prime} \in E^p$, we have $\left|\Omega_p(e)(e^{\prime})\right| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.
For all $e \in E^p$, we call the $p$-volume of $e$ the quantity
$$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$
(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 $\operatorname{vol}_p(e) = \left|\operatorname{det}\left(P_b^e\right)\right|$ where $P_b^e$ is the change of basis matrix from $b$ to $e$, i.e. $e_j = \sum_{i=1}^p \left(P_b^e\right)_{ij} b_i$ for all $j \in \llbracket 1, p \rrbracket$.
(b) Show that for all $e, e^{\prime} \in E^p$, we have $\left|\Omega_p(e)(e^{\prime})\right| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.