Let $p \in \llbracket 1, d \rrbracket$. We define a relation on the set of bases of a subspace $V$ of dimension $p$ of $E$ by: $e$ and $e^{\prime}$ are in relation if $\det_e(e^{\prime}) > 0$ where $\det_e(e^{\prime})$ is the determinant of $e^{\prime}$ in the basis $e$. We admit that this relation is an equivalence relation on the set of bases of $V$ for which there exist exactly two equivalence classes called orientations of $V$. An oriented subspace is a pair $(V, C)$ where $C$ is an orientation of $V$.
We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$.
(a) Show that if $e$ and $e^{\prime}$ are two free families of cardinal $p$ of $E$ then $\Omega_p(e)$ and $\Omega_p(e^{\prime})$ are collinear if and only if $\operatorname{Vect}(e) = \operatorname{Vect}(e^{\prime})$.
(b) Show that for all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, there exists a unique $\Psi(V, C) \in \mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.