For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form. Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We still denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$. Let $p \in \{1, \ldots, n\}$. We denote by $F$ the space spanned by $e_1, \ldots, e_p$. a) Show that $F^\perp$ is the preimage under $h$ of $\operatorname{Vect}(e_{p+1}^*, \ldots, e_n^*)$, where $h$ is defined in I.A.1. b) Show that $\operatorname{dim}(F) + \operatorname{dim}(F^\perp) = n$. c) Show that $(F^\perp)^\perp = F$.
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We still denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$. Let $p \in \{1, \ldots, n\}$. We denote by $F$ the space spanned by $e_1, \ldots, e_p$.
a) Show that $F^\perp$ is the preimage under $h$ of $\operatorname{Vect}(e_{p+1}^*, \ldots, e_n^*)$, where $h$ is defined in I.A.1.
b) Show that $\operatorname{dim}(F) + \operatorname{dim}(F^\perp) = n$.
c) Show that $(F^\perp)^\perp = F$.