Let $V$ be a finite-dimensional vector space $A$ be an ortogonal matrix and let $R:=R_1R_2 … R_n \in \mathrm{O}(V)$ be the product of $n$ reflections (it doesn’t matter if $n$ is even or odd). We know that each element of the special ortogonal group $\mathrm{SO}(V)$ can be expressed as the sum of a linear map $p_A$ and a antilinear map $q_A$. The map $p_A$ is not always invertible, so we define $\mathrm{SO}_*(V) \subset \mathrm{SO}(V)$ the set for which $p_A$ is invertible.
I was asked to prove that $AR \in \mathrm{SO}_*(V)$. If $n$ is even is very straightforward, but I’m stuck when $n$ is odd.
¿Any suggestions? Thanks :)
Oh, I almost forgot to mention. Each reflexion $R_k$ satisfies $R_k(e_k)=-ek$ and $R(v)=v$ if $d(v,e_k)=d(v, Je_k)=0$, where $\lbrace e_1, e_2, … e_n \rbrace$ is a orthonormal basis of the kernel of $p_A^t$, $J$ is a complex structure (a lineal application such that $J^2=-1$) and $d$ is a positive definite symmetric biliear form which preserves d, i.e. $d(Jv,Ju)=d(v,u)$, for every $v,u \in V$
Again, thanks in advance for any help that you can give me.