Show that there is at most one reflection $T$ of $V$ such that $T(a)=-a$ and $T(R)=R$.

Question

Let $R$ be a finite subset of real vector space $V$ consisting of nonzero vectors which spans $V$. Show that there is at most one reflection $T$ of $V$ such that $T(a)=-a$ and $T(R)=R$.

Suppose we have two reflections $T_1$ and $T_2$ which satisfy the required conditions.Consider $T=T_1T_2$ then $T(a)=a$ and $T(R)=R$.Since $R$ spans $V$ therefore order of $T$ is finite.I am stuck here,how to proceed further? I am trying to prove that $T$ is identity which will of course prove that $T_1=T_2$.

Answer 1

Hint: The composition of two reflection is either a rotation or a translation. Now compare the numer of fixed ponts.