Following Humphrey's Lie algebra, a root system $\Phi$ of the Eucludean space $E_n$ (with inner product $(\cdot,\cdot)$) is a set of vectors in $E_n$ satisfying following:
(1) $\Phi$ contains non-zero vectors, and generates the space $E_n$.
(2) If $\alpha\in \Phi$ then $-\alpha\in \Phi$ and these are the only multiples of $\alpha$ which are in $\Phi$.
(3) If $\alpha,\beta\in \Phi$ then $\langle \beta,\alpha\rangle:=2(\beta,\alpha)/(\alpha,\alpha)$ is an integer.
The subgroup of orthogonal group $O(n)$ generated by reflections $\sigma_{\alpha}$, $\alpha\in\Phi$is called Weyl group of $\Phi$.
For every $\alpha\in\Phi$, let $\alpha^*=2\alpha/(\alpha,\alpha)$. Then $\Phi^*=\{ \alpha^* : \alpha\in \Phi\}$ is called dual of $\Phi$.
Exercise: Show that $\Phi^*$ is a root system , whose Weyl group is naturally isomorphic to Weyl group of $\Phi$.
My answer: I verified the above axioms of root system for $\Phi^*$. Next, for $\alpha\in\Phi$, since $\alpha^*=2\alpha/(\alpha,\alpha)$, a multiple of $\alpha$. Hence reflection $\sigma_{\alpha}$ in $\alpha$ and the reflection $\sigma_{\alpha^*}$ in $\alpha^*$ is same(equal); so Weyl groups are equal.
Q.1. Am I right? I wondered why it is asked to prove that they are naturally isomorphic rather than to prove that they are equal(same)?
Q.2 Some places, the dual root system is defined by considering dual space of $E_n$; then is that notion equivalent to the Humphrey's notion of dual root system?