Definition: Let $w \in \mathbb{C}^n$ be a nonzero vector. The Householder matrix $U_w \in M_n$ is defined by $$U_w=I−2(w^∗w)^{−1}ww^∗$$
Exercise: Let $n \geq 2$ and let $x,y \in \mathbb{R}^n$ be unit vectors. If $x=y$ let $w$ be any real unit vector that is orthogonal to $x$. If $x \neq y$ let $w=x-y$. Show that $U_w x = y$.
Conclude that any $x \in \mathbb{R}^n$ can be transformed by a real Householder matrix into any $y \in \mathbb{R}^n$ such that $\|x\|_2=\|y\|_2$.
I've solved the first question but I'm stuck with the second one. I've written $y=U_w x$ and I have tried to compute $\|U_w x\|_2$ without success.
If you have a trouble with the conclusion while you proved the exercise statement, then most of your work is done.
Given two nonzero $x$ and $y$ such that $\rho:=\|x\|_2=\|y\|_2$, set $\tilde{x}:=x/\rho$ and $\tilde{y}:=y/\rho$. Then $\tilde{x}$ and $\tilde{y}$ are unit vectors and from the exercise you know how to make a $w$ in $U_w$ so that $U_w\tilde{x}=\tilde{y}$. Multiply by $\rho$ and get $U_wx=y$.