\documentclass[minion]{homework}

\newcommand{\Reals}{\mathbb{R}}
\doclabel{Math F490: Homework 6}
\docdate{Due: February 21, 2022}
\def\Quats{\mathbb{H}}
\def\ip<#1,#2>{\left<#1,#2\right>}
\newcommand{\vi}{\mathbf{i}}
\newcommand{\vj}{\mathbf{j}}
\newcommand{\vk}{\mathbf{k}}
\renewcommand{\Re}{\mathrm{Re}}

\begin{document}
\begin{problems}

\problem 2.2.4

\problem Suppose $T:\Reals\to \Reals$ is an isometry and that $T(0)=0$.  Give
a direct proof that either $T(x)=x$ for all $x$ or $T(x)=-x$ for all $x$.

\problem Let $u$ be a nonzero vector in $\Reals^n$.  Show that for every
$x\in\Reals^n$ there is a unique number $a$ and unique vector $w$ perpendicular 
to $u$ such that $x=au+w$.  Hint: take a dot product to determine what $a$ must be.

\problem Recall that a reflection through a hyperplane through the origin
in $\Reals^n$ is a linear map $T:\Reals^n\to\Reals^n$ such that
there exists a nonzero vector $u$ with
\begin{enumerate}
  \item $T(u)=-u$
  \item $T(w)=w$ whenever $w$ is perpendicular to $u$.
\end{enumerate}
Show that such a map is an isometry.

\problem Suppose $T:\Reals^n\to\Reals^n$ is an isometry and that $T(0)=0$.
Suppose moreover that $T(v)=w$ and that $w\neq v$. Show that the midpoint 
$(v+w)/2$ is perpendicular to the vector $u=v-w$.  Also, draw a diagram that contains
$0$, $v$, $w$, $u$ and $(v+w)/2$.

\end{problems}

\end{document}
