\documentclass[minion]{homework}

\newcommand{\Reals}{\mathbb{R}}
\doclabel{Math F490: Homework 5}
\docdate{Due: February 14, 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 Recal that for $p,q \in \Quats$, we define $\ip<p,q>=\Re(p\overline q)$,
where $\Re(q)$ is the real part of $q$.
\begin{subproblems}
% \item Show that if $p\in \Quats$, then $\ip<p,p>=|p|^2$.
% \item Using the facts that $\overline{ab}=\overline b \overline a$ and $\Re(\overline a) = \Re(a)$
% for all $a,b\in \Quats$, show that $\ip<p,q>=\ip<q,p>$.
% \item For $x=(x_1,x_2,x_3,x_4)\in \Reals^4$, let $Q(x)=x_1 + x_2\vi+ x_3\vj+x_4\vk$
% be its representation as a quaternion.
% Show that for $x,y\in \Reals^4$, $x\cdot y = \ip<Q(x),Q(y)>$.
% \item We say that $p$ and $q$ are orthogonal if $\ip<p,q>=0$. 
% Show that $p\in\Quats$ is orthogonal to $1$ if and only if $p$ is pure imaginary.
\item Suppose $s\in \Quats$.  Show that for all $p,q\in\Quats$,
$\ip<ps,qs> = |s|^2\ip<p,q>$.  
\item Show that for all $p,q\in\Quats$, $\ip<\overline p, \overline q>=\ip<p,q>$.
\item Suppose $s\in \Quats$.  Show that for all $p,q\in\Quats$,
$\ip<sp,sq> = |s|^2\ip<p,q>$.  
\item Suppose $u\in\Quats$ and $u\neq 0$.  Show that conjugation by $u$ takes the real quaternions to themselves.
\item Conclude quickly that if $u\in\Quats$ and $u\neq 0$, then
conjugation by $u$ takes the imaginary quaternions to themselves.
\end{subproblems}

\problem 2.1.1
\problem 2.1.2
\problem 2.2.1
\problem 2.2.2
\problem 2.2.5

\problem Extra Credit: 2.2.3
\end{problems}

\end{document}
