\documentclass[minion]{homework}

\newcommand{\Reals}{\mathbb{R}}
\newcommand{\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}{\mathbf{Re}}
\doclabel{Math F490: Homework 4}
\docdate{Due: February 13, 2019}

\begin{document}
\begin{problems}
\problem Stillwell 2.5.1

\problem Stillwell 2.5.2

\problem Stillwell 2.5.3

\problem A linear map $L:\Reals^n\to\Reals^n$ can be represented
as a matrix as follows.  Let $e_i$ be the standard basis vectors
and let $x_j = L(e_j)$.  We can write $x_j$ as a sum $\sum_i L_j^i e_i$.
The matrix of $L$ is the matrix $L^i_j$ where $i$ is the row index and $j$
is the column index.  

\begin{subproblems}
\item Consider the map $q->-\overline{q}$ from $\Quats$ to $\Quats$.  
Show that this map is linear.  Then compute its matrix representation.
What is the determinant of this matrix representation?

\item Let $u$ be a unit quaternion $u$.  Consider the map $q\mapsto uq$
from $\Quats$ to $\Quats$.  Show that this map is linear.  Then compute its 
matrix representation.  Then compute the determinant of this matrix representation.

\problem Challenge. Without citing that an isometry of $\Reals^n$ that fixes
the origin is linear, show that such a map takes lines through the origin
to lines through the origin.  Due in two weeks.
\end{subproblems}
\end{problems}

\end{document}
