\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 5}
\docdate{Due: February 20, 2019}

\begin{document}
\begin{problems}
Let $A$ be a subset of $\Reals^n$. A map $f:A\to \Reals^m$ if
whenever $x_j\to x$ in $A$, $f(x_j)\to f(x)$ in $\Reals^m$.

It is easy to show (you need not do this) that such a map
is continuous if and only if each component function $f_k:A\to
\Reals$, $1\le k\le m$ is continuous.

A path in a set $B\subseteq \Reals^m$ is a continuous map
$\gamma:[0,1]\to B$.  Given $b_0$ and $b_1$ in $B$, a path
from $b_0$ to $b_1$ is a path $\gamma$ in $B$ 
with $\gamma(0)=b_0$ and $\gamma(1)=b_1$.  The set is path connected if for all
$b_0$ and $b_1$ in $B$, there is a path from $b_0$ to $b_1$.

\problem By a concrete construction, show that $S^1\subseteq \Reals^2$ is
path connected.

\problem Use the intermediate value theorem to show that $\Reals\setminus\{0\}$
is not path connected.

\problem Show that $SU(2)$, the set of unit quaternions, 
is path connected.  You may use the fact that
a set $B$ is path connected if and only if there exists an $\hat b\in B$
such that for any $b$ in $B$ there is a path from $\hat b$ to $b$.

\problem Breifly explain why matrix multiplication is a continuous map from
$R^{n\times n}\times R^{n\times n}$ to $R^{n\times n}$.

\problem 3.2.1-3.2.3.  You are encouraged to have a brief solution; full
rigor is not required.

\end{problems}

\end{document}
