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{\bf Math 177 HW 1 Due Tuesday, January 23}

\begin{enumerate}
\item Show that every isometry $\phi:\mathbb R^n \to \mathbb R^n$ is a bijection. (One to one and onto.) Hence every isometry has an inverse.
\item An isometry is called {\it central} if it fixes 0. Show that every isometry is a central isometry followed by translation.
\item Thinking of $\mathbb R^n$ as a vector space over $\mathbb R$, show that every central isometry is a linear transformation, that is, 
\begin{enumerate}
\item For all $x, y, \in \mathbb R^n$, we have $\phi(x+y)=\phi(x)+\phi(y)$.
\item For all $x\in \mathbb R^n$ and $\lambda \in \mathbb R$, we have $\phi(\lambda x)=\lambda \phi(x)$.
\end{enumerate}
Hence every central isometry is given my multiplication by a matrix. Combined with the previous result, we  now know that that every isometry is of the form $\phi(x)=Ax+b$ for some matrix $A$ and some vector $b$.
\item Recall that a matrix $A$ with real entries is called {\it orthogonal} if $A^TA=I$. Show that if $A$ is orthogonal, then $\det A=\pm 1$. Show that $\phi(x)=Ax$ defines a central isometry if and only if $A$ is an orthogonal matrix.
\item What are all $1 \times 1$ orthogonal matrices? Use this to show that every isometry of $\mathbb R^1$ is either  a translation or a reflection.
\item Show that a  $2 \times 2$ orthogonal matrix must be of the form
$$\left( \begin{array}{cc} \cos \theta&-\sin \theta\\ \sin \theta&\cos \theta\end{array}\right )   \mbox{ or } 
\left( \begin{array}{cc} \cos \theta&\sin \theta\\ \sin \theta&-\cos \theta\end{array}\right ).$$
Show that the first matrix corresponds to rotation around the origin through the angle $\theta$ and the second to reflection through the line containing the origin that makes an angle of $\theta/2$ with the $x$-axis.
\item An isometry $\phi(x)=Ax+b$ is called {\it direct}, or {\it orientation preserving}, if $\det A=1$ and {\it opposite}, or {\it orientation reversing},  if $\det A=-1$. Show that for $\mathbb R^2$, a direct isometry is either a rotation or a translation and that an opposite isometry is either a reflection or a glide reflection. (A {\it glide reflection} is reflection in a line $\ell$ followed by translation in a  direction parallel to $\ell$.) 
\end{enumerate}
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