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\begin{center}
\begin{Large}
\textbf{MA243 Geometry}\\[.7em]
\textbf{Worksheet 1}\\[1.3em]
\end{Large}

\textbf{Please hand your solutions in by Thu 7th Oct 2004, 2:00pm.}
\end{center}
\hspace*{\fill}\\[-2em]

\begin{enumerate}
\renewcommand\theenumi{\Alph{enumi}}
\item 
\textbf{Points, vectors, and lines in $\R^3$}

Let $P=(1\c 0\c 0)$ and $Q=(2\c 1\c 1)\in\R^3$ (with Cartesian
coordinates $x_1\c x_2\c x_3$). Determine the vector $\vec{PQ}$ and the
line $PQ$ in parametric form: $PQ = \{ u + \lambda v \mid \lambda\in\R
\}$ with appropriate $u$ and $v$. Also describe $PQ$ by equations:
$PQ = \{ x\in\R^3 \mid f_i(x)=0\c i=1,\dots\c N\}$ with appropriate $N$
and $f_i$.
\item 
\textbf{Distance and isometries}

Consider $\R^2$ with Cartesian coordinates $x_1\c x_2$ and let
\[
d\colon\, \R^2\times\R^2\longrightarrow\R,
\quad
d\left( (x_1\c x_2), (y_1\c y_2)\right) :=
\max_{i\in\{1,2\}} \left\{ |x_i-y_i|\right\}.
\] 
Show that $(\R^2\c d)$ is a metric space. Draw the unit circle around the
origin in $(\R^2\c d)$ and prove that 
$g\left( (x_1\c x_2)\right):=(x_2\c x_1)$ preserves the distance $d$.

[Hint: You may use without proof that $|a-c|\leq|a-b|+|b-c|$ for all
$a\c b\c c\in\R$.]
\item 
\textbf{Eigenvectors and eigenvalues}

Consider the following map on $\R^3$ (with coordinates 
$x_1\c x_2\c x_3$):
\[
n:=(1\c 1\c 0)\in\R^3,\;\;
x=(x_1\c x_2\c x_3)\in\R^3: \quad
x \longmapsto 
x-(x_1+x_2)\, n.
\]
Determine the $3\times3$ matrix $A$ such that the map is given by
$x\mapsto Ax$. 

Give an orthonormal basis $e_1\c e_2\c e_3$ of
$\R^3$ such that each $e_i$ is an eigenvector of $A$, and determine the 
corresponding eigenvalue.

How does $A$ act on the cube with
vertices $(\pm1\c \pm1\c \pm1)$?

\item 
\textbf{Complex numbers}

Consider $(\R^2,d_{\mathrm{eucl}})$ with $d_{\mathrm{eucl}}$ the
Euclidean distance. The map 
\[
A\colon\R^2\longrightarrow\R^2,\quad
x\mapsto Ax,\quad
A=\left( \begin{array}{cc} 0&-1\\ 1&0 \end{array}\right)
\]
is an isometry of $(\R^2,d_{\mathrm{eucl}})$ (proof not required). Show
that $A$ does not have real eigenvalues. 

We extend $A$ to a map
$A\colon\C^2\longrightarrow\C^2$ by setting 
$A(x+iy):=Ax+iAy$ for $x\c y\in\R^2$. If $z=x+iy=(z_1\c z_2)\in\C^2$, let 
$\overline z:=x-iy=(\overline z_1\c \overline z_2)$ 
denote the complex conjugate of $z$,
and $z\cdot z:= (z_1)^2+ (z_2)^2$. 
Calculate two linearly independent eigenvectors $e\c e^\prime\in\C^2$
of the extended map. Show that without loss of generality you can
use $e^\prime=\overline{e}$, and that
$e\cdot e=\overline{e}\cdot \overline{e}=0$.
\end{enumerate}
\end{document}