% Exam, plain TeX

\magnification\magstep2
\def\Span#1{\langle#1\rangle}
\def\implies{\mathrel{\Longrightarrow}}
\def\Glide{\mathop{\rm Glide}}
\def\Rot{\mathop{\rm Rot}}
\def\Transl{\mathop{\rm Transl}}
\def\Refl{\mathop{\rm Refl}}
\def\SO{\mathop{\rm SO}}
\def\Cup{\bigcup}
\def\1{^{-1}}
\def\la{\lambda}
\def\La{\Lambda}
\def\Si{\Sigma}
\def\fie{\varphi}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\E{{\bf E}}
\def\proj{{\bf P}}
\def\aff{{\bf A}}

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\itemitem{1.\quad (a)} Define a motion of the Euclidean plane
$\E^2$. Describe in words and figures the different types of motion,
and explain without proof the theorem classifying motions into types.

\itemitem{(b)} Let $L$ be a line, $\bf a$ a vector along
$L$, and $P$ a point not on $L$, as in the diagram below. Consider
the glide reflection $g=\Glide(L,\bf a)$ and the rotation
$r=\Rot(P,2\theta)$. By considering its effect on points and lines
in the diagram below, or otherwise, determine the composite motion
$t=r\circ g$.

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\itemitem{2.\quad (a)} Let $X$ be a topological space and $Y\subset
X$ a subset with the subspace topology. Under the assumption that
$X$ is compact, prove that
$Y$ closed implies $Y$ compact. Under what condition on $X$ is the
converse true? (The proof is not required).

\itemitem{(b)} Let $X$ and $Y$ be compact metric spaces and
$\fie\colon X\to Y$ a continuous bijective map. Sketch the proof
that $\fie$ is a homeo\-morphism.

\itemitem{(c)} Let
$S^1=\bigl\{(x,y)\in\R^2\bigm|x^2+y^2=1\bigr\}$ be the unit circle,
and $\fie\colon S^1\to\R^n$ a continuous map with the property that
for all $(x,y)\in S^1$
 $$
\fie(x,y)=\fie(x',y')\iff (x,y)=\pm(x',y').
 $$ Prove that the image $\fie(S^1)$ is homeomorphic to $S^1$.

\medskip
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\itemitem{3.\quad (a)} Say what it means for a real $n\times n$
matrix $A$ to be orthogonal. If $A$ is a $3\times3$ orthogonal
matrix, prove from first principles that there exists a real vector
$x\in\R^3$ such that $Ax=\pm x$.

\itemitem{(b)} State without proof the theorem on the normal form of
an orthogonal matrix; explain what it means in the case of a
$3\times3$ orthogonal matrix with $\det A=1$.

\itemitem{(c)} Define the quaternion conjugate $q^*$ of a quaternion
$q$, and explain how to find the inverse $q\1$ if $q\ne0$. Let $p,q$
be quaternions. Prove that
 $$
\hbox{$p$ is pure imaginary}\implies\hbox{$qpq\1$ is pure imaginary.}
 $$ Identify the space of pure imaginary quaternions $p=xi+yj+zk$
with Euclidean space $\E^3$ (with coordinates $x,y,z$). Assuming
that $|q|^2=1$ for simplicity, prove that the map
$r_q\colon\E^3\to\E^3$ given by $r_q(p)=qpq\1$ is an element of
$\SO(3)$.

\itemitem{(d)} Find a unit quaternion $q$ for which $r_q$ fixes the
$z$-axis and rotates the $x,y$ plane through an angle $\theta$.

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\itemitem{4.\quad (a)} Define the notions of affine linear subspace
and linear span in affine space $\aff^n_\R$. State and prove the
theorem on dimension of intersection of affine linear subspaces
$E,F\subset\aff^n_\R$ (results on vector spaces may be used freely).
Explain briefly why the lecturer seems to think that this result is
more convenient to state in projective space $\proj^n_\R$.

\itemitem{(b)} Consider $\triangle PQR$ and $\triangle P'Q'R'$ in
projective 3-space $\proj^3$, where
 $$
\matrix{P=(0,0,0,1),&Q=(1,0,0,1),&R=(0,1,0,1),\cr\cr
P'=(0,0,1,1),&Q'=(1,0,1,1),&R'=(0,1,1,1).}
 $$ By writing out the lines $PP'$, $QQ'$, $RR'$, show that
$\triangle PQR$ and
$\triangle P'Q'R'$ are in perspectivity. [Hint: you may want to take
coordinates $x,y,z,t$ and to visualise
$PQRP'Q'R'$ as a triangular prism in the affine piece $t=1$.] Now
determine the points of intersection of the corresponding sides of
the triangles, for example:
 $$
\matrix {QR:(x+y=t,z=0),\quad Q'R':(x+y=t,z=t),\cr\cr
\qquad\implies QR\cap Q'R'=(1:-1:0:0).}
 $$

\itemitem{(c)} State and prove Desargues' theorem for two triangles
$\triangle PQR$ and $\triangle P'Q'R'$ spanning distinct planes in
$\proj^3_\R$ and in perspectivity from a point $O$. Verify it in the
example of (c).

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\item{5.\enspace} Give a brief mathematical discussion of the truth
or otherwise of each of the following statements (a--e). In each
section, credit will be given for the quality of your argument
rather than for guessing or remembering whether the statement is
true. The five sections carry equal marks.

\smallskip
\itemitem{(a)} The group of direct motions of Euclidean space
contains a non-Abelian finite subgroup.

\smallskip\itemitem{(b)} Let $(X,d)$ be a metric space and
$\sim$ an equivalence relation on $X$; then setting
 $$ d([x],[y])=\inf\bigl\{d(x',y')\bigr\}\quad\hbox{taken over all }
x'\in[x], y'\in[y].
 $$ defines a metric on the quotient space $Y=X/\sim$.

\smallskip\itemitem{(c)} Let $\fie\colon[0,1]\to\R^2$ be a loop with
$\fie(0)=\fie(1)$. Then there exists a point $Q\in\R^2$ such that
$\fie$ has winding number zero around $Q$ 

\smallskip\itemitem{(d)} There exists an affine linear map
$\aff^1\to\aff^1$ of the affine line taking any set of 3 distinct
points into any other.

\smallskip\itemitem{(e)} Given two lines $L$ and $M$ in
$\E^3$ with $L\cap M=\emptyset$, there exists a unique line
$N$ perpendicular to both $L$ and $M$.

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\end
