\chapter{Affine geometry}\label{ch!aff}\index{affine!geometry|(}
\markboth{\quad\thechapter\ Affine geometry\hfill}{}

\medskip

Affine geometry is the geometry of an $n$-dimensional vector space together
with its {\em in\-homo\-geneous linear structure}. Accordingly, this chapter
covers basic material on linear geometries and linear trans\-formations. The
in\-homo\-geneous linear maps that we allow as transformations of affine
space include translations such as $(x,y)\mapsto(x+a,y+b)$, dilations such
as $(x,y)\mapsto(2x,2y)$ and ``shear'' maps such as $(x,y)\mapsto(x,x+y)$.
It is impossible to define an origin, distances\index{distance!function}
between points, or angles\index{angle} between lines in a way which makes
them invariant under these transformations, or to compare ratios of
distances in different directions. However, the line $PQ$ through two points
$P$ and $Q$ of $\aff^n$ makes perfectly good sense; this is also called the
affine span\index{affine!span} $\Span{P,Q}$ of $P$ and $Q$. An affine line
is a particular case of an {\em affine linear
subspace}\index{affine!linear!subspace|(} $E\subset\aff^n$; I can view an
affine linear subspace as the
affine span $\Span{P_1,\dots,P_k}$ of a finite set of points, or as the set
of solutions of a system of inhomogeneous linear equations $M\bx=\bb$.
Arbitrary affine linear maps take affine linear subspaces into one another,
and also preserve collinearity of points, parallels\index{parallel!lines}
and ratios of distances along parallel lines; all of these are thus well
defined notions of affine geometry.

\medskip

\section{Motivation for affine space}
\index{affine!space $\aff^n$}
\markright{\hfill\thesection\ Motivation for affine space \quad}

As before, I write $\R^n$ for the set of $n$-tuples $(x_1,\dots,x_n)$ of
real numbers and $V\iso\R^n$ for an $n$-dimensional vector space over $\R$.
The rest of this chapter discusses the same set under the name of affine
$n$-space $\aff^n$; Chapter~\ref{ch!eucl} called it Euclidean $n$-space
$\E^n$. Before giving the formal definitions, let me explain briefly the
point of having so many alternative names and notations for what are
basically all the same thing.

The set $\R^n$ of $n$-tuples $(x_1,\dots,x_n)$ is an $n$-dimensional vector
space over the field $\R$ of real numbers: I can add two $n$-tuples and
multiply an $n$-tuple by a real number. These notions have a physical
meaning: in mechanics, for example, you could think of adding vectors in a
parallelogram of forces or velocities. A vector space $V$ is the abstract
structure in which the operations of linear algebra make sense: addition of
vectors and multiplication of vectors by scalars are defined in $V$, and
satisfy some rules. Once I know that $V$ has dimension $n$, I can choose a
basis $\{\bbe_1,\dots,\bbe_n\}$ and identify a vector
\[
\bv=\sum_{i=1}^n x_i\bbe_i \in V
\]
with the $n$-tuple $(x_1,\dots,x_n)$, so that $V=\R^n$. However, there may
be practical or theoretical reasons for not wanting to fix a basis at the
outset: a proof, or the answer to a calculation may turn out to be much
nicer in a well chosen basis. In mechanics, for example, you might want to
distinguish forces in the direction of motion from forces perpendicular to
the motion.

Similarly, working in coordinate geometry of $\R^n$ (even $\R^2$, of
course), there may be reasons to choose coordinates
\begin{equation}
x'_i=x_i-a_i \quad\hbox{for $i=1,\dots,n$}
\label{eq!trx'}
\end{equation}
centred at some point $P=(a_1,\dots,a_n)$. In mechanics, for example, if two
particles at points $P$ and $Q$ exert forces on one other, you may want to
take either $P$ or $Q$ as the origin of coordinates, or you may prefer to
take their centre of gravity, or some other point. The coordinate
change~(\ref{eq!trx'}) is however not a linear map or change of basis of the
vector space $V$; for example, it has the effect of changing the origin of
coordinates to make $P=(0,\dots,0)$. Indeed, two different choices of origin
differ by a translation of the form (\ref{eq!trx'}). Just as the laws of
physics should not depend on the choice of origin, we require that geometric
properties of affine space are invariant under affine coordinate changes,
which include maps of the form~(\ref{eq!trx'}).

The same issue commonly arises, from a slightly different point of view, in
problems where we are interested in some space that is clearly linear in
some sense, but has no preferred origin. The model case is the space of
solutions to a system of inhomogeneous linear equations $A\bx=\bb$: as you
know, the space of all solutions is given by a {\em particular} solution
$\bx_0$ plus the {\em general solution of the homogenised equations}
$A\bx=0$ (the kernel of the matrix $A$). Solutions of the homogeneous linear
equations form a vector space; the particular solution $\bx_0$ provides an
identification of the set of all solutions with a vector space $U$. There is
no preferred particular solution $\bx_0$, and a different particular
solution $\bx'_0$ gives another identification of the solution set with $U$,
differing by a translation as in (\ref{eq!trx'}), with $\ba=\bx'_0-\bx_0$.

\section{Basic properties of affine space}
\index{affine!space $\aff^n$}
\markright{\hfill\thesection\ Basic properties of affine
space\quad}\label{sec!affbasics}

This section lists basic properties that I take as the definition of affine
space $\aff^n$.

\begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})}
\item Affine space has a set of points $P\in \aff^n$ in one-to-one
correspondence with position vectors $\bp\in V$ in an $n$-dimensional vector
space~$V$ over~$\R$. The one-to-one correspondence $P\bij \bp$ between
points and vectors is not fixed; rather, I am always allowed to translate it
by a fixed vector $\bb$, so that the new identification is $P\bij \bp^\prime
=\bp+\bb$.
\item Further, a choice of basis of $V$ leads to an identification $V=\R^n$,
and thus to a {\em coordinate system} on $\aff^n$, in which
points $P\in\aff^n$ are represented by coordinates
\[
P\bij\bp= \begin{pmatrix} x_1\\ \vdots \\ x_n \end{pmatrix}\in\R^n
\quad\hbox{where $x_i\in\R$.}
\]
\item Two points $P,Q\in\aff^n$ determine a vector $\vec{PQ}\in V$ as in
Figure~\ref{points of An}. This vector is independent of the identifications
discussed in~(1).
\begin{figure}[ht]
\renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}}
\centering \input{aff_defn.pstex_t}
\caption{Points, vectors and addition}\index{parallel!hyperplanes}
\label{points of An}
\end{figure}
\item Conversely, a vector $\bx\in V$ can be added to a point $P\in\aff^n$
to get a new point $Q=P+\bx\in\aff^n$, and then $\vec{PQ}=\bx$; see again
Figure~\ref{points of An}. This operation is also independent of the
identifications discussed in~(1).
\end{enumerate}

As with the definition of $\E^n$ in \ref{En}, the definition of $\aff^n$
involves an identification $\aff^n=V$ or $\aff^n=\R^n$, followed by the
assurance that any other identification would do just as well provided that
it is related to the first by a suitable transformation, in this case an
affine linear transformation. How to define affine space in abstract
algebra (without explicit mention of any origin or coordinates) is a
slightly arcane issue, and is discussed in~\ref{subsec!affdef}.

\begin{rems}
In most of what follows, you can replace $\R$ by other fields. The most
obviously useful case is an $n$-dimensional vector space over $\C$, giving
rise to $\aff^n_\C$, but affine geometries\index{affine!geometry} over
finite fields $\FF_{p^n}$, or over other fields, also have applications in
many areas of math and science. I do not intend to labour this point,
because doing it properly would involve a lot of algebra of fields, and
because the course is directed more towards metric
geometries,\index{metric!geometry} which are ``real'' subjects.

Note also that I work here from the outset in a finite dimensional space~$V$.
However, in many areas of math, affine spaces appear as the set of
solutions of in\-homo\-geneous linear equations in infinite dimensional
spaces: there is no preferred solution, but the differences $\bx-\bx'$
between any two solutions form a vector space (finite dimensional or
otherwise). This happens, for example, in solving $\D x(t)=y(t)$ for
functions $x=x(t)$ in a suitable space of differentiable functions, where
$\D$ is a linear differential operator and $y(t)$ a given function. The
spaces of functions we work in, and sometimes also our affine space of
solutions, are often infinite dimensional.
\end{rems}

\section{The geometry of affine linear subspaces} \label{sec!afl}
\index{affine!linear!subspace}
\markright{\hfill \thesection\ The geometry of affine linear subspaces\quad}
\label{sec!affspaces}

An {\em affine linear subspace} $E\subset\aff^n$ is a nonempty subset of the
form
\[
E=P+U = \bigl\{ P+\bv \bigm | \bv\in U\bigr\},
\]
with $P\in\aff^n$ and $U\subset V$ a vector subspace. By Proposition~(i)
below, any point of $E$ will do equally well in place of $P$, so there is no
unique origin specified in $E$.

Let $P, Q\in\aff^n$ be two distinct points. The {\em line}\index{line}
spanned by $P$ and $Q$ is
\[
PQ =\bigl\{P +\la \vec{PQ}\bigm| \la\in\R\bigr\}
\]
The definition clearly shows that $PQ$ is an affine linear subspace, with
$U$ the one dimensional vector subspace of $V$ generated by $\vec{PQ}\in V$.
As in \ref{sec!col}, we have the\index{line!segment} {\em line segment} or
{\em interval}
\[
[P,Q]=\bigl\{P +\la \vec{PQ}\bigm| 0\le \la\le 1\bigr\}.
\]

It is useful to spell this out in vector notation. If $P,Q\in\aff^n$
correspond to position vectors $\bp, \bq$, their affine span is the set
\[
\bp\bq= \bigl\{\bp +\la(\bq-\bp) \bigm| \la\in\R\bigr\}=
\bigl\{(1-\la)\bp+\la \bq\bigm| \la\in\R\bigr\}.
\]
The latter is the form of the linear span construction most commonly used.
The line segment now becomes
\[
[\bp,\bq]=\bigl\{(1-\la)\bp+\la \bq\bigm| 0\le \la\le 1\bigr\},
\]
as shown in Figure~\ref{interval}.

Three points $P, Q, R$ are {\em collinear} if they lie on the same line. If
I represent the points by position vectors $\bp, \bq, \br$, this means that
$\br =(1-\la)\bp+\la \bq$; as we saw in \ref{sec!col}, there are three
subcases here:
\[
\renewcommand{\arraystretch}{1.2}
\left.
\begin{array}{r}
\la \le 0 \\
0\le \la \le1 \\
1\le\la
\end{array}
\right\} \iff
\left\{
\begin{array}{l}
\bp\in[\br,\bq] \mbox{ so } P\in[R,Q],\\
\br\in[\bp,\bq] \mbox{ so } R\in[P,Q],\\
\bq\in[\bp,\br]\mbox{ so } Q\in[P,R].
\end{array}
\right.
\]

\begin{figure}[ht]
\renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}a}
\centering \input{aff_span_x_y.pstex_t}
\caption{The affine construction of the line segment $[\bp,\bq]$}
\index{affine!span|(}
\label{interval}\end{figure}

\begin{prop}
\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{enumerate}
\item Let $E=P_0+U$ be an affine linear
subspace\index{affine!linear!subspace} of $\aff^n$. Then the vector space
$U$ is uniquely defined by $E$; explicitly
\[
U=\bigl\{\vec{PQ} \bigm| P,Q \in E\bigr\}.
\]
In other words, $E=P+U$ for any $P\in E$.

\item A necessary and sufficient condition for a nonempty subset
$E\subset \aff^n$ to be an affine subspace is that the line $PQ$ is
contained in $E$ for all $P,Q\in E$.

\item A necessary and sufficient condition for $E$ to be an affine subspace
is that it is nonempty, and defined by a set of inhomogeneous linear
equations in a coordinate system.

\end{enumerate}
\end{prop}

The proofs are easy exercises in linear algebra. (i) states that $E$ can be
translated back to the vector space $U$ choosing any point $P\in E$;
informally, any point $P\in E$ can serve as origin. (iii) spells out the
other easy way of specifying an affine linear subspace using coordinates;
examples can be found in~\exref{ex!3planes} and~\exref{aff_ex_dim}.

Write $\dim E=\dim U$ for the {\em dimension} of a nonempty affine linear
subspace $E$.\index{dimension} The only $n$-dimensional affine linear
subspace is $\aff^n$ itself; $\dim E=0$ means simply that $E$ consists of a
single point, whereas a 1-dimensional affine linear subspace is simply a
line. The last interesting case with a name of its own is an affine linear
subspace of dimension $n-1$ (that is, codimension one), a {\em
hyperplane}.\index{hyperplane} Two hyperplanes $E_1, E_2$ are {\em
parallel}\index{parallel!hyperplanes}, if they are translates of the same
vector subspace of $V$, that is $E_1=P_1+U$, $E_1=P_2+U$ with $\dim U=n-1$,
as in Figure~\ref{parallel_hyps}. An equivalent condition is to ask that the
two hyperplanes should either coincide or have no common point.

\begin{figure}[ht]
\renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}b}
\centering \input{aff_parallel_hplanes.pstex_t}
\caption{Parallel hyperplanes}\index{parallel!hyperplanes}
\label{parallel_hyps}
\end{figure}

\begin{defn} Let $\Si\subset\aff^n$ be any set; an {\em affine linear
combination} of $\Si$ is any point $P\in\aff^n$ of the form
\begin{equation}
P=P_0 + \sum_{i=1}^k\la_i\vec{P_1P_i},\quad\hbox{where $P_i\in\Si$ and
$\la_i\in\R$.}
\label{eq!comb}
\end{equation}

Using position vectors $\bp_i$ of points $P_i$ simplifies this expression
once more; an affine linear combination of $\Si$ is any point $P\in\aff^n$
of the form
\begin{equation}
\bp=\sum_{i=0}^k\mu_i\bp_i,\quad\hbox{where $\bp_i\in\Si$ and
$\mu_i\in\R$ with } \sum_{i=0}^k\mu_i=1.
\label{eq!aff_span}
\end{equation}
This generalises the expression $(1-\la)\bp+\la \bq$ used to parametrise
points of the affine line $PQ$. The points $P_i$ appear in the form
(\ref{eq!aff_span}) with $(\la_0,\dots,\la_k)=(0,\dots,1,\dots,0)$; this
confirms that I really mean $=1$ in (\ref{eq!aff_span}) rather than $=0$.

The {\em affine span} $\Span{\Si}$ of any subset is the set of affine linear
combinations of~$\Si$. By the previous remark, $\Span{\Si}$ contains all
lines spanned by pairs of points in $\Si$. If $P\in\Si$ then
$\Span{\Si}=P+U$, where $U\subset V$ is the vector subspace spanned by the
vectors $\vec{PQ}$ for $Q\in\Si$. Thus $\Span{\Si}\subset\aff^n$ is an
affine linear subspace,\index{affine!linear!subspace} in fact the smallest
one containing all the points of $\Si$.\index{affine!span|)}
\end{defn}

\section{Dimension of intersection} \label{dim inters}
\index{dimension!of intersection}
\markright{\hfill \thesection\ Dimension of intersection\quad}
The formula
\begin{equation}
\dim U+\dim W=\dim U\cap W+\dim(U+W)
\label{eq!dim.int}
\end{equation}
for vector subspaces $U,W$ of a finite dimensional vector space is familiar
from linear algebra. You remember the proof: pick a basis of $U\cap W$,
extend to two bases of $U$ and $W$, and the union is a basis of $U+W$.

\begin{thm} Let\/ $E,F\subset\aff^n$ be affine subspaces. Then
\begin{equation}\label{eq!aff.dim.int}
\dim E\cap F=\dim E+\dim F-\dim\Span{E,F},
\end{equation}
provided that\/ $E\cap F\ne\emptyset$.

The exceptional case $E\cap F=\emptyset$ happens if and only if $E$, $F$ are
contained in parallel hyperplanes\index{parallel!hyperplanes}. This can
happen essentially whatever the dimension\index{dimension} of\/ $E$ and $F$;
more precisely, there exist affine linear subspaces $E,F$ with $\dim E=a$,
$\dim F=b$, $E\cap F=\emptyset$ and\/ $\dim\Span{E,F}=c$ for any $a,b<n$ and
any $c$ with\/ $\max\{a,b\}+1\le c\le \min\{n,a+b+1\}$.
\end{thm}

\begin{pf} The proof of the first statement is almost trivial: if $P\in
E\cap F$ then the four affine subspaces in question are translates of the
four vector subspaces
\[
E',F',E'\cap F',E'+F'\subset V
\label{eq!E'+F'}
\]
so that the result follows at once from the linear algebra formula
(\ref{eq!dim.int}).

The counterexamples involve affine subspaces $E$, $F$ of $\aff^n$ contained
in parallel\index{parallel!hyperplanes} hyperplanes.\index{hyperplane} To be
specific, I choose coordinates and put
\[
E \subset \{x_1=0\} \quad \hbox{and} \quad F \subset \{x_1=1\}.
\]
Then certainly $E\cap F=\emptyset$. The converse is proved in
\exref{ex!empty_inters}.

Assume that $(0,\dots,0)\in E=E'$ and $(1,0,\dots,0)\in F$; then $F$ is the
translation\index{translation} by $(1,0,\dots,0)$ of a vector subspace
$F'\subset V$ contained in $\{x_1=0\}$. The equality (\ref{eq!dim.int})
holds, but the point is that $E\cap F=\emptyset$ takes no account of $\dim
E'\cap F'$. Now $E'$ and $F'$ are any vector subspaces contained in the
hyperplane given by $x_1=0$, so that $\dim E'$, $\dim F'$, $\dim(E'+F')$ can
be anything up to and including $n-1$. \QED \end{pf}

You will find it instructive to spell out the theorem in a few concrete
cases. For example, if $n=2$ and $E$, $F$ are distinct lines, then
$\dim\Span{E,F}=2$ and so the conclusion is that $E\cap F$ is zero
dimensional (that is, a point) unless it is empty,\index{empty set} the
standard dichotomy of intersecting and parallel lines.\index{parallel!lines}
For $n=3$, see Ex.~\ref{ch!aff}.\ref{ex!threesp}.
\index{affine!linear!subspace|)}

\section{Affine transformations} \label{sec!afflinA}
\index{affine!transformation|(}
\markright{\hfill \thesection\ Affine transformations\quad}

Recall the following definition, which I repeat here for completeness:

\paragraph{Definition} A map $T\colon \aff^n\to\aff^n$ is an {\em affine
transformation} if it is given in a coordinate system by $T(\bx)=A\bx+\bb$, where
$A=(a_{ij})$ is an $n\times n$ matrix with nonzero determinant and
$\bb=(b_i)$ a vector; in more detail,
\begin{equation}
\bx=(x_i)\mapsto \by=\Bigl(\sum_{j=1}^na_{ij}x_j+b_i\Bigr),
\quad\hbox{or}\quad
\begin{pmatrix}
x_1\\
\vdots\\
x_n
\end{pmatrix}
\mapsto
A
\begin{pmatrix}
x_1\\
\vdots\\
x_n
\end{pmatrix}
+
\begin{pmatrix}
b_1\\
\vdots\\
b_n
\end{pmatrix}.
\end{equation}

The set $\Aff(n)$ of affine transformations is the set of ``allowed
symmetries'' of affine space\index{affine!space $\aff^n$} $\aff^n$.
This set consists of invertible maps from $\aff^n$ to $\aff^n$ (because I
require $\det A\ne0$). It acts transitively on $\aff^n$; that is, a suitable
affine transformation maps any point to any other. In particular, there is no
distinguished origin, as I said before: every point is like every other.
Contrast this with the situation in linear algebra, where the allowed maps
$V\to V$ are the {\em homogeneous} linear maps, all mapping the origin
$0\in V$ to itself.

It is immediate that an affine transformation takes an affine linear
subspace\index{affine!linear!subspace} to an affine linear subspace; that
is, it preserves the incidence geometry of affine linear subspaces. In
Proposition~\ref{sec!afflinE}, I proved a converse statement, under the
additional assumption that $T$ restricts to an affine linear map on each
line. In fact, one can prove that, for $n\ge2$, a bijective map
$T\colon\aff^n\to\aff^n$ that preserves lines and is\index{continuous}
continuous is actually affine linear. (This is a point where working over
$\R$ is essential.)

\section{Affine frames and affine transformations}\label{aff_frames}
\index{affine!linear!map|(}
\index{affine!linear!dependence}
\markright{\hfill \thesection\ Affine frames and affine
transformations\quad}

\begin{defn}
A set of points $\{P_0,\dots,P_k\}$ of $\aff^n$ is {\em affine linearly
independent} if the $k$ vectors $\vec{P_0P_1},\dots,\vec{P_0P_k}$ are
linearly independent in $V$. In other words, a set $\Si\subset\aff^n$ is
{\em affine linearly dependent} if there exists a nontrivial relation
$\sum_{i=0}^k\la_i\bp_i=0$ between position vectors $\bp_0,\dots,\bp_k$ of
points in $\Si$, with $\la_i\in\R$ and $\sum_{i=0}^k\la_i=0$; $\Si$ is {\em
affine linearly independent} if no such relation exists.

A set $\Si\subset\aff^n$ is an {\em affine frame of
reference}\index{frame!affine} if it is affine linearly independent and
spans $\aff^n$. Compare the notion of Euclidean frame in \ref{Eframe}.
This means that every point $P\in\aff^n$ can be written in the form
(\ref{eq!comb}) of \ref{sec!afl} in a unique way; that is, no proper
subset of $\Si$ can span $\aff^n$. Equivalently,
$\Si=\{P_0,P_1,\dots,P_n\}$ where $P_0\in\Si$ is any point, and the vectors
$\vec{P_0P_1},\dots, \vec{P_0P_n}$ form a basis of $V$.
\end{defn}

In view of the correspondence between bases in a vector space and linear
maps, the last clause gives the following:

\begin{prop} Fix one affine frame\index{frame!affine} of reference
$P_0,\dots,P_n$. Then
\[
T\mapsto T(P_0),\dots,T(P_n)
\]
defines a one-to-one correspondence between affine transformations and
affine frames of reference of $\aff^n$.\index{affine!transformation|)}
\end{prop}
\index{affine!linear!map|)}

% \section{Affine linear maps and lines}
% \label{prop!lines_affinemaps}
% \markright{\hfill \thesection\ Affine linear maps and lines\quad}

% This is where the affine proposition was

\section{The centroid}
\label{aff_cent}
\index{centroid}
\markright{\hfill \thesection\ The centroid\quad}

The following proposition is usually thought of as part of (plane) Euclidean
geometry;\index{Euclidean!geometry} however, it only involves ratios along
lines and incidence of lines,\index{incidence of lines} so in fact it
belongs to affine geometry.\index{affine!geometry} The other ``famous''
centres of a triangle described in \ref{three centres} use notions such as
angle\index{angle} or distance that have no meaning in affine geometry.

\begin{prop} Let $P,Q,R$ be three points of $\aff^n$. Then the three medians
of $\triangle PQR$, that is, the three lines connecting each vertex to the
midpoint of the opposite side, meet in a common point $S$.
\end{prop}

\begin{figure}[ht]
\renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}}
\centering \input{aff_centroid.pstex_t}
\caption{The affine centroid}
\label{affine_centroid}
\end{figure}

\begin{pf} Write $\bp, \bq, \br$ for the
position vectors of $P, Q, R$. Write $\bp^\prime=\half (\bq+\br)$ for the midpoint
of $\bq$ and $\br$ and $\bs=\frac{2}{3}\bp^\prime+\frac{1}{3}\bp$ for the point
dividing the segment between $\bp$ and $\bp^\prime$ in ratio one to two. Then
$\bs=\frac{1}{3}(\bp+\bq+\br)$ is symmetric in $\bp,\bq,\br$, so lies on the
lines joining $\bq$ and $\bq^\prime=\half (\bp+\bq)$ and $\br$ and
$\br^\prime=\half (\bp+\bq)$.
Hence the point $S$ with position vector $\bs$ lies on all
medians of $\triangle PQR$.\QED \end{pf}

To reiterate the point: the statement that this is a theorem of {\em affine}
geometry means that applying any affine transformation takes
Figure~\ref{affine_centroid} to a figure with the same properties, and in
particular takes the centroid of a triangle to the centroid.\index{centroid}

\section*{Exercises to Chapter~\thechapter}
\markright{\hfill Exercises to Chapter~\thechapter\quad}
\addcontentsline{toc}{section}{Exercises to Chapter~\thechapter}
\begin{enumerate}
\renewcommand{\labelenumi}{\thechapter.\theenumi.}
\addtocounter{section}{1}

\item\label{ex!3planes} Consider the 3 planes
\[
\Pi_1:\{x-2=\textstyle\frac12(y-z)\},\enspace \Pi_2:\{x+2=y\},\enspace
\Pi_3:\{3(2x+z)=3y+1\}
\]
in affine space $\aff^3$. Calculate $\Pi_1\cap\Pi_2$ and
$\Span{\Pi_1,\Pi_2}$ and find out whether the dimension of
intersection\index{dimension!of intersection} formula works; if not, why
not? (Compare Theorem~\ref{dim inters}.) Ditto for $\Pi_1\cap\Pi_3$ and
$\Span{\Pi_1,\Pi_3}$.

\item\label{ex!threesp} Experiment with \ref{dim inters},
formula~(\ref{eq!aff.dim.int}) for $n=3$ and different $E$, $F$. For
example, classify pairs of lines of $\aff^3$ into three types, namely
intersecting, parallel\index{parallel!lines} and skew, drawing pictures for
each case.

\item\label{ex!empty_inters} Suppose that $E,F\subset\aff^n$ are disjoint
affine\index{empty set} linear subspaces; prove that there is a linear
form $\fie$ on $\aff^n$ such that $\fie(E)=0$ and $\fie(F)=1$. \begin{hint}
Let $P\in E$, $Q\in F$ and $\bv=\vec{PQ}$. Then $E=P+U$ for a vector
subspace $U\subset V$, and $\bv\notin U$. Deduce that there exists a linear
form on $V$ that is zero on $U$ but nonzero on $\bv$. \end{hint}

\item Write down the affine transformation\index{affine!transformation}
taking
\[
(0,0),(1,0),(0,1)\mapsto(2,1),(5,-1),(3,8).
\]
Can you map the same points $(0,0),(1,0),(0,1)$ to $(2,1),(5,-2),(3,0)$ by
an affine transformation? Why?

\item\label{aff_ex_dim} Determine the dimension\index{dimension} of the
affine linear subspace\index{affine!linear!subspace} $E$ of $\aff^5$ given
by the equations
\begin{eqnarray*}
x_1+x_3-2x_5 &=& 1 \\ x_2-2x_4 + x_5 & =&-2 \\
x_1 +2 x_2 + x_3- 4x_4 &=& -3.
\end{eqnarray*}
Find an affine transformation taking $E$ to an affine linear
subspace\index{affine!linear!subspace} given by $x_1=\cdots=x_k=0$ for some
value of $k$. \begin{hint} choose a suitable affine
frame\index{frame!affine} consisting of points on and off the subspaces,
compare \ref{aff_frames}.\end{hint}

\item Give a determinantal criterion in coordinates for $n+1$ points of
$\aff^n$ to be affine linearly dependent\index{affine!linear!dependence}
(Definition~\ref{aff_frames}). \begin{hint} start by saying how you tell if
3 points of $\aff^2$ are collinear.\end{hint}

\item In\index{centroid} $\triangle PQR$ of Figure~\ref{weighted_centroid},
\begin{figure}[t]
\renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}}
\centering \input{aff_weighted.pstex_t}
\caption{A weighted centroid}\index{centroid}
\label{weighted_centroid}
\end{figure}
take points dividing the three sides in the ratios $1:2$, $1:2$, $n:m$.
Assume that the three lines connecting the vertexes to the points on the
opposite sides have a common point. Calculate the value of the ratio $m:n$.
\begin{hint} follow the proof of Proposition~\ref{aff_cent}. Answer: the
ratio is $4:1$.\end{hint}

\item A general project: set up affine geometry over the finite field
$\FF_p$ of integers modulo the prime $p$. Count the number of points of
affine space $\aff^n$, and prove analogues of the theorems of the text.
Check that everything remains true, with a single exception [harder]: the
statement concerning the centroid {\em fails} for one value of $p$.

\end{enumerate}
\index{affine!geometry|)}

\endinput

% Throughout the book, I write $\R^n$ for the set of $n$-tuples
% $(x_1,\dots,x_n)$ of real numbers and $V\iso\R^n$ for an $n$-dimensional
% vector space over $\R$. The rest of this chapter discusses the same set
% under the name of affine $n$-space $\aff^n$, and the following
% Chapter~\ref{ch!eucl} will call it Euclidean $n$-space $\E^n$. Before
% discussing formal issues, let me explain briefly the point of having so many
% alternative names and notations for what are basically all the same thing.
%
% By convention, the empty set\index{empty set} is also considered to
% be an affine linear subspace.
%
