%
% file:         linalg4.tex
% purpose:      presentation of ``Linear Algebra 4'' course
% created:      pasha jul 5,11,18-23,29,31,aug 1-3 2018
% modified:     pasha may 7 2024
% modification: fixed an inaccuracy
%

\documentclass{beamer}

\usepackage{graphicx}

\newcommand{\vertbar}{\>|\>}
\newcommand{\set}[2]{\ensuremath{\{ #1 \vertbar #2 \}}}

\DeclareMathOperator{\id}{id}

\setcounter{framenumber}{-1}
\setbeamertemplate{navigation symbols}{}


\title{Linear Algebra 4}
%\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
Version of May 7, 2024
}


\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% from now on, display counters
\setbeamertemplate{headline} 
{
  \vskip2pt\hskip1pt\insertframenumber /\inserttotalframenumber 
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Literature}

\begin{itemize}
\item A.I. Kostrikin and Yu.I. Manin, \emph{Linear Algebra and Geometry},
Gordon and Breach, 1997 
\\
(referred in what follows as {\sc Kostrikin--Manin})

\item
S. Mac Lane and G. Birkhoff, \emph{Algebra}, 3rd ed., AMS Chelsea, 1999
\\
(referred as {\sc Mac Lane--Birkhoff})

\item
A. Onishchik and R. Sulanke, \emph{Projective and Cayley--Klein Geometries},
Springer, 2006 
\\
(referred as {\sc Onishchik--Sulanke})

\item
M. Postnikov, \emph{Lectures in Geometry. Semester I. Analytic Geometry},
Mir, Moscow, 1982
\\
(referred as {\sc Postnikov})

\end{itemize}

\vskip 0.8cm

\begin{center}
\footnotesize
(All images, unless specified otherwise, are courtesy of Wikipedia)
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Affine spaces}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Refresher: Vector space}

From the previous linear algebra courses, you (suppose to) know already what a 
vector (aka linear) space is.

\begin{center}
\includegraphics[scale=0.40]{Coord_system_CA_0.pdf}
\\
The vector space $\mathbb R^3$
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Refresher: Vector space (cont.)}

Abstractly, the vector space is a set equipped with operation of addition
and multiplication on scalars (elements of the base field) subject to the usual
condition satisfied by addition of vectors in $\mathbb R^n$ and multiplication
of a vector by a scalar:
\begin{gather*}
a + b = b + a \\
(a + b) + c = a + (b + c) \\
\lambda (a + b) = \lambda a + \lambda b \\
\text{etc ...}
\end{gather*}

Vector space is one of the central notions of mathematics and embodies the ideas
of \emph{linearity} and \emph{linearization}.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Refresher: Importance of vector spaces}

Most of the phenomena in the nature are described by nonlinear structures, but 
they can be approximated by linear ones; the paradigmatic examples are the 
notion of derivative in analysis (which is nothing but a slope of a tangent line -- i.e., 
an one dimensional space -- at a point to a curve), or, more generally, the 
notion of the tangent plane at a point to the manifold:

\begin{center}
\includegraphics[scale=0.22]{800px-Tangent-calculus.png}
\includegraphics[scale=0.43]{Image_Tangent-plane.pdf}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine space}

Now meet affine spaces, a specialization of the notion of vector spaces.

\begin{block}{Definition}
An \emph{affine space} over a field $K$ is a triple $(A, V, +)$, where $A$ is a
set, $V$ is a vector space over $K$, and $+$ is a binary operation 
$A \times V \to A$, $(a,v) \mapsto a+v$, satisfying the following axioms:
\begin{enumerate}
\item $(a + u) + v = a + (u + v)$ for any $a\in A$, $u,v \in V$
\item $a + 0 = a$ for any $a \in A$
\item 
for any $a,b\in A$ there exists a unique $v \in V$ such that $b = a + v$
\end{enumerate}
\end{block}

In other words: the additive group of $V$ acts on $A$ (conditions 
{\color{blue}1} and {\color{blue}2}) transitively and freely (condition 
{\color{blue}3}).

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine space (cont.)}

Elements of $A$ are called \emph{points}, and elements of $V$ are called 
\emph{vectors}. The unique $v$ from axiom {\color{blue}3} is called the
\emph{difference} of the corresponding points and is denoted by $b-a$.

\bigskip

The maps $t_v: A \to A$, $a \mapsto a + v$ are called \emph{translations}. 

\bigskip

The \emph{dimension} of the affine space is the dimension of its underlying 
vector space, $V$.

\bigskip

Often we will omit the symbol of operation ``+'', and denote an affine space
as merely a pair $(A,V)$.

\bigskip

Intuitively, affine space may be thought as a vector space whose origin $0$ is
``forgotten'': only operations of translation by a vector, summation of 
translations, and multiplication of a translation by a scalar are retained.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Examples of affine spaces}

\begin{itemize}
\item
A paradigmatic example: the ``points'' are points in $\mathbb R^3$, and
``vectors'' are vectors in $\mathbb R^3$, with vectors acting at a point.
\begin{center}
\includegraphics[scale=0.35]{Vector_from_A_to_B.pdf}
\end{center}

\item
More generally: $(V, V, +)$, where $V$ is a vector space, i.e. the sets of 
vectors and points are the same, with the action being the usual addition of 
vectors.

\item
$(P_2, P_1, +)$, where $P_2$ is the set of solutions of the inhomogeneous 
linear system $Ax = b$, $P_1$ is the set of solutions of the corresponding 
homogeneous system $Ax = 0$.
\begin{center}
\includegraphics[scale=0.10]{Affine_space_R3.png}
\end{center}

\end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Examples of affine spaces (cont.)}

\begin{block}{Exercise}
Try to give more examples of affine spaces. What is the general pattern for
constructing them?
\end{block}

\bigskip

\textbf{Hint}: see, for example, Wikipedia.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Properties of affine spaces}

\begin{block}{Lemma}
Let $(A, V, +)$ be an affine space.
\begin{enumerate}
\item
For any $v \in V$, the map $A \to A$, $a \mapsto a+v$ is a bijection.
\item
For any $a \in A$, the map $V \to A$, $v \mapsto a+v$ is a bijection.
\item
For any $a \in A$, $v \in V$, there is a unique $b \in A$ such that $b-a = v$.
\item
For any $a,b,c \in A$, it holds $(c - b) + (b - a) = c-a$.
\item
For any $a\in A$, it holds $a - a = 0$.
\item 
For any $a,b \in A$, $v,u\in V$, it holds $(a + v) - (b + u) = (a-b) + (v-u)$.
\end{enumerate}
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Warning}

Be careful! While $a + v$ is a legitimate expression, $v + a$ is not!
For example, in heading {\color{blue}4} from the previous Lemma we cannot merely
``open the brackets'' and write something like $(c - b) + b$, etc.

\bigskip

For the general rules and examples of manipulating with expressions containing 
multiple points and vectors with plus and minus signs, see 
{\sc Kostrikin--Manin}, pp.~196--197, and 
{\sc Mac~Lane--Birkhoff}, pp.~566--567.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Affine maps}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine maps}

\begin{block}{Definition}
Let $(A_1, V_1)$ and $(A_2,V_2)$ be affine spaces. The pair $(f, Df)$ of maps
$f: A_1 \to A_2$, $Df: V_1 \to V_2$ such that $Df$ is a linear map, and 
for any $a,b \in A_1$, 
$$
f(a) - f(b) = Df(a-b) ,
$$
is called an \emph{affine map} of the first affine space into the second one.
\end{block}

\begin{block}{Note}
Since $a,b$ run through all $A$, $Df$ is defined by $f$ uniquely, so often
(by abuse of language) we will call an affine map just $f$.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Examples of affine maps}

\begin{itemize}
\item
A linear map $f: V_1 \to V_2$ from a vector spaces $V_1$ to $V_2$ is an affine
map from $(V_1, V_1)$ to $(V_2, V_2)$.

\item
Any translation is an affine map of the affine space to itself. 

\item
An identity map $\id_A$ is an affine map.

\end{itemize}

\begin{block}{Exercise}
What will be the corresponding $Df$ in all these cases?
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Composition of affine maps}

\begin{block}{Theorem 1}
The composition of affine maps is an affine map.
\end{block}

\begin{block}{Note}
For those who knows (and appreciates) category theory: this Theorem, together 
with the fact that the identity map is an affine map, implies that the affine 
spaces form a category in which morphisms are affine maps.
\end{block}

\begin{block}{Theorem 2}
Let $f,g: (A,V) \to (B,W)$ be two affine maps. Then $Df = Dg$ if and only if one
of them is a composition of another one with a translation by some (unique) 
vector (i.e., say, $g = t_w \circ f$ for some $w \in W$).
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Affine coordinates}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine coordinates}

A \emph{system of affine coordinates} of an $n$-dimensional affine space $(A,V)$
over a field $K$ consists of a point $a_0 \in A$, and a basis 
$\{e_1, \dots, e_n\}$ of $V$.

The \emph{coordinates} of the point $a\in A$ in this system is an $n$-tuple
$(x_1, \dots, x_n) \in K^n$ such that 
$$
a = a_0 + x_1 e_1 + \dots + x_n e_n .
$$

\begin{block}{Theorem}
For any system of affine coordinates, every point has a unique coordinates in 
it.
\end{block}

\textbf{Hint}: the proof is not difficult, and follows from the previous 
theorems and facts.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Barycentric combination}

\begin{block}{Definition}
Let $(A,V)$ be an affine space over a field $K$, and $a_1, \dots, a_n \in A$. 
For any $x_1, \dots, x_n \in K$ such that $x_1 + \dots + x_n = 1$, define the
formal sum $\sum_{i=1}^n x_i a_i$ as 
$$
\sum_{i=1}^n x_i a_i = a + \sum_{i=1}^n x_i (a_i - a)
$$
for some $a\in A$. It is called the \emph{barycentric combination} of the
points $\{a_i\}$ with coefficients $\{x_i\}$.
\end{block}

\begin{block}{Theorem}
This definition does not depend on $a$.
\end{block}

\begin{block}{Warning}
For example, for two points $a$, $b$, the expression $\frac 12 a + \frac 12 b$
makes sense by above, while the summands, $\frac 12 a$ and $\frac 12 b$, are 
meaningless!
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A physical meaning of the barycentric combination}

In the affine space $(\mathbb R^3, \mathbb R^3)$, the barycentric combination 
represents the ``center of gravity'' of a system of masses located at the 
points $\{a_i\}$, weighted accordingly by coefficients $\{x_i\}$.

\begin{center}
\includegraphics[scale=0.40]{jupiter-sun.png}

{\tiny Image courtesy of NASA}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Barycentric combination and affine maps}

\begin{block}{Theorem}
Let $(A,V)$, $(B,W)$ be two affine spaces. A function $f: A \to B$ is an affine
map if and only if for any finite sets $\{a_i\}_{i=1}^n$ and $\{x_i\}_{i=1}^n$
of elements of $A$ and of the ground field respectively, such that 
$\sum_{i=1}^n x_i = 1$,
$$
f(\sum_{i=1}^n x_i a_i) = \sum_{i=1}^n x_i f(a_i) . 
$$
\end{block}

(I.e., a map is affine if and only if it ``preserves'' the barycentric 
combinations).

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Characterization of affine spaces}

\begin{block}{Theorem}
Any affine space is affine isomorphic to the affine space of the form $(V,V)$ 
for some vector space $V$.
\end{block}

\medskip

The proof uses the barycentric coordinates and the previous theorem.

\begin{block}{Corollary}
Any $n$-dimensional affine space over a field $K$ is affine isomorphic to
$(K^n, K^n)$.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Affine group}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine group}

\begin{block}{Theorem 1}
The set of bijective affine maps of an affine space $(A,V)$ forms a group with
respect to composition.
\end{block}

\begin{definition}
This group is called the \emph{affine group} of the affine space $(A,V)$, and is
denoted by $Aff(A,V)$.
\end{definition}

\begin{block}{Theorem 2}
For any affine space $(A,V)$, it holds $Aff(A,V) \simeq V \rtimes GL(V)$.
\end{block}

\bigskip

For an explicit isomorphism, and rules of computation in $V \rtimes GL(V)$,
see {\sc Kostrikin--Manin}, pp.~203--204.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Affine subspaces}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine subspace}

\begin{block}{Definition}
An \emph{affine subspace} of an affine space $(A,V)$ is an affine space $(B,W)$
satisfying one of the following conditions:
\begin{enumerate}
\item $B = \varnothing$ and $W = 0$;
\item
$B \subseteq A$, $W = \set{b_1 - b_2}{b_1,b_2 \in B}$ is a vector subspace 
of $V$, and $t_w(B) \subseteq B$ for any $w \in W$.
\end{enumerate}
\end{block}

\begin{block}{Note}
As $W$ is determined by $B$ uniquely, often (and again by abuse of terminology)
we will call an affine subspace just $B$.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Parallel affine subspaces}

\begin{block}{Definition}
Two affine subspaces of the same dimension of the same affine space are called 
\emph{parallel} if they have the common space $W$ from the previous definition.
\end{block}

\begin{block}{Theorem}
Parallel affine subspaces either do not have common points, or do coincide.
\end{block}

\begin{block}{Exercise}
What happens in the situation where $W$'s do not coincide, but one of them is 
contained in another?
\end{block}

See {\sc Kostrikin--Manin}, p.~208.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine span}

\begin{block}{Definition}
Let $S$ be a subset of the set of points of an affine space $(A,V)$. The 
smallest affine subspace of $(A,V)$ whose set of points contains $S$, is called
an \emph{affine span} of $S$.
\end{block}

Affine span always exists. (Why?)

\begin{block}{Theorem}
The set of points of the affine span of a set $S$ is equal to the set of 
barycentric combinations of elements of $S$.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine subspace (cont.)}

\begin{block}{Theorem 1}
Let $f_1, \dots, f_n$ be affine maps from affine space $A$ to the 
one-dimensional affine space $K$. Then the set of points 
$\set{a \in A}{f_1(a) = \dots = f_n(a) = 0}$ is an affine subspace of $A$. If 
$A$ is finite-dimensional, then any its affine subspace has this form.
\end{block}

\begin{block}{Theorem 2}
A subset of an affine space is an affine subspace if and only if it contains the
affine span of any its two points (or, in other words, the straight line passing
though those two points).
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Euclidean spaces}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Euclidean space}

Euclidean space = affine space + metric space

\begin{definition}
An \emph{(affine) Euclidean space} is an affine space $(A,V)$ over $\mathbb R$ 
such that the set of points $A$ is equipped with the metric $d$ satisfying the
following property: for any $a,b\in A$, $d(a,b) = |a-b|$, where $|\cdot|$ is the
standard Euclidean metric in the real vector space $V$. 

$|a-b|$ is called the \emph{distance} between two points $a$ and $b$.
\end{definition}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Motions of the Euclidean space}

\begin{block}{Definition 1}
A \emph{motion} of an Euclidean space $(A,V)$ is an affine map $(f,Df)$ such
that $f: A \to A$ preserves distances: $f(d(a,b)) = d(a,b)$ for any $a,b \in A$.
\end{block}

\begin{block}{Definition 2}
Let $(A,V)$ be an affine space, and $G$ a subgroup of $GL(V)$. The inverse image
of $G$ with respect to the canonical homomorphism $Aff(A,V) \to GL(V)$ is called
the \emph{affine extension} of $G$.
\end{block}

\begin{block}{Theorem}
The motions of an Euclidean space forms a group with respect to composition, 
which coincides with affine extension of the group of orthogonal isometries 
$O(V)$ of the Euclidean real space $V$.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Examples of motions}

Motions of an $n$-dimensional Euclidean space are:

\begin{itemize}
\item 
$n=1$: translations and reflections relative to a point.

\item
$n=2$: translations, rotations relative to a point, and composition of a 
reflection relative to a straight line, and translation along this line.

\end{itemize}

For a detailed study of these examples, as well as (much more complicated) 
$3$-dimensional and the general $n$-dimensional cases, see 
{\sc Kostrikin--Manin}, pp.~206--207.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Distance and angles in Euclidean space}

\begin{definition}
An \emph{angle} $\varphi$ between two vectors $v$, $w$ in an Euclidean vector 
space is defined by the formula
$$
cos\, \varphi = \frac{|v+w|^2 - |v|^2 - |w|^2}{2\, |v| \cdot |w|} .
$$
\end{definition}

\medskip

This conforms our usual geometric intuition about angles.

Exactly the same way as in the standard $2$-dimensional and $3$-dimensional 
Euclidean geometry, one can define the distance and angle between two lines, 
between a line and a plane, the notion of orthogonality, etc., in an abstract 
Euclidean space. 

See {\sc Postnikov}, p.~129--132,156--159,161--165 for details.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Basics of linear programming}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{An example}

A farmer has a piece of land with the area of 10 km$^2$, to be planted with 
either wheat, or barley, or some combination of the two. The farmer has a 
limited amount of fertilizer, 50 kilograms, and pesticide, 15 kilograms. Every
square kilometer of wheat requires 4 kilograms of fertilizer and 1 
kilogram of pesticide, while every square kilometer of barley requires 3 
kilograms of fertilizer and 2 kilograms of pesticide. Let 1000 EUR be the 
price of wheat per square kilometer, and 1500 EUR be the price of barley. How to
maximize the farmer's profit?

\medskip

{\tiny (adapted from Wikipedia)}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{An example (cont.)}

Denoting the area of land planted with wheat and barley by $x_1$ and $x_2$ 
respectively, the problem can be expressed as follows:

\bigskip

Maximize 
$$
1000 x_1 + 1500 x_2
$$ 
subject to constraints:
\begin{align*}
x_1 + x_2 &\le 10 \\
4 x_1 + 3 x_2 &\le 50 \\
x_1 + 2 x_2 &\le 15 \\
x_1 \ge 0, x_2 &\ge 0 . 	
\end{align*}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Linear programming problem}

Given row-vectors $b, c \in \mathbb R^n$, and $n \times n$ real matrix $M$, 
maximize (or minimize) $c^\top x$ over $x \in \mathbb R^n$, subject to 
$Ax \le b$ and $x \ge 0$.

\bigskip

In what follows, fix an affine space $(A,V)$ over $\mathbb R$. All affine maps 
are assumed to act from $A$ to itself.

\begin{definition}
A \emph{polyhedron} is an intersection of a finite number of sets of points the
form $\set{a\in A}{f(a) \ge 0}$, where $f$ is a non-constant affine map.
\end{definition}

\begin{block}{Theorem}
Assume that and affine map $f$ is bounded above on the polyhedron $S$. Then $f$
assumes its maximal value at all points of some faces of $S$, that is also a
polyhedron. If $S$ is bounded, then $f$ assumes its maximal value at some vertex
of $S$.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Affine quadrics}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Quadratic functions}

The notion of quadratic function is an ``affinization'' of the notion of 
quadratic form.

\begin{definition}
A \emph{quadratic function} on an affine space $(A,V)$ over a field $K$ is a map
$Q: A \to K$ of the form
\begin{equation}\label{eq-q}\tag{*}
Q(a) = q(a - a_0) + \ell(a - a_0) + c ,
\end{equation}
where $a_0 \in A$ is a point, $q: V \to K$ a quadratic form, $\ell: V \to K$ is
a linear form, and $c \in K$ is an element of the ground field.
\end{definition}

\begin{block}{Lemma}
The ``quadratic part'' $q$ of $Q$ is independent on the choice of $a_0$. 
Namely, if (\ref{eq-q}) holds, then for any $a_0^\prime \in A$
$$
Q(a) = q(a - a_0^\prime) + \ell^\prime(a - a_0) + c^\prime ,
$$
for some linear map $\ell^\prime: V \to K$ and $c^\prime \in K$.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Center}

\begin{definition}
The point $a_0$ is called a \emph{central point} of the quadratic function $Q$
if the linear part $\ell$ in the equality (\ref{eq-q}) is equal to zero. 

The set of all central points is called the \emph{center} of $Q$.
\end{definition}

\bigskip

The geometric meaning of the center: identifying $A$ with $V$ such that $a_0$
identifies with $0$, the function $Q$ becomes symmetric with respect to 
reflection $v \mapsto -v$.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Canonical forms of quadratic functions}

\begin{block}{Theorem}
Let $Q$ be a quadratic function on an $n$-dimensional affine space $(A,V)$ over
a field $K$. Then there exists an affine coordinate system in $(A,V)$ in which 
$Q$ has one of the following forms:
\begin{enumerate}
\item
If $q$ is non-degenerate, then 
$Q(x_1, \dots, x_n) = \sum_{i=1}^n \lambda_i x_i^2 + c$ for some 
$\lambda_i, c \in K$;
\item
If $q$ is degenerate of rank $r$, and the center of $Q$ is not empty, then
$Q(x_1, \dots, x_n) = \sum_{i=1}^n \lambda_i x_i^2 + c$ for some 
$\lambda_i, c \in K$;
\item
If $q$ is degenerate of rank $r$, and the center of $Q$ is empty, then
$Q(x_1, \dots, x_n) = \sum_{i=1}^n \lambda_i x_i^2 + x_{r+1}$ for some 
$\lambda_i \in K$.
\end{enumerate}
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Affine quadrics}

\begin{definition}
Let $(A,V)$ be an affine space. An \emph{affine quadric} is a set 
$\set{a\in A}{Q(a) = 0}$, where $Q$ is a quadratic function on $A$.
\end{definition}

\begin{block}{Theorem}
Let the affine quadric $X$ is given by equation $Q_1 = 0$ and $Q_2 = 0$ for some
quadratic functions $Q_1, Q_2$. Then either $X$ is an affine subspace, or
$Q_1 = \lambda Q_2$ for some element of the ground field $\lambda$.
\end{block}

(In other words, the quadratic function defining a quadric is unique up to
multiplication by a scalar).

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Conic sections}

In the $2$-dimensional real case, the quadrics are called \emph{conic sections}.
Using the theorem on canonical forms above, all conic sections are reduced
to either parabola, or hyperbola, or circle, or a pair of intersecting lines,
or a pair of parallel lines, with the center having the usual geometric meaning.

See {\sc Mac Lane--Birkhoff}, p.~585 and \\ 
{\sc Postnikov}, pp.~166--186,193--202.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Projective spaces}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Projective spaces}

\begin{definition}
Let $V$ be a vector space. The set $P(V)$ of straight lines in $V$ (i.e., 
one-dimensional subspaces) is called the \emph{projective space} of $V$. The 
\emph{dimension} of $P(V)$ is equal to $\dim V - 1$.
\end{definition}

\begin{definition}
Select a basis $\{e_0, \dots, e_n\}$ in $V$. Every point $p \in P(V)$ is 
uniquely determined by any nonzero vector on $p$. The coordinates 
$\{x_0, \dots, x_n\}$ of this vector are called \emph{homogeneous coordinates}
of $p$. They are defined up to nonzero scalar factor and denoted by 
$(x_0 : x_1 : \dots : x_n)$.
\end{definition}

In other words: the $n$-dimensional projective coordinate space $P(K^{n+1})$ is
the set of orbits of the multiplicative group $K^* = K \backslash \{0\}$ acting
on the set of nonzero vectors $K^{n+1} \backslash \{0\}$.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Visualization of a projective space}

Let $K = \mathbb R$ and $V = \mathbb R^3$. Any line in $\mathbb R^3$ meets
the sphere of radius $1$ centered at $0$ in two antipodal points. Identifying 
antipodal points, we get a realization of the projective space $P(\mathbb R^3)$.

\begin{center}
\includegraphics[scale=1.0]{Projektivna_rovina.png}
\end{center}

For more details, including higher-dimensional spaces and other ways of
visualization, see {\sc Kostrikin--Manin}, pp.224--225 and
{\sc Onishchik--Sulanke}, pp.~3--4.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The Fano plane}

An example of a $2$-dimensional projective space over $GF(2)$:

\begin{center}
\includegraphics[scale=0.30]{Fano_plane.pdf}
\end{center}

\begin{block}{Exercise}
Prove that this is the smallest $2$-dimensional projective space.
\end{block}

\textbf{Hint}: $2^3 - 1 = 7$.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Projective subspaces}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Projective subspaces}

\begin{definition}
A \emph{projective subspace} of a projective space $P(V)$ is a set of the form
$P(W)$, where $W$ is a vector subspace of $V$.
\end{definition}

\begin{block}{Lemma}
For two vector spaces $V, W$, we have $P(V \cap W) = P(V) \cap P(W)$.
\end{block}

Hence the family of the projective subspaces of a given projective space is 
closed with respect to intersection.

\begin{block}{Definition}
Let $S$ be a subset of the projective space $P(V)$. The smallest projective
subspace of $P(V)$ containing $S$ is called a \emph{projective span} of $S$, and
is denoted by $\overline S$.
\end{block}

\begin{block}{Theorem}
Let $P_1, P_2$ be finite-dimensional projective subspaces of the same 
projective space. Then 
$$
\dim P_1 \cap P_2 + \dim \overline{P_1 \cup P_2} = \dim P_1 + \dim P_2 .
$$
\end{block}

For an examples illustrating this Theorem, see {\sc Kostrikin--Manin}, p.226.

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Projective duality}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The dual projective space}

\begin{definition}
Let $V$ be a vector space, and $V^*$ its dual. Then the projective space
$P(V^*)$ is called the \emph{dual space} of the projective space $P(V)$. 
\end{definition}

\bigskip

Geometrical meaning: points in $P(V^*)$ are hyperplanes in $P(V)$.

\bigskip

When we pass to the projective dual:
\begin{itemize}
\item
The intersection of projective subspaces turned into the projective span of
their dual spaces, and vice versa.
\item
The inclusion of projective subspaces turned to containment, and vice versa.
\end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The principle of projective duality}

Suppose that there is a theorem about projective subspaces of projective spaces,
whose formulation involves the notions of dimension, inclusion, intersection, 
and projective span. Then there is a dual theorem, in which all terms are 
replaced by their duals according to the rules at the preceding slide.

\begin{block}{Note}
As stated, this is not a rigorous mathematical theorem. The exact 
formulation of this principle will require some mathematical logic!
\end{block}

\begin{block}{Example}
The theorem ``Two different planes in a 3-dimensional projective space 
intersect along one straight line'' is the dual of the theorem ``One straight 
line passes through any two points in a 3-dimensional projective space''.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}

\begin{center}
{\Huge Projective group}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The projective group}

\begin{block}{Lemma 1}
An injective linear map $f: V \to W$ between vector spaces $V,W$ induces a map
$P(f): P(V) \to P(W)$.
\end{block}

\begin{definition}
$P(f)$ is called the \emph{projectivization} of $f$.
\end{definition}

\begin{block}{Lemma 2}
The set of all projectivizations of invertible endomorphisms of $V$ forms a 
group with respect to composition.
\end{block}

\begin{definition}
This group is called the \emph{projective group} of $V$ and is denoted by 
$PGL(V)$.
\end{definition}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Three groups meet together (projective, general linear, affine)}

\begin{block}{Lemma }
For any vector space $V$, $GL(V) \to PGL(V)$, $f \mapsto P(f)$ is a surjective
homomorphism of groups.
\end{block}

\begin{block}{Exercise}
What is the kernel of this homomorphism?
\end{block}

See {\sc Kostrikin--Manin}, p.~234, or {\sc Onishchik--Sulanke}, p.~20,
or {\sc Mac Lane--Birkhoff}, p.~595.

\begin{block}{Theorem}
Let $(A,W)$ be an affine space, and $W$ a subspace of codimension $1$ in the 
vector space $V$. Then the subgroup of $PGL(V)$ consisting of projectivizations
which map $P(W)$ to itself, is isomorphic to $Aff(A,W)$.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Projective quadrics}

\begin{definition}
A \emph{projective quadric} in the projective space $P(V)$ over a field $K$,
is the set $\set{p \in P(V)}{q(p) = 0}$ for some quadratic form $q: V \to K$.
If $\dim P(V) = 2$, the projective quadrics are called \emph{projective conics}.
If the quadratic form $q$ has rank $r$, then the corresponding quadric is said
to have rank $r-1$.
\end{definition}

\begin{block}{Theorem}
In the real projective plane (i.e., in the $2$-dimensional projective space), 
any two nonempty projective conics of rank $2$ are equivalent under the 
projective group.
\end{block}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}

\begin{center}
{\Huge The End}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\setcounter{framenumber}{\value{finalframe}}

\end{document}
% eof
