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Agustí Reventós Tarrida - Affine Maps, Euclidean Motions and Quadrics

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Agustí Reventós Tarrida Affine Maps, Euclidean Motions and Quadrics

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Affine geometry and quadrics are fascinating subjects alone, but they are also important applications of linear algebra. They give a first glimpse into the world of algebraic geometry yet they are equally relevant to a wide range of disciplines such as engineering.

This text discusses and classifies affinities and Euclidean motions culminating in classification results for quadrics. A high level of detail and generality is a key feature unmatched by other books available. Such intricacy makes this a particularly accessible teaching resource as it requires no extra time in deconstructing the authors reasoning. The provision of a large number of exercises with hints will help students to develop their problem solving skills and will also be a useful resource for lecturers when setting work for independent study.

Affinities, Euclidean Motions and Quadrics takes rudimentary, and often taken-for-granted, knowledge and presents it in a new, comprehensive form. Standard and non-standard examples are demonstrated throughout and an appendix provides the reader with a summary of advanced linear algebra facts for quick reference to the text. All factors combined, this is a self-contained book ideal for self-study that is not only foundational but unique in its approach.

This text will be of use to lecturers in linear algebra and its applications to geometry as well as advanced undergraduate and beginning graduate students.

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Agust Revents Tarrida Springer Undergraduate Mathematics Series Affine Maps, Euclidean Motions and Quadrics 10.1007/978-0-85729-710-5_1 Springer-Verlag London Limited 2011
1. Affine Spaces
Agust Revents Tarrida 1
(1)
Department of Mathematics, Universitat Autnoma de Barcelona, Cerdanyola del Valls, 08193 Bellaterra, Barcelona, Spain
Agust Revents Tarrida
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Abstract
In this chapter we introduce the most fundamental concept of these notes: the affine space. It is a natural generalization of the concept of vector space but with a clear distinction between points and vectors. Often this distinction is not made, and it is frequent not to distinguish, for instance, between the point (1,2)2 and the vector v =(1,2)2. The problem is that 2 is at the same time a set of points and a vector space.
In the study of vector spaces, the vector subspaces and the relations among them (Grassmanns formula) play a central role. In the same way, in the study of affine spaces, the affine subspaces and the relations among them (affine Grassmanns formula) play also an important role.
The most simple figure that we can form with points and straight lines is the triangle. In this chapter we shall see two important results that refer to triangles and the incidence relation: the theorems of Menelaus and Ceva.
In the Exercises at the end of the chapter we verify Axioms 1, 2 and 3 of Affine Geometry given in the Introduction.
The subsections are
1.1
Introduction
1.2
Definition of affine space
1.3
Examples
1.4
Dimension of an affine space
1.5
First properties
1.6
Linear varieties
1.7
Examples of straight lines
1.8
Linear variety generated by points
1.9
Affine Grassmanns formulas
1.10
Affine frame
1.11
Equations of a linear variety
1.12
Barycenter
1.13
Simple ratio
1.14
Theorems of Thales, Menelaus and Ceva
Exercises
1.1 Introduction
In this chapter we introduce the concept of an affine space as an algebraic model in which the axioms of Affine Geometry listed in the introduction are all fulfilled. For this we will need to define the terms straight line , plane , etc. We shall also demonstrate some classical theorems, such as those of Thales and Menelaus.
1.2 Definition of Affine Space
Let k be a field. A review of the definitions of a field and of a vector space over a field, as well as their more elementary properties, can be found in [].
Definition 1.1
Let E be a k -vector space. An affine space over E is a set Affine Maps Euclidean Motions and Quadrics - image 1 together with a map
Affine Maps Euclidean Motions and Quadrics - image 2
such that:
(1)
Affine Maps Euclidean Motions and Quadrics - image 3 for all Picture 4 , where Picture 5 is the identity element of E ;
(2)
P +( v + w )=( P + v )+ w for all Picture 6 and v , w E ; and
(3)
given Picture 7 , there exists a unique v E such that P + v = Q .
1.2.1 Observations
Observation 1 .
Note that if Affine Maps Euclidean Motions and Quadrics - image 8 and v E , the notation P + v means only the image of the pair ( P , v ) via the above map Affine Maps Euclidean Motions and Quadrics - image 9 .
Hence, the four signs + appearing in condition 2 have different meanings: three of them represent the above map, and the other, ordinary vector addition in the vector space E .
Observation 2 .
The unique vector determined by the points P and Q is denoted by Affine Maps Euclidean Motions and Quadrics - image 10 . Hence, we have the fundamental relation
Affine Maps Euclidean Motions and Quadrics - image 11
Observation 3 .
Let G be a group. A map
Affine Maps Euclidean Motions and Quadrics - image 12
such that:
(1)
P + e = P for all Picture 13 , where e is the identity element of G ; and
(2)
P +( v + w )=( P + v )+ w for all Picture 14 and v , w G ,
is called an action of G on Picture 15 . If, in addition, the following holds:
(3)
for all pairs of points Picture 16 , there exists a unique v G such that P + v = Q ,
then the action is said to be simply transitive .
Hence, we can also define affine space as follows:
Definition 1.2
An affine space is a simply transitive action of the additive group of a k -vector space E on a set Affine Maps Euclidean Motions and Quadrics - image 17 .
Observation 4 .
Notice that, since the action of the additive group is transitive, for all Affine Maps Euclidean Motions and Quadrics - image 18 the map
Affine Maps Euclidean Motions and Quadrics - image 19
is a well-defined bijection. The injectivity and surjectivity of P are immediate; see Proposition 1.4.
1.3 Examples
Example 1 .
The standard example of an affine space is given by Affine Maps Euclidean Motions and Quadrics - image 20 , that is, the points of this affine space are the elements of the vector space. The action is
Affine Maps Euclidean Motions and Quadrics - image 21
where the sum is ordinary vector addition.
Let us verify that the three conditions of the definition of affine space are satisfied.
(1)
Affine Maps Euclidean Motions and Quadrics - image 22 . This is obvious, since Affine Maps Euclidean Motions and Quadrics - image 23
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