Groups of geometric transformations

The world of geometry is called affine space. It is a set of points, generally specified using a coordinate system consisting of an origin, often called O, and a number of axes: two for a plane, three for ordinary space… Its dimension is the dimension of the vector space associated with that affine space (see box).

Affine space and its associated vector space -----------------------------------------

A real vector space is an Abelian (commutative) group under addition: addition is associative and commutative, has an identity element (the zero vector), and every vector

\overrightarrow{V}

has an additive inverse

-\overrightarrow{V}

There is also another operation, called scalar multiplication (often denoted by a dot), which takes a real number λ and a vector

\overrightarrow{V}

to the vector λ .

\overrightarrow{V}

and has the following properties.

For all real numbers λ and μ, and all vectors

\overrightarrow{V}

and

\overrightarrow{W}

1. \overrightarrow{V} = \overrightarrow{V},

\lambda . \left(\overrightarrow{V} + \overrightarrow{W}\right) = \lambda . \overrightarrow{V} +\lambda . \overrightarrow{W} ,

\left( \lambda+ \mu \right) . \overrightarrow{V} = \lambda . \overrightarrow{V} + \mu . \overrightarrow{V} ,

$\left( \lambda \mu \right) . \overrightarrow{V} = \lambda . \left( \mu . \overrightarrow{V}\right) .$

An affine space

\mathcal{A}

is a set of points associated with a vector space E, subject to the following conditions:

- To each ordered pair (M, N) of points of

\mathcal{A}

, we can associate a vector

\overrightarrow{V}

in E (denoted

\overrightarrow{V}= \overrightarrow{MN}

);

- for every point M, the map taking N to

\overrightarrow{MN}

is a bijection from

\mathcal{A}

onto E;

- Chasles' relation holds: for all points M, N and P of

\mathcal{A}

\overrightarrow{MP}= \overrightarrow{MN} + \overrightarrow{NP}.

Every vector of E can be written uniquely as a linear combination* of the basis vectors, whose number is the dimension of the space. Thus, once the origin O and a basis of E have been chosen, a point M of

\mathcal{A}

is specified by the components of the vector

\overrightarrow{OM}

in that basis.

Once a basis has been chosen, the vectors associated with the "bipoints" of affine spaces (ordered pairs of points) can therefore be regarded as pairs of real numbers (in dimension 2), triples (in dimension 3), and so on.

If λ and μ are real numbers and $\overrightarrow{V}$ and $\overrightarrow{W}$ are vectors, then the vector $\lambda . \overrightarrow{V} +\lambda . \overrightarrow{W}$ is called a linear combination* of

\overrightarrow{V}

and

\overrightarrow{W}

Linear transformations ----------------------------

Transformations can be defined both on the affine space

\mathcal{A}

and on its associated vector space E. We begin with the transformations of E. We will study one specific class: the linear maps from E to itself, called endomorphisms of E, whose set is denoted by