5.3. Affine and vector spaces

This page is under construction.
Let us now apply the tools just introduced to two geometries (describing distinct but related categories of spaces), first affine geometry, whose models are called affine spaces and morphisms are the affine maps, then to the theory of vector spaces we shall call linear geometry, whose morphisms are the linear maps. The automorphism group of an affine space is called its affine group. The study will first focus on finite-dimensional spaces (infinite dimensional ones are harder to classify, and require to specify the geometry among non-equivalent possible formalizations). Finite dimensional spaces of each kind (affine and linear) will fall into one isomorphism class for each dimension, which can be any n∈ℕ, and all their subspaces (images of morphisms) are sections (thus spaces in their own right).

Affine subspaces and directions

In affine geometry, invariant structures in spaces E with dimension n = 2 or more, can be defined from the notion of their subspaces, which are intuitively understood as straight (and limitless, for those with nonzero dimension).
Affine subspaces are classified into types Ep by their dimension p between 0 and n ; each Ep is the set of images of injective affine maps from p-dimensional spaces to E : Equivalently, affine subspaces are the preimages of points by affine maps; p-dimensional subspaces are the preimages of points by surjective affine maps to (np)-dimensional spaces.

From this we can define an equivalence relation on each Ep : two p-dimensional subspaces are parallel if they are either equal or disjoint but both included in a common (p+1)-dimensional subspace. Equivalently, they are parallel if there is an affine map from E, by which they are the preimages of 2 points, i.e. they belong to a common partition of E by an affine map. Let us call affine quotient such a partition. In affine geometry, parallelism is an equivalence relation: for each subspace there exists a unique affine quotient to which it belongs, that is called its direction. So "to be parallel" can be phrased "to be in the same direction".

Intuition of affine geometry

Another invariant structure intuitively summing up affine geometry in dimension 2 or more, is the notion of parallelogram, conceived as 4-tuples (to directly fit in this presentation; their view as whole figures with sides and inside, definable using the order in ℝ, is also an affine invariant), and similarly parallelepipeds of any dimension. Here are intuitive descriptions of affine maps: As any function is surjective to its image (which is here a subspace), any affine map is a composite of a surjective one (a quotient) with an injective one (isomorphism to a subspace).
More structures of affine geometry (also preserved by its automorphisms), will be introduced later:

More detailed study of affine geometry in another page.

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