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In this case is a set of points of space, such that; therefore, it is affine hyperplane with the equation in basis. Endomorfizma the spaces meeting a condition are those endomorfizm which matrix in basis has an appearance

Back, if exist and, such that, it is possible to present in the form, where. Then the point determined by a condition belongs and as it is easy to see. It proves that belongs also, and thereby is not empty.

Back, in order that points in ℰ formed an affine reference point, it is necessary and enough that vectors formed basis, or (an equivalent condition that points did not belong to one affine hyperplane.

Matrix calculations would show that for this compliance rules of composition of displays are followed. On the other hand, an endomorphism with a matrix (it is reversible in only case when when the matrix is reversible (and then also equality is carried out. Thus, it turns out

Definition Let - vector space over any body. The set ℰ on which simply transitive action of abelevy group is defined is called as the affine space associated with.

Let ℰ - the affine space associated with vector space. Each vector subspace of space forms the subgroup of group operating on ℰ with broadcastings. By definition, action orbits on ℰ are called as linear affine varieties (in abbreviated form LAMAS) with the direction. The group operating it is simply transitive on each of these orbits determines by that on each of them the affine structure associated with; therefore we call these orbits (LAMAS) also affine subspaces in ℰ.

Lemma. Let the left vector space over a body and any set. Then the set of displays in are the left vector space over in relation to usual operations of addition of functions and their multiplication at the left by scalars:

At last, if - automorphism and - affine hyperplane in, inclusion attracts equalities. Really, there is an affine hyperplane in, and it is enough to apply a consequence of the theorem of II 2, having returned to a vector case by replacement of the beginning of century.

For such display any point is motionless; taking such point for the beginning, we come to a design case for vector space. From here existence of such displays, and also their following geometrical characterization follows:

where and. Thus a ratio (involves and therefore (see the offer. Back, if - a point from, there are points belonging and scalars (with the sum, unreliable equal, such that; this ratio also registers in a look

Short way of the proof of the offer is application of the offer: there is a crossing of all LAMAS containing. A lack of this reasoning that it is necessary to attract family" all LAMAS containing" of which a little that is known and which usually even is incalculable!

The theorem Let - the uniform space associated with group and for any let - group of an isotropy. Then there is the only bijection of a faktorprostranstvo on, such that for all is executed, where - an initial projection and - action on.

Let's notice that if is LAMAS of final dimension in ℰ and - an affine reference point in, that is a set of points of page. This way of parametrization is often useful. In particular, the affine straight line connecting two points in ℰ is a set of points.

These results are applicable, in particular, to a case, when, - vector continuations of affine spaces, and, - images, at initial immersions: any affine display in, is identified with the linear display of space in space meeting the requirement, and the group of affine bijections is identified with the subgroup keeping an affine giperplosklost

But there is an only linear display from in meeting these conditions (it is defined by the restrictions on additional runways and spaces); then restriction on - is affine display with the same linear part, as, and accepting in the same value, as, and thereby equal, the proved result from where follows.

Interpretation. We fix in ℰ some point and we will supply, vector structures, taking for the beginning in ℰ a point, and in - a point. Then will be semi-affine (according to affine) in that and only that case, if - semi-linear (respectively linearly display ℰ century.