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AFFINE SPACES
The most of the geometrical notions of this chapter will be studied in spaces for which the notions of point and vector are indispensable. The notion of affine space allows using the two notions in a welldefined environment.


§1. Definition and Examples
Let’s consider the nonempty set A = and agree to name each of its elements points, and bipoint of A any element (A, B) I A A . Point A will be named the origin of the bipoint (A, B). Bipoints (A, B) and (B, A) will be named symmetric bipoints.
1.1 Definition 
It is called affine space the triplet A, V, j), where A is a nonempty set of points, V an Kvector space, and the function j A_{}A A, _{},which satisfies the following conditions: A A, B, C I A j (A, B) + j (B, C) = j (A, C) A_{ } _{}A there is a point B I A, unique determined by the relation_{}. 
The set A is named support set of the affine space, and its elements will be named points of the affine space. The vector space V is named the directorial vector space of the affine space, and its elements will be named vectors of the affine space. The function j is named the affine structure function.
The elements of an affine space are points and vectors.
The affine space (A, V, j) is named real or complex by the vector space V which can be real or complex.
If we consider A = B = C in the axiom A_{1}) , then j (A, A) =_{}, A I A. Therefore, thru the structure function, to any bipoint (A, A) corresponds the null vector_{}.
The corresponding vectors of a pair of symmetrical bipoints are opposite vectors. Indeed, if we consider C = B, in axiom A_{1}), we have
j (A, B) =  j (B, A).
1.2 Consequence. 
Function j is surjective and moreover, for each fixed point O I A j_{O}_{ } A V , j_{O} (A) = j (O, A), A I A, is bijective. 
The proof followes considering axioms A_{1}) and A_{2}).
In an affine space (A, V, j), the function j determines an equivalence relation on the bipoints set of A, which we will name relation of equipollence.
We will say that the bipoint (A, B) is equipollent with the bipoint (C, D) if they have the same image thru j
(A, B) ~ (C, D) _{} j (A, B) = j (C, D) (1.1)
It is easy to verify that the relation “~” is reflexive, symmetric and transitive, meaning it is a relation of equipollence over A A.
The factor space A A_{ }is in a bijective correspondence with the vector space V. To each vector _{}_{}V, corresponds a single equivalence class of equipollent bipoints, and that is:
^{ }(_{}) = (1.2)
When we identify the factor space A A_{ } with the vector space V thru this bijection, the class of the bipoint (A, B) denoted by _{}, is named free vector of the affine space.
Considering this, the axioms A_{1}) and A_{2}) can be written as follows:
A, B, C I A _{} (1.3)
_{}A,_{} BI A unique such that _{}
Let O IA be a fixed point and A A = the set of the bipoints with O as origin.
Taking into consideration the Consequence 1.2 and that the relation A I A _{} (O, A)IA° is a bijective correspondence, it results that A° can identify itself as well with A as with the directorial vector space V.
When A° is identified with the vector space V, a vector structure from V is induced on A°. The vectors of this space are called tied vectors of the affine space, or tangent vectors in O to A, and will be denoted by _{}.
When A is identified with the vector space A° thru the bijection A I A (O, A) I A^{o}, it means that A was considered a vector space having as origin the point O.
The vector _{} will be named position vector.
Actually in any point O I A of an affine space (A V j), a vector space A°, which identifies with A°, can be constructed.
Following these identifications, the notion of dimension of an affine space is justified as being the dimension of the directorial vector space V.
If dimV= n, then the affine space of dimension n will be denoted by (A_{ n}, V_{n}), shortly A_{ n}
° If the vector space V is an euclidean vector space, then the affine space (A, V, j) is called euclidean punctual space. If dimV=n, then we will denote by E_{n} the corresponding euclidean vector space. The euclidean structure of the directorial vector space V will allow studying the metric properties of certain subsets of the euclidean punctual space E_{n}
There are affine spaces that are not vector spaces. But, any vector space is an affine space, because the function j: V V V, _{} verifies the axioms A_{1}) and A_{2}). The affine space (V, V, j) such defined is named canonical affine space associated to the vector space V.
1° The standard affine space
Let’s consider the standard space K^{n}. This space can be organized as a vector space for which we can associate the canonical affine space (K^{n}, K^{n}, j ), where the affine structure function j is defined by the relation j ( A, B) = ( b_{1 } a_{1}, b_{2 } a_{2 }, , b_{n } a_{n }) , for A = ( a_{1}, a_{2 }, , a_{n }) and B = (b_{1}, b_{2}, , b_{n}). This affine space is named standard affine space and will be denoted with K^{n} as well.

Particularly for K = R, we have the affine standard space (R^{n}, R^{n}, j) where the directorial vector space R^{n} is an euclidean space, therefore the affine space (R^{n}, R^{n}, j) becomes an euclidean punctual space.
2° The geometrical affine space of the free vectors
Let’s consider as, support set, the punctual space of the elementary geometry, denoted by E_{ }, as directorial vector space , the vector space of the free vectors V_{ } and as affine structure function j_{ }: E_{3} E_{3} V_{ } j(A,B) =_{}_{}V_{ }
In this way we obtain the geometrical affine space of the free vectors A_{ } = (E_{3}, V_{ } j). This space constituted the model for the affine spaces. We will study in detail this space in the next chapter.
3° The linear variety of a vector space V are affine spaces.
A linear variety of a vector space V is a subset L, _{}, where V is a vector subspace of V.
If we consider the function:
j : L L V , _{},
then the axioms A_{1}) and A_{2}) are satisfied and, therefore, the triplet (L, V j is an affine space.
In particular, any vector subspace is an affine space.
2.5 Definition. 
A cartesian frame is a pair R , where O is a fixed point in A_{n}, and B is a basis of the directorial vector space. 
Let B be a basis of the vector space V_{n}. Then, for each point P I A_{n}, the position vector _{} can be written in an unique way as follows:
_{} (2.8)
The scalars x_{1}, x_{2},, x_{n} are called cartesian coordinates of the point P with respect to frame R , and the bijection _{} is called function of coordinates with respect to frame R
Let R be a cartesian frame in A_{n }. Another frame R of A_{n} will be determined in an unique way if we know the position vector of the point O with respect to the initial frame R and the relation between the basis B and the initial basis
B , that is:
_{} (2.9)
If P I A_{n} is a point and (x_{i}), (x’_{j}), i,j = _{} are its coordinates in the frame R respectively R , then from the relation _{} we obtain the formulas:
_{} (2.10)
called the coordinate transformation formula.
By denoting X = ^{t}[x_{1}, x_{2}, , x_{n}], X = ^{t}[x _{ }, x _{ }, , x _{n}], A_{0} = (a_{i}_{0}),
A = (a_{ij}) we can write the coordinate transformation formula in the following form:
_{} (2.11)
The matrix _{} of n + 1 rank is called the matrix of change of frames from R to R
In particular, if B B then A = I and the equations (2.11) are written as:
_{} (2.11)
The changing of the frame R with R governed by the coordinate transformation formula (11) is called translation.
If O = O, then changing the frame R with the frame R , meaning a_{i}_{0} = 0, _{}, is called centroaffinity and is characterised by the following equations:
_{} (2.11)
Let (A , V, j) be an affine space, A a nonempty subset of A and j its restriction to A A If V j A A ) is a vector subspace of V then the axioms A_{1}) and A_{2}) are satisfied for the triplet
A , V j
3.1 Definition. 
We call affine subspace of the affine space A , V, j a triplet (A , V j ) , with the properties: A A is a nonempty subset  V j A A ) is a vector subspace of V j is the restriction of j to A A

An affine subspace of the affine space (A , V, j) is determined either by the subset A A for which j A A ) = V V is a vector subspace, or by a point P_{0} I A and a vector subspace V V. In this case the support set is given by _{}.
Let us consider A and _{} two affine spaces.
3.7 Definition 
An application t: A _{} with the property t aP bQ at(P) + bt(Q), P, Q I A and a b I K, a b=1 is called an affine application. 
An affine application t:A _{} uniquely determines the morphism T: V _{} between the associated vector spaces. Knowing that for _{} and A I A, B I A such that j (A, B) = _{}, we can define the associated linear map T : V _{} by _{}, where _{} is affine structure function of the space _{}. The definition does not depend on the choice of the point A.
The set of all bijective affine applications on A represents a group GA(A), called the affine group.
We will name object of the affine space A any subset of points belonging to A
By affine geometry of the affine space A, means the study of all objects, and their invariant properties with respect to the affine group.
The simplest and yet most important affine properties are:
the property collinearity of three points
the property of two affine spaces to be parallel
These mentioned affine properties aid in determining other properties, therefore they will be called fundamental affine properties.
§4. The geometrical affine space of the free vectors
Let us consider E_{3 }the punctual space of the elementary geometry and V_{ } the vector space of the free vectors.
If we associate to any bipoint (A, B) I E_{3} E_{3 }the free vector_{}I V_{ }, then the application j : E_{3} E_{3} V_{ } j (A, B) = _{} satisfies properties A_{1}) and A_{2}) of the affine space definition:
A_{1}) A, B, C I E_{3 }, _{}
A_{2}) _{}I V_{ } A I E_{3} exists B I E_{3} uniquely determined by _{}.
4.1 Definition. 
The triplet A_{ } (E_{3}, V_{ } j) is called the affine geometrical space of the free vectors. 
The elements of the affine space A_{ } are points and vectors. The points of the affine space A_{ }are points of the support set E_{3 }and we will denote them by capital letters A, B, C, , O, P, , and the vectors of the affine space A_{ } are vectors of the directorial vector space V_{ }, free vectors that we’ll denote by _{}, _{}, , or by _{}. The application j : E_{3} E_{3} V_{ } which satisfies axioms A_{1}) and A_{2}) represents the affine structure function, and the equivalence relation defined by it on the set E_{3 }is exactly the relation of equipollence “~” of oriented segments, like it was defined in the euclidean geometry.
Consider O I E_{3 }a fixed point. The application j : E_{3} E_{3} V_{ } defined by j_{ }(A) = j(O, A), A I E_{3} is bijective allowing the identification of the punctual space E_{3 }with the vector space of the free vectors.
The point O I E_{3}, corresponding to the null vector _{}, will be considered as origin of the affine space A_{ }. Furthermore, _{}IV_{3} there exists an unique point A I E_{3} determined by the relation _{}. The vector _{} is called the position vector of point A.
The set of the position vectors forms an isomorphic vector space with the vector space of the free vectors.
Lets consider now two distinct points A and B of the space E_{3}. The affine subspace generated by A and B, L() = = d is a subspace of dimension one named affine line, shortly  line, having the directorial vector space the vector line
V_{ }
For each point P I d , collinear with A and B, the system of points is affinedependent, that means the vectors _{} and _{}are linearly dependent.
We remind that two vectors with the same direction are named collinear vectors. The vectors of the subspace V_{ }have the same direction, which justifies the definition of collinearity from an affine space.
4.2 Proposition. 
Two vectors _{} and _{} I V_{ } are collinear if and only if they are linearly dependent, meaning l m I R, _{} such that _{}.

Proof. Consider O I E_{3 }a fixed point. Consider A, B I E_{3} such that _{} and _{}. If _{}, _{} are collinear then the points O, A, B are collinear, meaning that the system of points is affine dependent. This is equivalent with the linear dependence of the vectors _{} and _{}.
The oriented segments _{} and _{} are the representation of vectors _{} and _{} in O, meaning that _{}are _{} linearly dependent for any choice of point O.
Conversely, if _{}and _{} I V_{ } are linearly dependent, then O I E_{3}, the vectors _{} and _{} are linearly dependent, meaning that the system of points is affine dependent. The collinearity of the points O, A, and B is equivalent with the fact that the vectors _{}and _{} have the same direction. (q.e.d.)
If _{}is collinear with _{}then we write:
_{} , l I R (the condition of collinearity) (4.1)
4.3 Consequence. 
The subset _{} of all collinear vectors with the nonnull vector _{} is a onedimensional vector subspace. 
Any three noncollinear points are affine dependent, meaning that any two noncollinear vectors are linearly dependent.
Three noncollinear points A, B, C I E_{3} determine a plane. The plane generated by these three affine independent points is determined by:
A_{ } p
affine subspace having as directorial vector space the vector plane:
_{}
For any point PI p , the system is affine dependent, which means that the vectors _{}, _{} and _{} are linearly dependent.
Three vectors _{}, _{} and _{} are considered to be coplanar if they are parallel with a plane.
4.4 Proposition. 
Three vectors _{}, _{}, _{} I V_{ } are coplanar if and only if they are linearly dependent, namely, l m n I R, l^{ } m^{ } n^{ } such that _{}. 
If _{}, _{} I V_{ } are two vectors, then _{}I V_{ } coplanar with _{} and _{}, can be written as follows:
_{} = _{}, l m I R , (the condition of coplanarity) (4.2)
4.5 Consequence. 
The subset _{} of all vectors coplanar with the nonnull vectors _{} and _{}, is a twodimensional vector space. 
Consider now four noncoplanar points A, B, C, D I E_{3 } . The system of points is affine independent, which means that any three noncoplanar vectors are linearly independent. The affine space generated by four noncoplanar points is of dimension three and any five points of this space will be affine dependent.
4.6 Theorem. 
The vector space V_{ } has three dimensions. 
Fig. 1
representatives of the vectors _{}, _{}, _{}, _{} in the point O (fig. 1).
Using twice the parallelogram rule for summing two free vectors in the parallelogram OA_{1}X_{1}B and respectively OX_{1}XC_{1}, it results:
_{}
Because _{} and _{}, _{} and _{}, _{} and _{} are collinear then we have
_{} or
_{},
namely is a basis for the vector space of the free vectors, therefore dimV_{ } = 3. q.e.d.
For any point O I E_{3} and a given basis in V_{ }, the ensemble R (O; _{}) represents a cartesian frame in the affine space A_{ } ( E_{3}, V_{ } j
For any point P I E_{3} we have the position vector _{} given by:
_{} ; x_{1}, x_{2}, x_{3} I R
The scalars x_{1}, x_{2}, x_{3} I R are called cartesian coordinates of point P.
If _{} I V_{ }is a free vector, then P I E_{3} is unique, such that _{}, and in the frame R _{}is written as:
_{},
where the scalars x_{1}, x_{2}, x_{3} I R, the coordinates of point P, will be called the coordinates of vector _{} in the frame R
If R(O; _{}) is a fixed cartesian frame, A_{ }is a geometrical affine space and x_{1}, x_{2}, x_{3} I R are the coordinates of vector _{} I V_{ }, we will write that _{} = ( x_{1}, x_{2}, x_{3}) or briefly _{}( x_{1}, x_{2}, x_{3}).
If we have _{} ( x_{1}, x_{2}, x_{3}) and _{} ( y_{1}, y_{2}, y_{3}) two free vectors, then:
1° _{} is collinear with _{} (_{}_{}) if and only if their coordinates are proportional (or equal in the particular case of _{} = _{}).
2° _{}, _{}, _{} are coplanar if and only if the coodinates of one vector represents a linear combination of the other two.


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