# Space and Geometry - In the Light of Physiol., Psych. and by E. Mach By E. Mach

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2 shows that every afﬁne subspace of E is of the form u + U , for a − → − → − → subspace U of E . The subspaces of E are the afﬁne subspaces of E that contain 0. − → The subspace V associated with an afﬁne subspace V is called the direction of − → − → V . It is also clear that the map + : V × V → V induced by + : E × E → E confers − → to V, V , + an afﬁne structure. 7 illustrates the notion of afﬁne subspace. − → E E − → V a − → V = a+ V − → Fig. 7 An afﬁne subspace V and its direction V . − → By the dimension of the subspace V , we mean the dimension of V .

Note that the barycenter x of the family of weighted points ((ai , λi ))i∈I is the unique point such that − → → ax = ∑ λi − aa i for every a ∈ E, i∈I and setting a = x, the point x is the unique point such that → = 0. xa i ∑ λi − i∈I In physical terms, the barycenter is the center of mass of the family of weighted points ((ai , λi ))i∈I (where the masses have been normalized, so that ∑i∈I λi = 1, and negative masses are allowed). Remarks: (1) Since the barycenter of a family ((ai , λi ))i∈I of weighted points is deﬁned for families (λi )i∈I of scalars with ﬁnite support (and such that ∑i∈I λi = 1), we might as well assume that I is ﬁnite.

V This should dispell any idea that afﬁne spaces are dull. Afﬁne spaces not already equipped with an obvious vector space structure arise in projective geometry. 1 that the complement of a hyperplane in a projective space has an afﬁne structure. 4. 4 Afﬁne Combinations, Barycenters 17 − → E E b − → ab a c − → ac − → bc Fig. 4 Points and corresponding vectors in afﬁne geometry. → Since a = a + − aa and by (A1) a = a + 0, by (A3) we get − → aa = 0. Thus, letting a = c in Chasles’s identity, we get − → → − ba = −ab. 