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Lecture 1: Affine Subspaces, Affine Combinations, Convex Sets, Convex Combinations

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Lecture 1: Affine Subspaces, Affine Combinations, Convex Sets, Convex Combinations LECTURE 1: AFFINE SUBSPACES, AFFINE COMBINATIONS, CONVEX SETS, CONVEX COMBINATIONS In this lecture we introduce the basic concepts used throughout the semester. We deal with only nite dimensional Euclidean spaces. We regard an n-dimensional Euclidean spaces as an ane space whose vectors are the elements of the n-dimensional vector space Rn over the set of real numbers. Fixing an arbitrary point of an ane space, the elements of the corresponding vector space and the points of the space can be identied in a natural way, in which a point is associated to the vector that moves the xed point to this one. In this case the xed point is usually called origin. As it often appears in the literature, during the term we identify the Euclidean space with the vector space Rn (in high school language: we identify points and their position vectors). We will usually denote the points/vectors of the space Rn by small Latin letters, while its subsets by capital Latin letters. We denote the usual inner (scalar) product of Rn by h:; :i. The length p jjvjj of a vector v 2 Rn is the quantity hv; vi. For the coordinates of the vector/point v in the standard orthonormal basis of Rn we use the notation v = (v1; v2; : : : ; vn). We denote the origin by o. The distance of the points and 0 0 0 , denoted p = (x1; x2; : : : ; xn) q = (x1; x2; : : : ; xn) r n by , is the quantity P 0 2, which coincides with the dist(p; q) (xi − xi) i=1 value of jjq − pjj. The interior, boundary, closure and cardinality of a set X ⊆ Rn will be denoted by int(X); bd(X); cl(X); jXj, respectively. Denition 1. Let and be two point sets, and . Then V1 V2 λ 2 R V1 + V2 = fv1 + v2 : v1 2 V1; v2 2 V2g is called the Minkowski sum of the two sets, and λV1 = fλv1 : v1 2 V1g the multiple of V1 by λ. Denition 2. Let p 2 Rn be an arbitrary point, and L and arbitrary (linear) subspace in the vector space Rn. Then the set p + L ⊆ Rn is called an ane subspace of the space Rn. 1 2 The next remark is a straightforward consequence of the properties of linear subspaces. Remark 1. Let p; q 2 Rn and let L; L0 be linear subspaces in Rn. Then p + L = q + L0 is satised if and only if L = L0 and q 2 p + L. Proof. Assume that p + L = q + L0. Then L = (q − p) + L0 by the denition of Minkowski sum, which yields, in particular, that q−p 2 L, from which we have q 2 p + L. But as linear subspaces are closed with respect to addition, q − p 2 L implies (q − p) + L = L, from which (q − p) + L = (q − p) + L0, yielding L = L0. On the other hand, if q 2 p + L, then (q − p) 2 L =) (q − p) + L = L =) q + L = p + L. 2 Theorem 1. A nonempty intersection of ane subspaces is an ane subspace. Proof. Consider the ane subspaces Ai (i 2 I), where I is an arbitrary T index set. Let A = Ai. Consider a point p 2 A. Then, due to the i2I previous remark, for any i 2 I we have Ai = p + Li for some suitable linear subspace of n. The intersection of linear subspaces is a linear Li R T subspace, and thus, L = Li is a linear subspace. On the other hand, i2I we clearly have A = p + L, from which the assertion follows. 2 By the dimension of an ane subspace we mean the dimension of the corresponding linear subspace. We call the 0-, 1-, 2-, (n − 1)- dimensional subspaces points, lines, planes and hyperplanes. The next property readiéy follows from the denition of ane subspaces and the properties of inner product. Remark 2. If u 2 Rn and t 2 R arbitrary, then the set fv 2 Rn : hv; ui = tg is a hyperplane. Furthermore, for any hyperplane H there is some vector u 2 Rn and scalar t 2 R for which H = fv 2 Rn : hv; ui = tg. Since inner product is a continuous map from Rn to R, the previous remark implies that for any hyperplane H decomposes the space into two connected, open components, which we call open half spaces. The unions of open half spaces with the bounding hyperplane we call closed half spaces. Denition 3. Let G1 = p1 + L1 and G2 = p2 + L2 be ane subspaces. If for any vectors v1 2 L1, v2 2 L2 we have hv1; v2i = 0, then we say that G1 and G2 are perpendicular or orthogonal. Two ane subspaces are parallel, if they can be written in the form p1 +L and p2 +L, where L is a linear subspace. 3 Denition 4. Let X ⊂ Rn be a nonempty set. Then the ane hull of X, denoted by aff(X), is dened as the intersection of all ane subspaces containing X. The linear hull of X is dened as the ane hull aff(X [ fog). We denote the linear hull of X by lin(X). The relative interior and relative boundary of X is dened as the interior and boundary of X, respectively, with respect to the induced topology in aff(X). We denote them by relint(X) and relbd(X), respectively. We remark that by Theorem 1, the ane hull of a set is an ane subspace. Denition 5. A point set X is called anely independent if for any x 2 X we have aff(X n fxg) 6= aff X. The points sets that are not anely independent are called anely dependent. Denition 6. Let n nitely many points, and let p1; p2; : : : ; pk 2 R k be real numbers satisfying P . Then the point λ1; λ2; : : : ; λk 2 R λi = 1 i=1 k P λipi is called an ane combination of the points p1; p2; : : : ; pk. i=1 Proposition 1. The ane hull of a set X is the set of the ane combinations of all nite point sets from X. Proof. Let Y denote the set of all ane combinations of nitely many points in X, and let p 2 X be an arbitrary point. Consider the points and numbers for which p1 = p; p2; : : : ; pk 2 X λ1; λ2; : : : ; λk 2 R Pk is satised. According to our conditions: i=1 λi = 1 k k X X λipi = p1 + λi(pi − p1): i=1 i=1 Thus the ane combination can be written as a translate of the point p with a linear combination of the vectors pi − p. Hence, if L denotes the linear subspace formed by the linear combinations of the vectors q − p, q 2 X, then Y = p + L. As it is clearly an ane subspace, we have aff(X) ⊆ Y . On the other hand, if an ane subspace contains X, then it can be written in the form p + L for some linear subspace L. The subspace L contains all vectors of the form q − p, q 2 L, and thus it contains their linear combinations as well. Hence, p + L contains all ane combinations of points of X in the case that p is one of the points. Since any k-point an combination is also a (k + 1)-point ane combination in which one of the points is p, we have that p + L contains all ane 4 combinations of the points of X. Thus, Y ⊆ p + L, implying Y ⊆ aff(X). 2 Corollary 1. A point set X is anely independent if and only if there is no point of X that can be written as an ane combination of some other points from X. Theorem 2. Let n. Then is anely inde- X = fp1; p2; : : : ; pkg ⊂ R X k k P P pendent if and only if λipi = 0 and λi = 0 implies λi = 0 for all i=1 i=1 values of i. Proof. Assume that a pont, say pk, can be written as an ane com- k−1 k−1 P P bination of the other points; that is, pk = λipi, where λi = 1. i=1 i=1 k k P P Then, setting λk = −1, we have 0 = λipi and λi = 0. i=1 i=1 On the other hand, assume that for some values of the coecients k k−1 P P λi, not all of them zero, we have 0 = λipi and λi = 0. Without i=1 i=1 loss of generality, we may assume that λk 6= 0. For any 1 ≤ i ≤ k − 1, let λ0 = − λi . Then i λk k−1 P k−1 λi X i=1 −λk λ0 = − = − = 1; i λ λ i=1 k k and k−1 k−1 X 1 X 1 λ0 p = − λ p = − (−λ p ) = p ; i i λ i i λ k k k i=1 k i=1 k and the point set is anely dependent. 2 2 Corollary 2. If X ⊂ Rn is anely independent, then every point of aff(X) can be uniquely written as an ane combination of some points in X. Theorem 3. If jXj ≥ n + 2, then X is anely dependent. Proof. Assume that p1; p2; : : : ; pn+2 2 X. Consider the vectors p2 − p1; : : : ; pn+2 − p1. Since the n-dimensional Euclidean space is an n- dimensional vector space, the above vectors are linearly dependent, that is one of them, say pn+2 − p1, can be written as a linear combination n+1 of the other vectors: Pn+1 .
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