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convex_sets - Notes by David Groisser Copyright c 2006 for...

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Unformatted text preview: Notes by David Groisser, Copyright c 2006, for use in MAS 4105 Convex Sets in Vector Spaces Throughout these notes, “vector space” means “real vector space”. Although vector spaces are defined purely algebraically, we often visualize them ge- ometrically. Geometry in Euclid’s sense—in which lengths and angles play key roles— requires not just a vector-space structure, but an inner product , about which you have learned or will learn in this course. However, certain more primitive geometric notions, such as point , line , and another that is the chief topic of these notes— convex set —make sense in any vector space, not just in inner-product spaces. Definition 1. Let V be a vector space and let v , u ∈ V with u 6 = . The line through v in the direction of u (or parallel to u ) is the set ‘ = { v + t u | t ∈ R } . (1) A subset of V is a line (or straight line ) if it is a line through some element of V in some direction. Exercise 1. To make sure you have a feel for where the terminology comes from, draw examples of lines in R 2 and R 3 through various points in various directions. The next several exercises develop some simple but important features of lines. Exercises. Below, V is a fixed but arbitrary vector space. 2. Show that if ‘ is a line in V then for all v 1 , v 2 , v 3 , v 4 ∈ ‘ , the vectors v 2- v 1 and v 3- v 4 are parallel (i.e. one is a multiple of the other). 3. Show that every one-dimensional subspace of V is a line, but that if dim( V ) ≥ 2, the converse is false. 4. Show that for any line ‘ in V , there are infinitely many vectors v and nonzero vectors u such that ‘ is the line through v in the direction of u . 5. Let v ∈ V and let S be a nonempty subset of V . The translate of S by v (or S translated by v ) is the set v + S := { v + w | w ∈ S } . If S = v + S for some v ∈ V , we say that S is a translate of S . 1 (a) Show that “is a translate of” is an equivalence relation. I.e. show that if S,S ,S 00 are nonempty subsets of V , then (i) S is a translate of S ; (ii) if S is a translate of S then S is a translate of S , and (iii) if S is a translate of S , and S 00 is a translate of S , then S 00 is a translate of S . (b) Show that for every line ‘ in V , there is a unique one-dimensional subspace L ⊂ V such that ‘ is a translate of L . Further, show that in the context of Exercise 4, the vectors u are exactly the nonzero elements of L . 6. Let H be a subspace of a vector space, and let v ∈ H . Show that v + H = H . 7. Let V be a vector space v 1 , v 2 ∈ V , let H 1 ,H 2 be subspaces of V , and suppose that v 1 + H 1 = v 2 + H 2 . Show that H 1 = H 2 . Definition 2. A translated subspace 1 in a vector space V is any set of the form v + H where v ∈ V and H is a subspace of V . Exercise 7 shows that if A is a translated subspace in V , then there is a unique subspace H = H A of which A is a translate. (However, unless A consists of a single element, there are infinitely many vectors by which...
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