v2010.10.26 - Convex Optimization

52 CHAPTER 2. CONVEX GEOMETRYR 2 R 3(a)(b)Figure 17: (a) Cube in R 3 projected on paper-plane R 2 . Subspace projectionoperator is not an isomorphism because new adjacencies are introduced.(b) Tesseract is a projection of hypercube in R 4 on R 3 .is convex. It is easy to show directly that convex combination of elementsfrom Y remains an element of Y . 2.12 Instead given convex set Y , C mustbe convex consequent to inverse image theorem 2.1.9.0.1.More generally, vw T in (43) may be replaced with any particular matrixZ ∈ R p×k while convexity of set 〈Z , C〉⊆ R persists. Further, by replacingv and w with any particular respective matrices U and W of dimensioncompatible with all elements of convex set C , then set U T CW is convex bythe inverse image theorem because it is a linear mapping of C . 2.2.1 Frobenius’2.2.1.0.1 Definition. Isomorphic.An isomorphism of a vector space is a transformation equivalent to a linearbijective mapping. Image and inverse image under the transformationoperator are then called isomorphic vector spaces.△2.12 To verify that, take any two elements C 1 and C 2 from the convex matrix-valued set C ,and then form the vector inner-products (43) that are two elements of Y by definition.Now make a convex combination of those inner products; videlicet, for 0≤µ≤1µ 〈vw T , C 1 〉 + (1 − µ) 〈vw T , C 2 〉 = 〈vw T , µ C 1 + (1 − µ)C 2 〉The two sides are equivalent by linearity of inner product. The right-hand side remainsa vector inner-product of vw T with an element µ C 1 + (1 − µ)C 2 from the convex set C ;hence, it belongs to Y . Since that holds true for any two elements from Y , then it mustbe a convex set.