Analysis and Linear Algebra for Finance: Part II Patrick Roger Download free books at Patrick Roger Analysis and Linear Algebra for Finance: Part II Patrick ROGER LaRGE Research Center EM Strasbourg Business School University of Strasbourg Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part II First edition © 2013 Patrick Roger & bookboon.com (Ventus Publishing ApS) ISBN 978-87-403-0429-9 Download free eBooks at bookboon.com Deloitte & Touche LLP and affiliated entities Analysis and Linear Algebra for Finance: Part II Contents Contents Introduction Vector spaces and linear mappings 1.1 Vector spaces: definitions and general properties 1.2 Linear mappings 25 1.3 Finite-dimensional spaces and matrices 33 1.4 Norms and inner products 55 1.5 Hilbert spaces 61 1.6 Separation theorems and Farkas lemma 64 Functions of several variables 2.1 Metric spaces 2.2 Continuity and differentiability 2.3 Implicit and homogeneous functions 360° thinking 360° thinking 73 74 84 106 360° thinking Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities Discover the truth at www.deloitte.ca/careers Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities at www.deloitte.ca/careers Discover the truth Click on the ad to read more © Deloitte & Touche LLP and affiliated entities Dis Analysis and Linear Algebra for Finance: Part II Contents Optimization without constraints 113 3.1 Preliminaries 114 3.2 Optimizing a single-variable function 120 3.3 Optimizing a function of two variables 124 3.4 Functions of n variables 131 Constrained optimization 137 4.1 Functions of two variables and equality constraint 138 4.2 Functions of p variables with m equality constraints 145 4.3 Functions of p variables with mixed constraints 150 Index 154 Increase your impact with MSM Executive Education For almost 60 years Maastricht School of Management has been enhancing the management capacity of professionals and 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                                                                                                                                                                                                                                                 Download free eBooks at bookboon.com Introduction Analysis and Linear Algebra for Finance: Part II Vector spaces and linear mappings                                                                                                                                                                                                               Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part II  Vector spaces and linear mappings                                                                                                                                                                                                                                              Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part II Vector spaces and linear mappings                                                                                                                                                     GOT-THE-ENERGY-TO-LEAD.COM We believe that energy suppliers should be renewable, too We are therefore looking for enthusiastic new colleagues with plenty of ideas who want to join RWE in changing the world Visit us online to find out what we are offering and how we are working together to ensure the energy of the future Download free eBooks at bookboon.com Click on the ad to read more Analysis and Linear Algebra for Finance: Part II  Vector spaces and linear mappings                                                                                                                                                                                                                                                                                                          Download free eBooks at bookboon.com 10      Constrained optimization Analysis and Linear Algebra for Finance: Part II                                                                                                                                                                                                                                               Increase your impact with MSM Executive Education For almost 60 years Maastricht School of Management has been enhancing the management capacity of professionals and organizations around the world through state-of-the-art management education Our broad range of Open Enrollment Executive Programs offers you a unique interactive, stimulating and multicultural learning experience Be prepared for tomorrow’s management challenges and apply today For more information, visit www.msm.nl or contact us at +31 43 38 70 808 or via admissions@msm.nl For more information, visit www.msm.nl or contact us at +31 43 38 70 808 the globally networked management school or via admissions@msm.nl Executive Education-170x115-B2.indd Download free eBooks at bookboon.com 18-08-11 15:13 142 Click on the ad to read more Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                      Download free eBooks at bookboon.com 143 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                                                                                                                                                                                                                       Download free eBooks at bookboon.com 144 Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                                                                                                  Download free eBooks at bookboon.com 145 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                        GOT-THE-ENERGY-TO-LEAD.COM We believe that energy suppliers should be renewable, too We are therefore looking for enthusiastic new colleagues with plenty of ideas who want to join RWE in changing the world Visit us online to find out what we are offering and how we are working together to ensure the energy of the future Download free eBooks at bookboon.com 146 Click on the ad to read more Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                                                                                                  Download free eBooks at bookboon.com 147 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                                                                                                                                                                                                                                                                                                                  Download free eBooks at bookboon.com 148 Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                                                                                                                       Download free eBooks at bookboon.com 149 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                                                                                                                                                                                Download free eBooks at bookboon.com 150 Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                                                                                                                                                                                                                                        Download free eBooks at bookboon.com 151 Analysis and Linear Algebra for Finance: Part II  Constrained optimization                                                                                                                                                                                                                                                                                 Download free eBooks at bookboon.com 152 Analysis and Linear Algebra for Finance: Part II Constrained optimization                                                                                                                 With us you can shape the future Every single day For more information go to: www.eon-career.com Your energy shapes the future Download free eBooks at bookboon.com 153 Click on the ad to read more Analysis and Linear Algebra for Finance: Part II Index Index convex, 104 homogeneous, 109 Arrow-Debreu security, 22 gradient, 88 ball Hessian, 97 closed, 79 open, 78 identity basis, 20 implicit function change, 45 element, theorem, 109 closed, 79 Implicit function theorem, 106 combination indifference curve, 93 convex, 103 inner product linear, 15 norm, 56 curve interior, 79 isomorphism, 28 indifference, 93 derivative Lagrangian, 140 partial, 86 multiplier, 140 determinant, 41, 42 lemma differential, 91 alternative, 66 distance Farkas, 66 limit, 84 metric, 74 linear eigenspace, 50 combination, 15 eigenvalue, 49 linear functional, 31 eigenvector, 49 endomorphism, 40 mapping linear, 25 Farkas lemma, 66 compound, 35 form functional, 32 isomorphism, 28 quadratic, 60 formula kernel, 26 matrix, 33 Taylor, 99 function C1, 86 marginal rate of substitution, 93 concave, 104 matrix continuous, 85 Download free eBooks at bookboon.com 154 quadratic form, 60 change-of-basis, 45 Analysis and Linear Algebra for Finance: Part II Index cofactor, 41 sequence definite, 57 Cauchy, 84 diagonalizable, 54 convergent, 84 diagonalization, 49 set eigenvalue, 49 bounded, 83 eigenvector, 49 closed, 79 Hessian, 97 closure, 81 identity, 40 compact, 83 orthogonal, 55 convex, 62, 103 principal minor, 41 dense, 82 semidefinite, 57 exterior, 81 maximum frontier, 81 global, 121, 150 interior, 81 local, 121, 122, 129, 145, 149 open, 79 metric solution distance, 74 Euclidian, 75 space space, 75 Hilbert, 61 minimum isomorphic, 29 global, 150 metric, 75 local, 129, 145, 149 separable, 82 corner, 116 multiplier sum direct, 13 vectors, 12 Kuhn-Tucker, 152 net present value, 108 norm, 55 theorem notation Euler, 111 mean value, 97 Landau, 100 numéraire, 71 vector optimum Arrow-Debreu, 22 global, 150 direct sum, 13 local, 126, 132, 147 family option rank, 24 spanning, 19 call, 15 linearly polynomial dependent, 16 independent, 17 characteristic, 50 Download free eBooks at bookboon.com 155 orthogonal, 59 Analysis and Linear Algebra for Finance: Part II rank, 24 space, Index basis, 20 finite-dimensional, 22 identity element, subspace, 11 supplementary, 13 sum, 12 vector space Hilbert, 61 inner product, 56 norm, 55 Download free eBooks at bookboon.com 156 ... entities Analysis and Linear Algebra for Finance: Part II Contents Contents Introduction Vector spaces and linear mappings 1.1 Vector spaces: definitions and general properties 1.2 Linear mappings... Analysis and Linear Algebra for Finance: Part II Patrick ROGER LaRGE Research Center EM Strasbourg Business School University of Strasbourg Download free eBooks at bookboon.com Analysis and Linear. .. more and apply online Visit accenture.com/bootcamp Download free eBooks at bookboon.com 27 Click on the ad to read more Analysis and Linear Algebra for Finance: Part II  Vector spaces and linear