Fundamentals of Linear Algebra and Optimization

Problem B1 (20 +∞2/∞1 ≈ 40 pts). (1) Let H be the affine hyperplane in R given by the equation a1x1 + · · ·+ anxn = c, with ai 6= 0 for some i, 1 ≤ i ≤ n. The linear hyperplane H0 parallel to H is given by the equation a1x1 + · · ·+ anxn = 0, and we say that a vector y ∈ R is orthogonal (or perpendicular) to H iff y is orthogonal to H0. Let h be the intersection of H with the line through the origin and perpendicular to H. Prove that the coordinates of h are given by

[1]  J. Hadamard.  Leçons de géométrie élémentaire II : (Géométrie dans l'espace) , 1901 .

[2]  Jianbo Shi,et al.  Grouping with Bias , 2001, NIPS.

[3]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[4]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[5]  Jianbo Shi,et al.  Solving Markov Random Fields with Spectral Relaxation , 2007, AISTATS.

[6]  David Salesin,et al.  Wavelets for computer graphics: theory and applications , 1996 .

[7]  Bernard Gostiaux,et al.  Géométrie différentielle : variétés, courbes et surfaces , 1992 .

[8]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[9]  Revaz Valerianovich Gamkrelidze,et al.  Topology and Geometry , 1970 .

[10]  M. Carter.  Computer graphics: Principles and practice , 1997 .

[11]  Bernd Gärtner,et al.  Understanding and using linear programming , 2007, Universitext.

[12]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[13]  Charles L. Epstein,et al.  Introduction to the mathematics of medical imaging , 2003 .

[14]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[15]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[16]  Emanuele Trucco,et al.  Introductory techniques for 3-D computer vision , 1998 .

[17]  James R. Munkres,et al.  Topology; a first course , 1974 .

[18]  Jean Gallier,et al.  Geometric Methods and Applications: For Computer Science and Engineering , 2000 .

[19]  J. Miller.  Numerical Analysis , 1966, Nature.

[20]  Christopher J. BISHOPAbstra,et al.  Orthogonal Functions , 2022 .

[21]  S. Axler.  Linear Algebra Done Right , 1995, Undergraduate Texts in Mathematics.

[22]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[23]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[24]  D. Ruppert.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction , 2004 .

[25]  R. Cooke.  Real and Complex Analysis , 2011 .

[26]  G. W. STEWARTt.  ON THE EARLY HISTORY OF THE SINGULAR VALUE DECOMPOSITION * , 2022 .

[27]  Truong Q. Nguyen,et al.  Wavelets and filter banks , 1996 .

[28]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[29]  G. Golub.  Some modified eigenvalue problems , 1971 .

[30]  Robert J. Vanderbei,et al.  Linear Programming: Foundations and Extensions , 1998, Kluwer international series in operations research and management service.

[31]  A. Grothendieck,et al.  Espaces vectoriels topologiques , 1954 .

[32]  D. Pedoe,et al.  Geometry, a comprehensive course , 1988 .

[33]  Jean Ponce,et al.  Computer Vision: A Modern Approach , 2002 .

[34]  Calcul diff´erentiel , 2013 .

[35]  R. A. Silverman,et al.  Introductory Real Analysis , 1972 .

[36]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[37]  Leon M. Hall,et al.  Special Functions , 1998 .

[38]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[39]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[40]  J. O´Rourke,et al.  Computational Geometry in C: Arrangements , 1998 .

[41]  Bernhard Schölkopf,et al.  New Support Vector Algorithms , 2000, Neural Computation.

[42]  F. W. Warner.  Foundations of Differentiable Manifolds and Lie Groups , 1971 .

[43]  M. Bertin,et al.  Algèbre linéaire et géométrie classique , 1981 .

[44]  Klaus Nordhausen,et al.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition by Trevor Hastie, Robert Tibshirani, Jerome Friedman , 2009 .

[45]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[46]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[47]  Jean Gallier,et al.  Geometric Methods and Applications , 2011 .

[48]  S. P. Mudur,et al.  Three-dimensional computer vision: a geometric viewpoint , 1993 .

[49]  Bernhard Schölkopf,et al.  Estimating the Support of a High-Dimensional Distribution , 2001, Neural Computation.

[50]  J. Milnor.  Topology from the differentiable viewpoint , 1965 .

[51]  A. Banerjee.  Convex Analysis and Optimization , 2006 .

[52]  Nello Cristianini,et al.  Kernel Methods for Pattern Analysis , 2003, ICTAI.

[53]  J. Gallier.  Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations , 2008, 0805.0292.

[54]  Chih-Jen Lin,et al.  Training ν-Support Vector Classifiers: Theory and Algorithms , 2001 .

[55]  P J Fox,et al.  THE FOUNDATIONS OF MECHANICS. , 1918, Science.

[56]  Ernst Hairer.  Analyse II (Calcul Différentiel et Equations Différentielles) , 1999 .

[57]  Gilbert Strang,et al.  Introduction to applied mathematics , 1988 .

[58]  G. Ziegler.  Lectures on Polytopes , 1994 .

[59]  P. Lockhart.  INTRODUCTION TO GEOMETRY , 2007 .

[60]  Danièle Lino,et al.  Calcul différentiel , 2015, Mathématiques.

[61]  David S. Dummit.  Abstract Algebra / by David S. Dummit, Richard M. Foote , 1999 .

[62]  Gene H. Golub,et al.  Matrix computations , 1983 .

[63]  Gene H. Golub,et al.  Some modified matrix eigenvalue problems , 1973, Milestones in Matrix Computation.

[64]  J. Dieudonné,et al.  Algèbre linéaire et géométrie élémentaire , 1967 .

[65]  Dimitris N. Metaxas.  Physics-Based Deformable Models , 1996 .

[66]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[67]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[68]  D. Serre.  Matrices: Theory and Applications , 2002 .

[69]  J. Munkres.  Analysis On Manifolds , 1991 .

[70]  Mogens Flensted‐Jensen,et al.  GROUPS AND GEOMETRIC ANALYSIS Integral Geometry, Invariant Differential Operators, and Spherical Functions (Pure and Applied Mathematics: A Series of Monographs and Textbooks) , 1985 .

[71]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[72]  Alexander J. Smola,et al.  Learning with kernels , 1998 .

[73]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[74]  Richard M. Wilson,et al.  A course in combinatorics , 1992 .

[75]  A. R. Rao,et al.  Linear algebra , 1961 .

[76]  S. Lang.  Differential and Riemannian Manifolds , 1996 .

[77]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[78]  A. Gray.  Modern Differential Geometry of Curves and Surfaces , 1993 .

[79]  Panos M. Pardalos,et al.  Convex optimization theory , 2010, Optim. Methods Softw..

[80]  Dimitri P. Bertsekas,et al.  Convex Optimization Algorithms , 2015 .

[81]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[82]  S. Lang.  Real and Functional Analysis , 1983 .

[83]  W. Gander,et al.  A constrained eigenvalue problem , 1988 .