The Linear Algebra Survival Guide: Illustrated with Mathematica PDF
Preview The Linear Algebra Survival Guide: Illustrated with Mathematica
The Linear Algebra Survival Guide: Illustrated with Mathematica The Linear Algebra Survival Guide Illustrated with Mathematica Fred E. Szabo, PhD Concordia University Montreal, Canada AMSTERDAM • BOSTON • HEIDELBERG • LONDON• NEW YORK • OXFORD PARIS • SAN DIEGO •SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an Imprint of Elsevier Academic Press is an imprint of Elsevier 125, London Wall, EC2Y 5AS. 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Copyright © 2015 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangement with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-12-409520-5 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress For information on all Academic Press visit our website at http://store.elsevier.com/ Printed and bound in the USA 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 About the Matrix Plot The image on the previous page is a Mathematica matrix plot of a random 9-by-9 matrix with integer elements between -9 and 9. Random matrices are used throughout the book where matrix forms are required to illustrate concepts, properties, or calculations, but where the numerical content of the illustrations is largely irrelevant. The presented image shows how matrix forms can be visualized as two-dimensional blocks of color or shades of gray. MatrixForm[A=RandomInteger[{-−9,9},{9,9}]] -−4 -−9 1 6 8 5 6 7 -−8 1 -−5 -−5 2 -−8 8 8 1 -−8 -−7 -−4 2 -−1 9 3 1 7 6 -−3 -−1 6 4 -−9 1 -−2 0 9 7 -−8 -−4 -−1 -−6 -−8 5 5 1 7 -−5 -−3 -−3 -−1 -−2 -−9 8 -−1 -−1 6 5 6 -−5 4 7 -−9 5 -−7 8 -−7 5 4 -−1 1 -−5 4 8 -−2 8 7 -−8 -−9 4 -−3 -−7 MatrixPlot[A] Preface The principal goal in the preparation of this guide has been to make the book useful for students, teachers, and researchers using linear algebra in their work, as well as to make the book sufficiently complete to be a valuable reference source for anyone needing to understand the computational aspects of linear algebra or intending to use Mathematica to extend their knowledge and understanding of special topics in mathematics. This book is both a survey of basic concepts and constructions in linear algebra and an introduction to the use of Mathematica to represent them and calculate with them. Some familiarity with Mathematica is therefore assumed. The topics covered stretch from adjacency matrices to augmented matrices, back substitution to bilinear functionals, Cartesian products of vector spaces to cross products, defective matrices to dual spaces, eigenspaces to exponential forms of complex numbers, finite-dimensional vector spaces to the fundamental theorem of algebra, Gaussian elimination to Gram–Schmidt orthogonalization, Hankel matrices to Householder matrices, identity matrices to isomorphisms of vector spaces, Jacobian determinants to Jordan matrices, kernels of linear transformations to Kronecker products, the law of cosines to LU decompositions, Manhattan distances to minimal polynomials, vector and matrix norms to the nullity of matrices, orthogonal complements to overdetermined linear systems, Pauli spin matrices to the Pythagorean theorem, QR decompositions to quintic polynomials, random matrices to row vectors, scalars to symmetric matrices, Toeplitz matrices to triangular matrices, underdetermined linear systems to upper- triangular matrices, Vandermonde matrices to volumes of parallelepipeds, well-conditioned matrices to Wronskians, and zero matrices to zero vectors. All illustrations in the book can be replicated and used to discover the beauty and power of Mathematica as a platform for a new kind of learning and understanding. The consistency and predictability of the Wolfram Language on which Mathematica is built are making it much easier to concentrate on the mathematics rather than on the computer code and programming features required to produce correct, understandable, and often inspiring mathematical results. In addition, the included manipulations of many of the mathematical examples in the book make it easy and instructive to explore mathematical concepts and results from a computational point of view. The book is based on my lecture notes, written over a number of years for several undergraduate and postgraduate courses taught with various iterations of Mathematica. I hereby thank the hundreds of students who have patiently sat through interactive Mathematica-based lectures and have enjoyed the speculative explorations of a large variety of mathematical topics which only the teaching and learning with Mathematica makes possible. The guide also updates the material in the successful textbook “Linear Algebra: An Introduction Using Mathematica,” published by Harcourt/Academic Press over a decade ago. The idea for the format of this book arose in discussion with Patricia Osborn, my editor at Elsevier/Academic Press at the time. It is based on an analysis of what kind of guide could be written that meets two objectives: to produce a comprehensive reference source for the conceptual side of linear algebra and, at the same time, to provide the reader with the computational illustrations required to learn, teach, and use linear algebra with the help of Mathematica. I am grateful to the staff at Elsevier/Academic Press, especially Katey Birtcher, Sarah Watson and Cathleen Sether for seeing this project through to its successful conclusion and providing tangible support for the preparation of the final version of the book. Last but not least I would like to thank Mohanapriyan Rajendran (Project Manager S&T, Elsevier, Chennai) for his delightful and constructive collaboration during the technical stages of the final composition and production. xii | The Linear Algebra Survival Guide Many students and colleagues have helped shape the book. Special thanks are due to Carol Beddard and David Pearce, two of my teaching and research assistants. Both have helped me focus on user needs rather than excursions into interesting but esoteric topics. Thank you Carol and David. Working with you was fun and rewarding. I am especially thankful to Stephen Wolfram for his belief in the accessibility of the computable universe provided that we have the right tools. The evolution and power of the Wolfram Language and Mathematica have shown that they are the tools that make it all possible. Fred E Szabo Beaconsfield, Quebec Fall 2014 Dedication To my family: Isabel, Julie and Stuart, Jahna and Scott, and Jessica, Matthew, Olivia, and Sophie About the Author Fred E. Szabo Department of Mathematics, Concordia University, Montreal, Quebec, Canada Fred E. Szabo completed his undergraduate studies at Oxford University under the guidance of Sir Michael Dummett, and received a Ph.D. in mathematics from McGill University under the supervision of Joachim Lambek. After postdoctoral studies at Oxford University and visiting professorships at several European universities, he returned to Concordia University as a faculty member and dean of graduate studies. For more than twenty years, he developed methods for the teaching of mathematics with technology. In 2012 he was honored at the annual Wolfram Technology Conference for his work on "A New Kind of Learning" with a Wolfram Innovator Award. He is currently professor and Provost Fellow at Concordia University. Professor Szabo is the author of five Academic Press publications: - The Linear Algebra Survival Guide, 1st Edition - Actuaries' Survival Guide, 1st Edition - Actuaries' Survival Guide, 2nd Edition - Linear Algebra: Introduction Using Maple, 1st Edition - Linear Algebra: Introduction Using Mathematica, 1st Edition Introduction How to use this book This guide is meant as a standard reference to definitions, examples, and Mathematica techniques for linear algebra. Complementary material can be found in the Help sections of Mathematica and on the Wolfram Alpha website. The main purpose of the guide is therefore to collect, in one place, the fundamental concepts of finite-dimensional linear algebra and illustrate them with Mathematica. The guide contains no proofs, and general definitions and examples are usually illustrated in two, three, and four dimen- sions, if there is no loss of generality. The organization of the material follows both a conceptual and an alphabetic path, whichever is most appropriate for the flow of ideas and the coherence of the presentation. All linear algebra concepts covered in this book are explained and illustrated with Mathematica calculations, examples, and additional manipulations. The Mathematica code used is complete and can serve as a basis for further exploration and study. Examples of interactive illustrations of linear algebra concepts using the Manipulate command of Mathematica are included in various sections of the guide to show how the illustrations can be used to explore computational aspects of linear algebra. Linear algebra From a computational point of view, linear algebra is the study of algebraic linearity, the representation of linear transforma- tions by matrices, the axiomatization of inner products using bilinear forms, the definition and use of determinants, and the exploration of linear systems, augmented matrices, matrix equations, eigenvalues and eigenvectors, vector and matrix norms, and other kinds of transformations, among them affine transformations and self-adjoint transformations on inner product spaces. In this approach, the building blocks of linear algebra are systems of linear equations, real and complex scalars, and vectors and matrices. Their basic relationships are linear combinations, linear dependence and independence, and orthogonality. Mathematica provides comprehensive tools for studying linear algebra from this point of view. Mathematica The building blocks of this book are scalars (real and complex numbers), vectors, linear equations, and matrices. Most of the time, the scalars used are integers, playing the notationally simpler role of real numbers. In some places, however, real numbers as decimal expansions are needed. Since real numbers may require infinite decimal expansions, both recurring and nonrecurring, Mathematica can represent them either symbolically, such as ⅇ and π𝜋, or as decimal approximations. By default, Mathematica works to 19 places to the right of the decimal point. If greater accuracy is required, default settings can be changed to accommodate specific computational needs. However, questions of computational accuracy play a minor role in this book. In this guide, we follow the lead of Mathematica and avoid the use of ellipses (lists of dots such as "...") to make general statements. In practically all cases, the statements can be illustrated with examples in two, three, and four dimensions. We can therefore also avoid the use of sigmas (Σ) to express sums. The book is written with and for Mathematica 10. However, most illustrations are backward compatible with earlier versions of Mathematica or have equivalent representations. In addition, the natural language interface and internal link to Wolfram/Al- pha extends the range of topics accessible through this guide. Mathematica cells Mathematica documents are called notebooks and consist of a column of subdivisions called cells. The properties of notebooks and cells are governed by stylesheets. These can be modified globally in the Mathematica Preferences orcell- by-cell, as needed. Theavailablecelltypes in adocumentarerevealed byactivating thetoolbars in theWindow >Show Toolbarmenu.UnlessMathematica isusedexclusively for input–outputcalculations, it isadvisable toshow the toolbar immediately aftercreating a notebook or tomakeShow Toolbar a default notebooksetting.
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