The Linear Algebra Survival Guide: Illustrated with Mathematica

432 Pages·2015·10.23 MB·English

by  Fred Szabo

The Linear Algebra Survival Guide: Illustrated with Mathematica
The Linear Algebra
Survival Guide
Illustrated with Mathematica
Fred E. Szabo, PhD
Concordia University
Montreal, Canada
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ISBN: 978-0-12-409520-5
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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.