Undergraduate Texts in Mathematics Editors S.Axler FWo Gehring K.A Ribet Springer Science+Business Media, LLC Undergraduate Texts in Mathematics Editors S.Axler FWo Gehring K.A Ribet Springer Science+Business Media, LLC Undergraduate Texts in Mathematics Anglin: Mathematics: A Concise History and Philosophy Readings in Mathematics Anglin/Lambek: The Heritage of Thales Readings in Mathematics Apostol: Introduction to Analytic Number Theory Second edition Armstrong: Basic Topology Armstrong: Groups and Symmetry Axler: Linear Algebra Done Right Second edition Beardon: Limits: A New Approach to Real Analysis Bak/Newman: Complex Analysis Second edition Banchoff/Wermer: Linear Algebra Through Geometry Second edition Berberian: A First Course in Real Analysis Bix: Comics and Cubics: A Concrete Introduction to Algebraic Curves Readings in Mathematics Bremaud: An Introduction to Probabilistic Modeling Bressoud: Factorization and Primality Testing Bressoud: Second Year Calculus Readings in Mathematics Brickman: Mathematical Introduction to Linear Programming and Game Theory Browder: Mathematical Analysis: An Introduction Buskes/van Rooij: Topological Spaces: From Distance to Neighborhood Cederberg: A Course in Modem Geometries Childs: A Concrete Introduction to Higher Algebra Second edition Chung: Elementary Prob ability Theory with Stochastic Processes Third edition Cox/Little/O'Shea: Ideals, Varieties, and Algorithms Second edition Croom: Basic Concepts of Algebraic Topology Curtis: Linear Algebra: An Introductory Approach Fourth edition DevIin: The Joy of Sets: Fundamentals of Contemporary Set Theory Second edition Dixmier: General Topology Driver: Why Math? Ebbinghaus/Flum/Thomas: Mathematical Logic Second edition Edgar: Measure, Topology, and Fractal Geometry Elaydi: Introduction to Difference Equations Exner: An Accompaniment to Higher Mathematics Fine/Rosenberger: The Fundamental Theory of Algebra Fischer: Intermediate Real Analysis Flanigan/Kazdan: Ca1culus Two: Linear and Nonlinear Functions Second edition Fleming: Functions of Several Variables Second edition Foulds: Combinatorial Optimization for Undergraduates Foulds: Optimization Techniques: An Introduction FrankIin: Methods of Mathematical Economics Gordon: Discrete Probability Hairer/Wanner: Analysis by Its History Readings in Mathematics Halmos: Finite-Dimensional Vector Spaces Second edition Halmos: Naive Set Theory Hämmerlin/Hoffmann: Numerical Mathematics Readings in Mathematics Hijab: Introduction to Calculus and Classical Analysis Hilton/Holton/Pedersen: Mathematical Reflections: In a Room with Many Mirrors Iooss/Joseph: Elementary Stability and Bifurcation Theory Second edition Isaac: The Pleasures of Probability Readings in Mathematics (continued after index) LarrySmith Linear Algebra Third Edition With 23 Illustrations , Springer LarrySmith Mathematisches Institut Universität Göttingen Bunsenstrasse 3-5 Gottingen, D-37073 Germany Editorial Board S.Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA FW Gehring Mathematics Department EastHall University ofMichigan AnnArbor, MI 48109 USA K.A Ribet Department of Mathematics Universi ty of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (1991): 15-01 LibraryofCongress Cataloging-in-Publication Data Smith, Larry Linear Algebra / Larry Smith - 3rd ed cm - (Undergraduate texts in mathematics) p Inc1udes bibliographical references and index ISBN 978-1-4612-7238-0 ISBN 978-1-4612-1670-4 (eBook) DOI 10.1007/978-1-4612-1670-4 Algebras, Linear Title II Series QA184.S63 1998 512 5-dc21 98-16278 3rd Printed on acid-free paper © 1978, 1984, 1998 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Ine in 1998 Softcover reprint ofthe hardcover 3rd edition 1998 All rights reserved This work may not be translated or copied in whole or in part wi thout thewrittenpermissionofthepublisher Springer Scienee+Business Media, LLC, except for brief excerpts in connection with reviews or scholarlyanalysis Use in connection with any form of information storage and retrieval, electranic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former are notespeciallyidentified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Anthony Guardiola; manufacturing supervised by JacquiAshri Typeset by the author using f S TEX 987654321 ISBN 978-1-4612-7238-0 Fri,\ ·ch Wilhelm/.B~~sel (1784 1846) (KOnigsberg, Prussia) ;~t; Lewis C~Jlj(l832-;}898) (Oxford, England) 1\~stin Cauc9~{(if789 1857)(Paris, France) Artliu.fCayley ('t~21,:r1895) (Cambridge, England) Ga~~~l C~~erh70tt1752)(Switzerland) Rent~~~cifrtes (1596l'~650) (Paris, France) Euclid of Alexaftdria (365 B~Qt:lBC) (Alexandria, Asia Minor) Joseph Fouri~J;;~:lf6s.: 1830)(Paris, France) Abraha4j,~ " (16§,'l,7""1744) (London, England) Jcergen Pede~ ram (185Q+l91 ,Copenhagen, Denmark) William Rowan Hamilton.~~~~5) (Dublin, Ireland) Charles Hermi 22-1901} (Paris, France) Camille JQJ!~' 838-1922) (Paris, France) Joseph Louis',~grange(1~~.~J,~~~) (Turin, Italy) Adrien Marie Legenare (17&~1~~~) (mUlo se, Paris, France, Berlit1\ GeJ:itany) Marc-Antoine des Chenes Par~e'V81 (1755-1 ) (Paris, France) ,Hungary) Friedrich Riesz (1880-1956) (.0 1.0 Rodrigues (born at the doh-century) (Paris, France) P.F Sarms (born a d of thQ.,"leth -century) (Perpignan, trasbourg, Ftahce Erhard Schmidt (1876-1959) (Ber '~y) Hermann Amandus Schwarz (184 (Halle, G6ttingen, Berlin, Ge y) James Joseph Sylvester (1814 1J)7) (UniveI:§jtyl){Vrrginia, Charlottesville, VA, Johns HopkinaiUnivers~~8altimo MD) Preface This text was originally written for a one semester course in linear algebra at the (U.S.) sophomore undergraduate level, preferably directly following a one variable calculus course, so that linear algebra could be used in a course on multidimensional calculus and/or differential equations Students at this level generally have had little contact with complex numbers or abstract mathematics, so the book deals almost exclusively with real finite-dimensional vector spaces, but in a setting and formulation that permits easy generalization to abstract vector spaces The parallel complex theory is developed in part in the exercises The goal of the first two editions was the principal axis theorem for real symmetric linear transformations Twenty years of teaching in Germany, where linear algebra is a one year course taken in the first year of study at the university, has modified that goal The principal axis theorem becomes the first of two goals, and to be achieved as originally planned in one semester, a more or less direct path is followed to its proof As a consequence there are many subjects that are not developed, and this is intentional: this is only an introduction to linear algebra As compensation, a wide selection ofexamples of vector spaces and linear transformations is presented, to serve as a testing ground for the theory Students with a need to learn more linear algebra can so in a course in abstract algebra, which is the appropriate setting Through this book they will be taken on an excursion to the algebraic/analytic zoo, and introduced to some of the animals for the first time Further excursions can teach them more about the curious habits of some of these remarkable creatures In the second edition of the book I added, among other things, a safari into the wilderness of canonical forms, where the hardy student could vii viii Preface pursue the Jordan form, which has become the second goal ofthis book, with the tools developed in the preceding chapters In this edition I have added the tip of the iceberg of invariant theory to show that linear algebra alone is not capable of solving these canonical forms problems, even in the simplest case of x complex matrices Gottingen, Germany, February 1998 Larry Smitfi Contents vii Preface Vectors in the Plane and in Space 1.1 First Steps 1.2 Exercises 1 12 Vector Spaces 15 2.1 Axioms for Vector Spaces 2.2 Cartesian (or Euclidean) Spaces 2.3 Some Rules for Vector Algebra 2.4 Exercises 15 18 21 22 Examples of Vector Spaces 25 Subspaces 35 Linear Independence and Dependence 47 3.1 Three Basic Examples 3.2 Further Examples of Vector Spaces 3.3 Exercises 4.1 Basic Properties of Vector Subspaces 4.2 Examples of Subspaces 4.3 Exercises 25 27 30 35 41 42 5.1 Basic Definitions and Examples 47 5.2 Properties of Independent and Dependent Sets 50 5.3 Exercises 53 ix B.1 The Complex Numbers 439 We can next prove the Fundamental Theorem of Algebra, which tells us that polynomials with complex coefficients always have complex roots, justifying our use of complex numbers in the development of the Jordan normal form As we will see, the proof uses essentially nonalgebraic ideas: in this case a topological idea THEOREM B.I.S (Fundamental Theorem of Algebra): Every polynomial ofpositive degree with complex coefficients has a complex root PROOF: Let p(z) =amz m + + alZ + ao be a polynomial with complex coefficients of degree m, i.e., am ~ O By dividing by am we obtain a new polynomial q(z) = zm + Cm_1Z m - + + CIZ + co, also of degree m, with the same roots as p(z) (if there are any at all!) and leading coefficient Let Cr C ce =m,2 be the circle centered at the origin of radius r ~ O Then the image of Cr under the polynomial map q : ce - > ce is a smooth curve in the plane Let us assume that q(z) has no roots Then none of the curves q(Cr), r Em, pass through the origin Therefore, we can consider the number of times CJ(r), the curve q(Cr ), winds around the origin y x Figure 19.1.3 This can be defined as follows: let the points of Cr be parameterized by their argument ~ arg(z) < 27T Starting at B = we draw the vector from the origin to the point q(r(cos(B) + sin(B» on the curve q(Cr), and record the angle between this vector and the positive x-axis This is just the argument of the point on the curve regarded as a complex number As B varies from to 27T this angle varies, and the number of As all proofs I know 440 B Complex Numbers times it passes through 27T (to become again) is the winding number c >(r) In Figure B.1.3 the curve winds times around the origin Clearly, as r varies, the winding number of q(Cr) remains constant, so long as the curves not pass through the origin, as is our case Again, so long as the curves not pass through the origin, the winding number c >r depends continuously on r This says that for our family of curves q(Cr), the winding number is a constant For r =0 the curve q(Cl) is just the point q(O) = Co, and hence the winding number is O On the other hand, we may write q(z) in the form q(z) = zm (1 t C;t) , + k=1 and so by de Moivre's representation and the preceding lemma arg(q(z» =m arg(z) + arg (1 t C;~k) + k=1 Hence as we move around the curve, letting e vary from to 21T, we find that the net change in the value of arg(q(z» is m(21T) plus the net change of arg (1 + 2.:;:=1 C;;;'k ) If we choose r large enough, then It C;~k I