[Larry_Smith__(auth.)]_Linear_Algebra(BookZZ.org).pdf

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Unformatted text preview: 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 1. Algebras, Linear. 1. 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 do 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 2 x 2 complex matrices. Gottingen, Germany, February 1998 Larry Smitfi Contents Preface vii 1. Vectors in the Plane and in Space 1.1 First Steps 1.2 Exercises 1 1 12 2. 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 3. Examples of Vector Spaces 25 4. Subspaces 35 5. 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 x Contents 6. Finite-Dimensional Vector Spaces and Bases 57 6.1 Finite-Dimensional Vector Spaces..................... 6.2 Properties of Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.3 Using Bases........................................... 6.4 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. The Elements of Vector Spaces: A Summing Up 57 61 65 71 75 7.1 Numerical Examples.................................. 75 7.2 Exercises 82 8. Linear Transformations 8.1 Definition of Linear Transformations 8.2 Examples of Linear Transformations. . . . . . . . . . . . . . . . .. 8.3 Properties of Linear Transformations 8.4 Images and Kernels of Linear Transformations 8.5 Some Fundamental Constructions 8.6 Isomorphism of Vector Spaces 8.7 Exercises 9. Linear Transformations: Examples and Applications 9.1 Numerical Examples 9.2 Some Applications 9.3 Exercises 85 85 89 91 94 98 102 109 113 113 123 124 10. Linear Transformations and Matrices 129 11. Representing Linear Transformations by Matrices 159 12. More on Representing Linear Transformations by Matrices 185 10.1 Linear Transformations and Matrices in m.3 10.2 Some Numerical Examples 10.3 Matrices and Their Algebra 10.4 Special Types of Matrices 10.5 Exercises 129 134 136 141 151 11.1 Representing a Linear Transformation by a Matrix .. 159 11.2 Basic Theorems 165 174 11.3 Change of Bases 11.4 Exercises 178 12.1 Projections 185 Contents xi 191 12.2 Nilpotent Transfonnations 193 12.3 Cyclic Transfonnations 12.4 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 195 13. Systems of Linear Equations 199 14. The Elements of Eigenvalue and Eigenvector Theory 227 15. Inner Product Spaces 267 16. The Spectral Theorem and Quadratic Forms 307 17. Jordan Canonical Form 343 13.1 Existence Theorems 13.2 Reduction to Echelon Fonn 13.3 The Simplex Method 13.4 Exercises 14.1 The Rank of an Endomorphism 14.2 Eigenvalues and Eigenvectors 14.3 Detenninants 14.4 The Characteristic Polynomial 14.5 Diagonalization Theorems 14.6 Exercises 15.1 Scalar Products 15.2 Inner Product Spaces 15.3 Isometries 15.4 The Riesz Representation Theorem 15.5 Legendre Polynomials 15.6 Exercises 16.1 Self-Adjoint Transfonnations 16.2 The Spectral Theorem 16.3 The Principal Axis Theorem for Quadratic Fonns 16.4 A Proof of the Spectral Theorem in the General Case 16.5 Exercises 17.1 Invariant Subspaces 17.2 Nilpotent Transfonnations 17.3 The Jordan Nonnal Fonn 17.4 Square Roots 17.5 The Hamilton-Cayley Theorem 17.6 Inverses 17.7 Exercises 201 209 216 224 227 230 238 245 253 260 268 274 288 291 298 301 308 316 324 335 338 345 350 357 372 374 376 377 xii Contents 18. Application to Differential Equations 381 19. The Similarity Problem 405 18.1 Linear Differential Systems: Basic Definitions 18.2 Diagonalizable Systems 18.3 Application of Jordan Form 18.4 Exercises 19.1 The Fundamental Problem of Linear Algebra 19.2 A Bit of Invariant Theory 19.3 Exercises 381 386 395 402 405 406 409 A. Multilinear Algebra and Determinants 411 B. Complex Numbers 433 A.1 Multilinear Forms 411 415 A.2 Determinants A.3 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 428 B.1 The Complex Numbers B.2 Exercises Font Usage Notations Index 433 441 443 445 447 Chapter 1 Vectors in the Plane and in Space In physics certain quantities such as force, displacement, velocity, and acceleration possess both a magnitude and a direction and they are most usually represented geometrically by drawing an arrow with the magnitude and direction of the quantity in question. Physicists refer to the arrow as a vector, and call the quantity so represented a vector quantity. In the study of the calculus the student has no doubt also encountered what are called vectors, particularly in connection with the study of lines and planes and the differential geometry of space curves. We begin by reviewing these ideas and codifying the algebra of vectors. 1.1 First Steps Vectors as they appear in physics and the study of curves and surfaces can be described as ordered pairs of points (P, Q) which we call the vector from P to Q and often denote by PQ. This is substantially the same as the physics definition, since all it amounts to is a technical description of the word "arrow". P is called the initial point and Q the terminal point. For our purposes it will be convenient to regard two vectors as being equal if they have the same magnitude and the same direction. In other words, we will regard PQ and RS as determining the same vector ifRS results by moving PQ parallel to itself. N.B. Vectors that conform to this definition are called free vectors, since we are "free to pick" their initial point. Not all "vectors" that occur in nature conform to this convention. If the vector quantity depends not 1 L. Smith, Linear Algebra © Springer Science+Business Media New York 1998 2 1. Vectors in the Plane and in Space only on its direction and magnitude but as well on its initial point it is called a bound vector. For Example, torque is a bound vector. In the force vector diagram represented by Figure 1.1.1 PQ does not have the same effect as RS in pivoting a bar. In this book we will consider only free vectors. p R Q e 1-1 ----' s Figure 1.1.1 With this convention of equality of vectors in mind it is clear that if we fix a point 0 in space called the origin, then we may regard all our vectors as having their initial point at O. The vector OP will very ---> often be abbreviated to P if the point 0 which serves as the origin of ---> all vectors is clear from context. The vector P is called the position vector of the point P relative to the origin O. In physics vector quantities such as force vectors are often added together to obtain a resultant force vector. This process may be described as follows. Suppose an origin 0 has been fixed. Given vectors P and ij their sum is defined by the Figure 1.1.2. That is, draw the parallelogram determined by the three points P, 0 and Q. Let R be the fourth ---> ---> ---> vertex and set P + Q = R. P o b?7 Q R Q Figure 1.1.2 The following basic rules of vector algebra may be easily verified by elementary Euclidean geometry. ~ ~ -7 ~ (1) P+Q=Q+P. ~-7 -7-7-7-7 (2) (P + Q) + R = P + (Q + R). (3) P+ 0 =P = 0 + P. It is also possible to define the operation of multiplying a vector by a number. Suppose we are given a vector P and a number a. If a> 0, we 1.1 First Steps ~ 3 ~ let aP be the vector with the same direction as P only a times as long (see Figure 1.1.3). 1.5P o Figure 1.1.3 ~ If a < 0, we set aP equal to the vector of magnitude a times the magnitude ofP but having direction opposite to that ofP (see Figure 1.1.4). ~ ~ Figure 1.1.4 If a =0 we set aP equal to O. It is then easy to show that vector algebra satisfies the following additional rules. (4) P + (-lP) =O. ~ ~ ~ ~ ~ = aP + aQ. (a + b)P = aP+ bP. ~ ~ ~ (5) a(P + Q) (6) ~ ~ ~ ~ ~ ~ (7) (a b)P = a(bP). ---7 (8) OP = 0, 1P =P. ---7 ---7 ---7 Note that Rule 6 involves two types of addition, namely addition of numbers and addition of vectors. Vectors are particularly useful in studying lines and planes in space. Suppose that an origin 0 has been fixed and L is the line through the two points P and Q as in Figure 1.1.5. Suppose that R is any other point on L. Consider the position vector R. Since the two points P, Q completely determine the line L, it is quite reasonable to look for some ~ 4 1. Vectors in the Plane and in Space -7 -7 -7 relation among the vectors P, Q, and R. L o.,.:::... ~ Figure 1.1.5 One such relation is provided by Figure 1.1.6. L o Figure 1.1.6 Observe that -7 -7 -7 S+P=Q. -7 -7 Therefore, if we write -P for (-I)P, we see that -7 -7 -7 S=Q-P. 1.1 First Steps 5 Notice that there is a number t such that -> -> T= tS. Moreover, -> -> -> R=P+T, and hence we find ~ R =P+ t(Q-P). -7 -7---7 Equation (*) is called the vector equation of the line L. To make practical computations with this equation it is convenient to introduce in addition to the origin 0 a Cartesian coordinate system as in Figure 1.1.7. Every point P then has coordinates (x, Y, z), and if we have two points P and Q with coordinates (xp, yp, zp) and (xQ, YQ, zQ) then it is -> ---'> quite easy to check that the vector P + Q is the position vector of the point with components (xp + xQ> yp + YQ, Zp + zQ). z y x Figure 1.1.7 ---'> Likewise, for a number a the vector aP is the position vector of the point with coordinates (axp, ayp, azp). Thus we find by considering the coordinates of the points represented equation (*) that (x, Y, z) lies on the line L through P, Q if and only if there is a number t such that x = Xp + t(xQ -xp), Y = yp+t(YQ-Yp), Z = Zp + t(zQ - zp). EXAMPLE 1: Does the point (1, 2, 3) lie on the line passing through the points (4, 4, 4) and (1, 0, I)? 6 1. Vectors in the Plane and in Space SOLUTION: Let L be the line through P = (4, 4, 4) and Q = (1, 0, 1). Then the points of L must satisfy the equations x = 4+ t(1-4) = 4-3t, =4+ t(0-4) = 4-4t, z =4+ t(1-4) =4-3t, y where t is a number. Let us check whether this is possible: namely, does there exist a number t such that 1 =4-3t, 2 = 4-4t, 3=4-3t. The first equation gives -3 = -3t t = 1. Putting this in the last equation gives 3 = 4-3 = 1, which is impossible. Therefore (1,2,3) does not lie on the line through (4,4,4) and (1, 0, 1). EXAMPLE 2: Let L 1 be the line through the points (1, 0, 1) and (1,1, 1). Let L 2 be the line through the points (0,1,0) and (1,2,1). Determine whether the lines L 1 and L 2 intersect. If so find their point of intersection. SOLUTION: The equations of L 1 are x = 1 + t1(1- 1) = 1, y = 0 + t1(1- 0) = t1, z = 1 + t1(1- 1) = 1. The equations of L 2 are =0 + (1- 0)t2 = t2, y = 1 + (2 - 1)t2 = 1 + t2, z =0 + (1- 0)t2 =t2· x Ifa point lies on both of these lines we must have 1 = t2, =1 + t2, 1 = t2. t1 1.1 First Steps 7 Therefore t2 = 1 and tl = 2. Hence (1,2,1) is the only point these lines have in common. EXAMPLE 3: Determine whether the lines L 1 and L 2 with equations X L1 { y z = 1- 3t, = 1 + 3t, = t, =-2-3t, =4+3t, z = 1 + t, X L2 { y have a point in common. SOLUTION: If a point (x, y, z) lies on both lines, it must satisfy both sets of equations, so there is a number tl such that = 1- 3tt, Y = 1 + 3tt, x and a number t2 with x = -2-3t2, Y =4+ 3t2, Z = 1 + t2, and the answer to the problem is reduced to determining whether in fact two such numbers can be found, that is if the simultaneous equations =-2 - 3t2, 1 + 3tl =4 + 3t2, tl = 1 + t2, 1 - 3tl have any solutions. Writing these equations in the more usual form they become 3 =3tl-3t2, =-3tl + 3t2, -1 =-tl + t2. -3 By dividing the first equation by 3, the second by -3, and multiplying the third by -1 we get 1 = tl - t2, 1 = tl - t2, 1 = tl - t2, 8 1. Vectors in the Plane and in Space giving as the only requirement on tl and t2 that = 1 + t2...
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