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LINEAR ALGEBRA Second Edition KENNETH Professor of Mathematics Massachusetts Institute HOFFMAN of Technology RAY KUNZE Professor of Mathematics University of California, Irvine PRENTICE-HALL, INC., Englewood Cliffs, New Jersey

@ 1971, 1961 by Prentice-Hall, Englewood Inc. Cliffs, New Jersey All rights reproduced permission in any in writing reserved. No part of this book may be form or by any means without from the publisher. PRENTICE-HALL~NTERNATXONAL,INC., PRENTICE-HALLOFAUSTRALIA,PTY. PRENTICE-HALL PRENTICE-HALLOFINDIA PRENTICE-HALL OF CANADA, OFJAPAN,INC., London LTD., Sydney LTD., PRIVATE Toronto LIMITED, New Delhi Tokyo Current printing (last digit) : 10 9 8 7 6 Library Printed of Congress Catalog Card No. 75142120 in the United States of America

Pf re ace Our original purpose in writing from ranged linear algebra graduate course was designed fourths of the students were drawn and M.I.T. the the country basic material sity audience first edition for the text have seen and have afforded to a variety freshmen and course at the Massachusetts at for mathematics this book was to provide Institute junior and the from other scientific majors for a text of Technology. level, although the under- This three- disciplines through remains the proliferation graduate generally students. accurate technological This description today. The the ten years since of of linear algebra courses throughout one of the authors the opportunity of groups at Brandeis University, to Washington teach the Univer- (St. Louis), Our principal the University in revising of California (Irvine). Linear Algebra has been to increase the variety aim especially the way, allowing to exercise a considerable of courses which can easily be taught chapters, the more difficult ping points along course other hand, we have used reference book The major comprehensive for mathematicians. changes have been In Chapter the for a rather increased spaces. theory of canonical the amount the amount one-year from ones, so that it. On one hand, we have structured there are several natural the stop- instructor in a one-quarter or one-semester of choice of material course in in the subject matter. On in the linear the it can be and even as a text, so that algebra in our treatments of canonical forms and 6 we no longer begin with the general forms. We first handle spatial characteristic then build triangulation and diagonalization theorems and the general theory. We have split Chapter 8 so that the basic material spaces and unitary diagonalization is followed by a Chapter forms and operators the more sophisticated on real inner product properties spaces. of normal product which underlies in relation to up inner product treats tors, to sesqui-linear including We have also made a number normal first edition. But the basic philosophy We have made no particular course should students may not be primarily matics of hodgepodge basic mathematical techniques, concepts. of small behind concession changes and the to fact text the interested in mathematics. improvements from the is unchanged. the majority that of the For we believe a mathe- not give science, engineering, or social science students but should provide them with an understanding inner theory values our way on 9 which opera- a of . . . am

Preface On grounds which students For this the very beginning presents it most profitable to read students such basic Throughout theorem, important importance tion, concepts. hundred), best students. the other hand, we have been keenly the students may possess and, of the experience with abstract mathematical aware of the wide in particular, range of back- the fact that reasoning. have had very reason, we have avoided little the introduction of too many abstract In addition, we have included an Appendix relation. We have independently, but to advise ideas at which found the of the book. ideas as set, function, not to dwell on the Appendix when the book we have and equivalence ideas ideas arise. a great these these included concepts which occur. The study of such examples and the number to minimize tends proof in logical order without The book also contains ranging routine from These exercises are applications intended 1 deals with systems of linear equations row operations this material. on matrices. It provides of the more abstract ideas occurring the computational variety of examples is of fundamental of the of students who can repeat defini- of the abstract the meaning grasping to ones which will extend to be an important and their solution It has been our practice the student with technique the later in some picture necessary (about part of the six the very text. by means to spend about of the to under- chapters. Chap- Chapter 3 treats a wide variety of graded exercises spaces, subspaces, bases, and dimension. their algebra, representation their and dual spaces. Chapter ideals in that algebra, roots, Taylor’s and formula, and formula. 5 develops being viewed as an alternating Chapter determinants of square matrices, n-linear function of the on modules to multilinear functions the concept of determinant in elementary a discussion of the concepts which are basic on modules is usually setting places found than on a finite-dimensional vector space; on the (eigen) functionals, over a field, transformation then proceeds It also deals with transformations, linear Chapter of elementary six lectures origins of linear algebra and with stand examples ter 2 deals with vector linear isomorphism, polynomials a polynomial. polation minant and The material comprehensive and 7 contain linear teristic cepts of the diagonalizable and the rational theorems admissible domain, and cussion of semi-simple Chapter the basic geometry, to a vector’ onto a subspace and unitary operators. self-adjoint of normal operators forms, emphasizing well as groups preserving pseudo-orthogonal Chapter operators operators on real or complex canonical play a central subspaces. Chapter operators, finite-dimensional and culminates 9 introduces relating to the development the computation on an and and Lorentz and Jordan operators 8 treats values, leading role, then and triangulable by matrices, 4 defines as well as the algebra of of factorization the prime the Lagrange inter- the deter- rows of a matrix, ring. in a wider and more 6 Chapters textbooks. as well as the Grassman to the analysis of a single the analysis of charac- the con- transformations; transformation, and cyclic decomposition at the study through of of matrices over a polynomial divisors of a matrix, parts of a more general and diagonalizable and nilpotent canonical forms. The primary being arrived latter the 7 includes a discussion of invariant factors and elementary of the Smith canonical form. The chapter ends with a dis- to round out inner product the analysis of a single operator. spaces the to in some detail. It covers idea of ‘best approximation of the orthogonal projection orthogonalization the concepts the orthogonal complement of a subspace. The chapter of a vector treats and normal and theory normal 10 discusses bilinear the spectral concerning to positive in the diagonalization sesqui-linear forms, of self-adjoint relates them inner product to more space, moves on to results sophisticated inner product forms for symmetric spaces. Chapter and skew-symmetric forms, especially the orthogonal, forms, as unitary, non-degenerate groups. We feel that any course which uses this text should cover Chapters 1, 2, and 3

Preface V the thoroughly, and determinants, polynomial sketchily without inclination because mentary ter 6, together with be included, possibly excluding Sections 3.6 and 3.7 which deal with transpose of a linear transformation. Chapters may be treated with varying degrees of ideals and basic properties is to deal with serious damage to the these chapters so well illustrates the material course may now be concluded (the new) Chapter of determinants logic flow of the (except carefully ideas of linear the basic nicely with the the rational 8. If first 4 and 5, on polynomials thoroughness. In may be covered the double dual and fact, quite text; however, our in the the results on modules), algebra. An ele- four sections of Chap- forms are to and Jordan a more extensive remains indebtedness coverage of Chapter to those who contributed 6 is necessary. to the first edition, espe- Our cially to Mrs. In addition, ceptive patience tion. Lastly, and her to Professors Harry Furstenberg, Judith Bowers, Mrs. Betty Ann we would comments led in dealing with special thanks tireless efforts thank to this like to two authors are due revision, (Sargent) Rose and Miss Phyllis Louis Howard, Daniel Kan, Edward Thorp, Ruby. and colleagues whose per- their students the staff of Prentice-Hall throes of academic administra- the many and for caught to Mrs. Sophia Koulouras the for both her skill in in typing the revised manuscript. K. M. H. / R. A. K.

Contents Chapter 1. Linear Equations Fields Systems of Linear Equations 1.1. 1.2. 1.3. Matrices 1.4. Row-Reduced 1.5. Matrix Multiplication 1.6. Matrices Invertible Echelon Matrices and Elementary Row Operations Chapter 2. Vector Spaces Vector Spaces Subspaces Bases and Dimension 2.1. 2.2. 2.3. 2.4. Coordinates 2.5. 2.6. Computations Summary of Row-Equivalence Concerning Subspaces Chapter 3. Linear Transformations Linear Transformations The Algebra Isomorphism 3.1. 3.2. 3.3. 3.4. Representation 3.5. 3.6. 3.7. Linear Functionals The Double Dual The Transpose of a Linear Transformation of Linear Transformations of Transformations by Matrices Vi 1 1 3 6 11 16 21 28 28 34 40 49 55 58 67 67 74 84 86 97 107 111

Chapter 4. Polynomials 4.1. 4.2. 4.3. 4.4. 4.5. Algebras The Algebra Lagrange Polynomial The Prime of Polynomials Interpolation Ideals Factorization of a Polynomial Rings Functions and the Uniqueness of Determinants Properties of Determinants Chapter 5. Determinants 5.1. Commutative 5.2. Determinant Permutations 5.3. 5.4. Additional 5.5. Modules 5.6. Multilinear 5.7. Functions The Grassman Ring Chapter 6. Elementary Canonical Forms 6.1. 6.2. 6.3. 6.4. 6.5. Introduction Characteristic Annihilating Invariant Simultaneous Diagonalization 6.6. Direct-Sum 6.7. 6.8. Invariant The Primary Values Polynomials Subspaces Triangulation; Simultaneous Decompositions Direct Sums Decomposition Theorem Chapter 7. The Rational and Jordan Forms 7.1. Cyclic Subspaces and Annihilators 7.2. Cyclic Decompositions 7.3. 7.4. Computation 7.5. The Jordan Form Semi-Simple of Invariant Factors Operators Summary; and the Rational Form Chapter 8. Inner Product Spaces 8.1. 8.2. 8.3. 8.4. Unitary 8.5. Normal Inner Products Inner Product Linear Functionals Operators Operators Spaces and Adjoints Contents vii 117 117 119 124 127 134 140 140 141 150 156 164 166 173 181 181 182 190 198 206 209 213 219 227 227 231 244 251 262 270 270 277 290 299 311

. . . 0222 Contents Chapter 9. Operators on Inner Product Spaces Introduction Forms on Inner Product Positive 9.1. 9.2. 9.3. Forms 9.4. More on Forms 9.5. Theory Properties 9.6. Spectral Further Spaces Chapter 10. Bilinear Forms of Normal Operators Forms Bilinear Forms Bilinear Forms Forms Bilinear Symmetric Skew-Symmetric Groups Preserving Bilinear Sets Functions Equivalence Quotient Equivalence The Axiom Relations Spaces Relations of Choice in Linear Algebra 10. I. 10.2. 10.3. 10.4 A.1. A.2. A.3. A.4. A.5. A.6. Appendix Bibliography Index 319 319 320 325 332 335 349 359 359 367 375 379 386 387 388 391 394 397 399 400 401

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