Sheldon Axler Linear Algebra Done Right - Sharif

1 undergraduate texts in MathematicsLinear Algebra Done RightSheldon AxlerThird EditionUndergraduate texts in MathematicsUndergraduate texts in MathematicsSeries Editors: Sheldon AxlerSan Francisco State University, San Francisco, CA, USAK enneth RibetUniversity of California, Berkeley, CA, USAA dvisory Board:Colin Adams,Williams College, Williamstown, MA, USAA lejandro Adem,University of British Columbia, Vancouver, BC, CanadaRuth Charney,Brandeis University, Waltham, MA, USAI rene M. Gamba,The University of Texas at Austin, Austin, TX, USAR oger E. Howe,Ya l e U n i v e rs i t y, N e w H a v e n , C T, U S ADavid Jerison,Massachusetts Institute of Technology, Cambridge, MA, USAJ effrey C.

2 Lagarias,University of Michigan, Ann Arbor, MI, USAJill Pipher,Brown University, Providence, RI, USAF adil Santosa,University of Minnesota, Minneapolis, MN, USAAmie Wilkinson,University of Chicago, Chicago, IL, USAU ndergraduate texts in Mathematicsare generally aimed at third- and fourth-year undergraduate mathematics students at North American universities. Thesetexts strive to provide students and teachers with new perspectives and novelapproaches. The books include motivation that guides the reader to an appreciationof interrelations among different aspects of the subject.

3 They feature examples thatillustrate key concepts as well as exercises that strengthen further volumes: Sheldon AxlerLinear Algebra Done RightThird edition123 ISSN 0172-6056 ISSN 2197-5604 (electronic)ISBN 978-3-319-11079-0 ISBN 978-3-319-11080-6 (eBook)DOI Cham Heidelberg New York Dordrecht LondonLibrary of Congress Control Number: 2014954079 mathematics Subject Classification (2010): 15-01, 15A03, 15A04, 15A15, 15A18, 15A21c Springer International Publishing 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights oftranslation,reprinting,reuseofillustra tions,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.

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5 Violations are liable to prosecution under the respective Copyright use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors northe publisher can accept any legal responsibility forany errors or omissions that may be made.

6 The publisher makes no warranty, express or implied, withrespect to the material contained by the a uthor in L aTeXPrinted on acid-free paperSpringer is part of SpringerScience+BusinessMedia( ) Sheldon AxlerSanFranciscoStateUniversityDepartme nt of MathematicsSan Francisco, CA, USAC over figure:ForastatementofApolloniusexercise in Section s Identity and its connection to Linear Algebra , see the lastContentsPreface for the InstructorxiPreface for the StudentxvAcknowledgmentsxvii1 Vector Numbers2 Lists5Fn6 Digression on Fields10 Exercises Definition of Vector Space12 Exercises Subspaces18 Sums of Subspaces20 Direct Sums21 Exercises Vector Span and Linear Independence28 Linear Combinations and Span28 Linear Independence32 Exercises Bases39 Exercises Dimension44 Exercises The Vector Space of Linear Maps52 Definition and Examples of Linear Maps52 Algebraic Operations.

7 W /55 Exercises Null Spaces and Ranges59 Null Space and Injectivity59 Range and Surjectivity61 Fundamental Theorem of Linear Maps63 Exercises Matrices70 Representing a Linear Map by a Matrix70 Addition and Scalar Multiplication of Matrices72 Matrix Multiplication74 Exercises Invertibility and Isomorphic Vector Spaces80 Invertible Linear Maps80 Isomorphic Vector Spaces82 Linear Maps Thought of as Matrix Multiplication84 Operators86 Exercises Products and Quotients of Vector Spaces91 Products of Vector Spaces91 Products and Direct Sums93 Quotients of Vector Spaces94 Exercises Duality101 The Dual Space and the Dual Map101 The Null Space and Range of the Dual of a Linear Map104 The Matrix of the Dual of a Linear Map109 The Rank of a Matrix111 Exercises Conjugate and Absolute Value118 Uniqueness of Coefficients for Polynomials120 The Division Algorithm for Polynomials121 Zeros of Polynomials122 Factorization of Polynomials overC 123 Factorization of Polynomials overR 126 Exercises 41295 Eigenvalues, Eigenvectors.

8 And Invariant Invariant Subspaces132 Eigenvalues and Eigenvectors133 Restriction and Quotient Operators137 Exercises Eigenvectors and Upper-Triangular Matrices143 Polynomials Applied to Operators143 Existence of Eigenvalues145 Upper-Triangular Matrices146 Exercises Eigenspaces and Diagonal Matrices155 Exercises Product Inner Products and Norms164 Inner Products164 Norms168 Exercises Orthonormal Bases180 Linear Functionals on Inner Product Spaces187 Exercises Orthogonal Complements and Minimization Problems193 Orthogonal Complements193 Minimization Problems198 Exercises on Inner Product Self-Adjoint and Normal Operators204 Adjoints204 Self-Adjoint Operators209 Normal Operators212 Exercises The Spectral Theorem217 The Complex Spectral Theorem217 The Real Spectral Theorem219 Exercises Positive Operators and Isometries225 Positive Operators225 Isometries228 Exercises Decomposition and Singular Value Decomposition233 Polar Decomposition233 Singular Value Decomposition236 Exercises on Complex Vector Generalized Eigenvectors and Nilpotent Operators242 Null Spaces of Powers of an Operator242 Generalized Eigenvectors244

9 Nilpotent Operators248 Exercises Decomposition of an Operator252 Description of Operators on Complex Vector Spaces252 Multiplicity of an Eigenvalue254 Block Diagonal Matrices255 Square Roots258 Exercises Characteristic and Minimal Polynomials261 The Cayley Hamilton Theorem261 The Minimal Polynomial262 Exercises Jordan Form270 Exercises on Real Vector Complexification276 Complexification of a Vector Space276 Complexification of an Operator277 The Minimal Polynomial of the Complexification279 Eigenvalues of the Complexification280 Characteristic Polynomial of the Complexification283 Exercises Operators on Real Inner Product Spaces287 Normal Operators on Real Inner Product Spaces287 Isometries on Real Inner Product Spaces292 Exercises and Trace296 Change of Basis296 Trace: A Connection Between Operators and Matrices299 Exercises Determinant307 Determinant of an Operator307 Determinant of a Matrix309 The Sign of the Determinant320 Volume323 Exercises Credits333 Symbol Index335 Index337 Preface for the InstructorYou are about to teach a course that will probably give students their secondexposure to Linear Algebra .

10 During their first brush with the subject, yourstudents probably worked with Euclidean spaces and matrices. In contrast,this course will emphasize abstract vector spaces and Linear audacious title of this book deserves an explanation. Almost alllinear Algebra books use determinants to prove that every Linear operator ona finite-dimensional complex vector space has an eigenvalue. Determinantsare difficult, nonintuitive, and often defined without motivation. To prove thetheorem about existence of eigenvalues on complex vector spaces, most booksmust define determinants, prove that a Linear map is not invertible if and onlyif its determinant equals0, and then define the characteristic polynomial.