Advanced-linear-algebra

Graduate Texts\nin Mathematics\n听雨抚火 @ 寂寞状\nGTM 系列电子书下载\nSpringer 版权所有\n仅供学习，请支持正版书籍\nhttp://realking1980.boke.com Graduate Texts\nin Mathematics\nSteven Roman\nAdvanced Linear Algebra\nThird Edition\nSpringer Graduate Texts in Mathematics\n135\nEditorial Board\nS. Axler\nK.A. Ribet Graduate Texts in Mathematics\n\n1 TAKEUT/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.\n2 OXLEY. Measure and Category. 2nd ed.\n3 SCHLAIFER. Topological Vector Spaces. 2nd ed.\n4 HILTON/STAMMBACH. A Course in Homological Algebra. 2nd ed.\n5 MAC LANE. Categories for the Working Mathematician. 2nd ed.\n6 HUGHES/PIPER. Projective Planes.\n7 J. P. SIRET. A Course in Arithmetic.\n8 TAKEUT/ZARING. Axiomatic Set Theory.\n9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory.\n10 COHEN. A Course in Simple Homotopy Theory.\n11 CROWELL/FOX. Introduction to Knot Theory.\n12 DUNFORD/SHWARTZ. Linear Operators. Vol. I.\n13 BEALS. Advanced Mathematical Analysis.\n14 ANDERSON/FILER. Rings and Categories of Modules. 2nd ed.\n15 GARDINER/GUILLAIN. Stable Mappings and Their Singularities.\n16 BEGBEDER. Lectures in Functional Analysis and Operator Theory.\n17 WINTER. The Structure of Fields.\n18 ROSENBLUM/ROSENBLUM. Fields. 2nd ed.\n19 HALMOS. Measure Theory.\n20 HAINES. A Hilbert Space Problem Book. 2nd ed.\n21 HUSEMOLLER. Fibre Bundles. 3rd ed.\n22 HUMPHREYS. Linear Algebraic Groups.\n23 BARNES/MACK. An Algebraic Introduction to Mathematical Logic.\n24 GRIER. Linear Algebra. 4th ed.\n25 HOLMES. Geometric Functional Analysis and Its Applications.\n26 HEWITT/STROMBERG. Real and Abstract Analysis.\n27 MENN. Algebraic Theories.\n28 KIRLEY. General Topology.\n29 ZARISK/SAMUEL. Commutative Algebra. Vol. II.\n30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts.\n31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra.\n32 JACOBSON. Lectures in Abstract Algebra III. The Theory of Galois Theory.\n33 HIKSCH. Differential Topology.\n34 SPITZER. Principles of Random Walk. 2nd ed.\n35 ALEXANDER/WERMKE. Several Complex Variables and Banach Algebras. 3rd ed.\n36 KIELY/MAMOIKA et al. Linear Topological Spaces.\n37 MONK. Mathematical Logic.\n38 GRAEBER/FIRTZSCH. Several Complex Variables.\n39 ARVESON. An Invitation to C*-Algebras.\n40 KEMNITZ/SPEIL/KNAP. Denumerable Markov Chains. 2nd ed.\n41 AXIOTI. Modular Functions and Dirichlet Series in Number Theory. 2nd ed.\n42 J. P. SIRET. Linear Representations of Finite Groups.\n43 GILLMAN/JERISON. Rings of Continuous Functions.\n44 KENDIG. Elementary Algebraic Geometry.\n45 LOVEL. Probability Theory I. 4th ed.\n46 LOVEL. Probability Theory II. 4th ed.\n47 MONES. Geometric Topology in Dimensions 2 and 3.\n48 SIEGEL/WI. General Relativity for Mathematicians.\n49 GUBERNICK/WEIR. Linear Geometry. 2nd ed.\n50 REYNOLDS. Fermat's Last Theorem.\n51 KLEINBERG. A Course in Differential Geometry.\n52 HAKSHTON. Algebraic Geometry.\n53 MANIN. A Course in Mathematical Logic.\n54 GIBBONS/WATKINS. Combinatorics with Emphasis on the Theory of Graphs.\n55 BRYA/PHIKV. Introduction to Operator Theory I: Elements of Functional Analysis.\n56 MASSEY. Algebraic Topology: An Introduction.\n57 CROWELL/FOX. Introduction to Knot Theory.\n58 KOBRITZ. p-adic Numbers, p-adic Analysis, and Zeta-Functions. 2nd ed.\n59 LANG. Cyclotomic Fields.\n60 ANCONI. Mathematical Methods in Classical Mechanics. 2nd ed.\n61 WITHEM. Elements of Homotopy Theory.\n62 KAC/ADOM. Fundamental of the Theory of Groups.\n63 BOLLOBAS. Graph Theory.\n64 EDWARDS. Fourier Series. Vol. I. 2nd ed.\n65 WEILS. Differential Analysis on Complex Manifolds. 3rd ed.\n66 WATERSHIRE. Introduction to Affine Group Schemes.\n67 SIRET. Local Fields.\n68 WIDEMAN. Linear Operators in Hilbert Spaces.\n69 LANG. Cyclotomic Fields.\n70 MASSEY. Singular Homology Theory.\n71 FRANK/WRAK. Riemann Surfaces. 2nd ed.\n72 SWIRBLE. Classical Topology and Combinatorial Group Theory. 2nd ed.\n73 HUNGFORD. Algebra.\n74 DUNWORTH. Multiplicative Number Theory. 3rd ed.\n75 HOCKSHILD. Basic Theory of Algebraic Groups and Lie Algebras. (continued after index) Steven Roman\n\nAdvanced Linear Algebra\nThird Edition\n\nSpringer Steven Roman\n\n8 Night Star\nIrvine, CA 92603\nUSA\nsroman@romanpress.com\n\nEditorial Board\nS. Axler\nMathematics Department\nSan Francisco State University\nSan Francisco, CA 94132\nUSA\naxler@sfsu.edu\n\nK.A. Ribet\nMathematics Department\nUniversity of California at Berkeley\nBerkeley, CA 94720-3840\nUSA\nribet@math.berkeley.edu\n\nISBN-13: 978-0-387-72828-5 e-ISBN-13: 978-0-387-72831-5\nLibrary of Congress Control Number: 2007934001\nMathematics Subject Classification (2000): 15-01\n© 2008 Springer Science+Business Media, LLC\nAll rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.\nThe use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.\nPrinted on acid-free paper.\n\n9 8 7 6 5 4 3 2 1\n\nspringer.com To Donna\nand to\nRashelle, Carol and Dan Preface to the Third Edition\n\nLet me begin by thanking the readers of the second edition for their many helpful comments and suggestions, with special thanks to Joe Kidd and Nam Trang. For the third edition, I have corrected all known errors, polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products) and upgraded some proofs that were originally done only for finite-dimensional/rank cases. I have also moved some of the material on projection operators to an earlier position in the text.\n\nA few new theorems have been added in this edition, including the spectral mapping theorem and a theorem to the effect that dim(V) ≤ dim(V*), with equality if and only if V is finite-dimensional.\n\nI have also added a new chapter on associative algebras that includes the well-known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem). The reference section has been enlarged considerably, with over a hundred references to books on linear algebra.\n\nSteven Roman\nIrvine, California, May 2007 Preface to the Second Edition\n\nLet me begin by thanking the readers of the first edition for their many helpful comments and suggestions. The second edition represents a major change from the first edition. Indeed, one might say that it is a totally new book, with the exception of the general range of topics covered.\n\nThe text has been completely rewritten. I hope that an additional 12 years and roughly 20 books worth of experience has enabled me to improve the quality of my exposition. Also, the exercise sets have been completely rewritten.\n\nThe second edition contains two new chapters: a chapter on convexity, separation and positive solutions to linear systems (Chapter 15) and a chapter on the QR decomposition, singular values and pseudoinverses (Chapter 17). The treatments of tensor products and the umbral calculus have been greatly expanded and I have included discussions of determinants (in the chapter on tensor products), the complexification of a real vector space, Schur's theorem and Gershgorin disks.\n\nSteven Roman\nIrvine, California February 2005 Preface to the First Edition\n\nThis book is a thorough introduction to linear algebra, for the graduate or advanced undergraduate student. Prerequisites are limited to a knowledge of the basic properties of matrices and determinants. However, since we cover the basics of vector spaces and linear transformations rather rapidly, a prior course in linear algebra (even at the sophomore level), along with a certain measure of \"mathematical maturity,\" is highly desirable.\n\nChapter 0 contains a summary of certain topics in modern algebra that are required for the sequel. This chapter should be skimmed quickly and then used primarily as a reference. Chapters 1-3 contain a discussion of the basic properties of vector spaces and linear transformations.\n\nChapter 4 is devoted to a discussion of modules, emphasizing a comparison between the properties of modules and those of vector spaces. Chapter 5 provides more on modules. The main goals of this chapter are to prove that any two bases of a free module have the same cardinality and to introduce Noetherian modules. However, the instructor may simply skim over this chapter, omitting all proofs. Chapter 6 is devoted to the theory of modules over a principal ideal domain, establishing the cyclic decomposition theorem for finitely generated modules. This theorem is the key to the structure theorems for finite-dimensional linear operators, discussed in Chapters 7 and 8.\n\nChapter 9 is devoted to real and complex inner product spaces. The emphasis here is on the finite-dimensional case, in order to arrive as quickly as possible at the finite-dimensional spectral theorem for normal operators, in Chapter 10. However, we have endeavored to state as many results as is convenient for vector spaces of arbitrary dimension.\n\nThe second part of the book consists of a collection of independent topics, with the one exception that Chapter 13 requires Chapter 12. Chapter 11 is on metric vector spaces, where we describe the structure of symplectic and orthogonal geometries over various base fields. Chapter 12 contains enough material on metric spaces to allow a unified treatment of topological issues for the basic