Linear algebra
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Linear algebra
LINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra f California, IrvinePrentice-Hall, Inc., Englewood Cliffs, New Jersey© 1971, 1961 byPrentice-Hall, Inc.Englewood Cliffs, New JerseyAll rights reserved. No part, of this book may be reproduced in any form or by any means without permission in writing from the publisher.PRENTICE-HALL INTERNATIONAL, IN Linear algebra C., LondonPRENTICE-HALL OF AUSTRALIA, PTY. LTD., SydneyPRENTICE-HALL OF CANADA, LTD., TorontoPRENTICE-HALL OF INDIA PRIVATE LIMITED, New DelhiPRENTICELinear algebra
-HALL OF JAPAN, INC., TokyoCurrent printing (last digit):10 9 8 7 6Library of Congress Catalog Card No. 75-142120Printed in the United States of AmeriLINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra Technology. This course was designed for mathematics majors at the junior level, although three-fourths of the students were drawn from other scientific and technological disciplines anil ranged from freshmen through graduate students. This description of the M.I.T. audience for the text remains ge Linear algebra nerally accurate today. The ten years since the first edition have seen the proliferation of linear algebra courses throughout the country and have afLinear algebra
forded one of the authors the opportunity to teach the basic material to a variety of groups at Brandeis University, Washington University (St. Louis)LINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra be taught from it. On one hand, we have structured the chapters, especially the more difficult ones, so that there are several natural stopping points along the way, allowing the instructor in a one-quarter or one-semester course to exercise a considerable amount of choice in the subject matter. On Linear algebra the other hand, we have increased the amount of material in the text, so that it can be used for a rather comprehensive one-year course in linear algeLinear algebra
bra and even as a reference book for mathematicians.The major changes have been in our treatments of canonical forms and inner product spaces. In ChapLINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra ion to triangulation and diagonalization theorems and then build our way up to the general theory. We have split Chapter 8 so that the basic material on inner product, spaces and unitary diagonalization is followed by a Chapter 9 which treats sesqui-linear forms and the more sophisticated properties Linear algebra of normal OỊXĩra-tors, including normal operators on real inner product spaces.We have also made a number of small changes and improvements from theLinear algebra
first edition. But the basic philosophy behind the text is unchanged.We have made no particular concession to the fact that the majority of the studenLINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra s a hodgepodge of techniques, but should provide them with an understanding of basic mathematical concepts.Hiiv PrefaceOn the other hand, we have been keenly aware of the wide range of backgrounds which the students may possess and, in particular, of the fact that the students have had very little e Linear algebra xperience with abstract mathematical reasoning. For this reason, we have avoided the introduction of too many abstract ideas at the very beginning ofLinear algebra
the book. In addition, we have included an Appendix which presents such basic ideas as set, function, and equivalence relation. We have found it most LINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra ve included a great, variety of examples of the important concepts which occur. The study of such examples is of fundamental importance and tends to minimize the number of students who can repeat definition, theorem, proof in logical order without grasping the meaning of the abstract concepts. The b Linear algebra ook also contains a wide variety of graded exercises (about six hundred), ranging from routine applications to ones which will extend the very best stLinear algebra
udents. These exercises are intended to be an important part of the text.Chapter 1 deals with systems of linear equations and their solution by means LINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra ture of the origins of linear algebra and with the computational technique necessary to understand examples of the more abstract ideas occurring in the later chapters. Chapter 2 deals with vector spaces, subspaces, bases, and dimension. Chapter 3 treats linear transformations, their algebra, their r Linear algebra epresentation by matrices, as well as isomorphism, linear functionals, and dual spaces. Chapter 4 defines the algebra of polynomials over a field, theLinear algebra
ideals in that algebra, and the prime factorization of a jKilynomial. It also deals with roots, Taylor’s formula, and the Lagrange interpolation formLINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra then proceeds to multilinear functions on modules as well as the Grassman ring. The material on modules places the concept of determinant in a wider and more comprehensive setting than is usually found in elementary textbooks. Chapters 6 and 7 contain a discussion of the concepts which are basic to Linear algebra the analysis of a single linear transformation on a finite-dimensional vector space; the analysis of characteristic (eigen) values, triangulable andLinear algebra
diagonalizable transformations; the concepts of the diagonalizable and nilpotent parts of a more general transfer J nation, and the rational and JordaLINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra paces. Chapter 7 includes a discussion of matrices over a polynomial domain, the computation of invariant factors and elementary divisors of a matrix, and the development of the Smith canonical form. The chapter ends with a discussion of semi-simple operators, to round out the analysis of a single o Linear algebra perator. Chapter 8 treats finite-dimensional inner product spaces in some detail. It covers the basic geometry, relating orthogonalization to the ideaLinear algebra
of ‘best approximation to a vector’ and leading to the concepts of the orthogonal projection of a vector onto a subspace ami the orthogonal complemenLINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of Linear algebra s sesqui-linear forms, relates them to positive and self-adjoint operators on an inner product space, moves on to the spectral theory of normal operators and then to more sophisticated results concerning normal operators on real or complex inner product spaces. Chapter 10 discusses bilinear forms, e Linear algebra mphasizing canonical forms for symmetric and skew-symmetric forms, as well as groups preserving non-degenerate forms, especially the orthogonal, unitaLinear algebra
ry, pseudo-orthogonal and Lorentz groups.We feel that any course which uses this text should cover Chapters 1, 2, and 3LINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity ofLINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity ofGọi ngay
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