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Linear algebra

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Nội dung chi tiết: Linear algebra

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 DelhiPRENTICE

Linear 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 Ameri

LINEAR 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 scienti

fic 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 af

Linear 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 alge

Linear 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 Chap

LINEAR 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 the

Linear 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 studen

LINEAR 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 of

Linear 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 m

inimize 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 st

Linear 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 th

e 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, the

Linear algebra

ideals in that algebra, and the prime factorization of a jKilynomial. It also deals with roots, Taylor’s formula, and the Lagrange interpolation form

LINEAR 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 and

Linear algebra

diagonalizable transformations; the concepts of the diagonalizable and nilpotent parts of a more general transfer J nation, and the rational and Jorda

LINEAR 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 idea

Linear 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 complemen

LINEAR 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 operat

ors 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, unita

Linear algebra

ry, pseudo-orthogonal and Lorentz groups.We feel that any course which uses this text should cover Chapters 1, 2, and 3

LINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of

LINEAR ALGEBRASecond EditionKENNETH HOFFMANProfessor of MathematicsMassachusetts Institute of TechnologyRAY KUNZEProfessor of MathematicsUniversity of

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