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An Introduction to Linear Algebra and Tensors PDF

194 Pages·2010·9.1 MB·English
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DOVER BOOKS ON ADVANCED MATHEMATICS Asymptotic Expansions of Integrals, Norman Bleistein and Richard A. Handelsman. (65082-0) $10.95 Elementary D ecision T heory, Herman Chemoff and Lincoln E. Moses. (65218-1) $8.95 A H istory of Vector Analysis, Michael J. Crowe. (64955-5) $7.00 Some T heory of Sampling, W. Edwards Deming. (64684-X) $14.95 Statistical Adjustment of D ata, W. Edwards Deming. (64685-8) $7.95 Introduction to Linear Algebra and D ifferential Equations, John W. Dettman. (65191-6) $8.95 Calculus of Variations with Applications, George M. Ewing. (64856-7) $8.50 Introduction to D ifference Equations, Samuel Goldberg. (65084-7) $6.95 Probability: An Introduction, Samuel Goldberg. (65252-1) $7.95 U nbounded Linear O perators, Seymour Goldberg. (64830-3) $7.00 G roup T heory, W. R. Scott. (65377-3) $10.95 Elements of R eal Analysis, David A. Sprecher. (65385-4) $8.95 An Introduction to Linear Algebra and T ensors, M.A. Akivis and V.V. Goldberg. (63545-7) $4.50 Introduction to D ifferentiable Manifolds, Louis Ausländer and Robert E. MacKenzie. (63455-8) $6.00 T heory of G ames and Statistical D ecisions, David Blackwell and M.A. Girshick. (63831-6) $7.95 Non-Euclidean G eometry, Roberto Bonola. (60027-0) $8.00 Contributions to the Founding of the T heory of T ransfinite N umbers, Georg Cantor. (60045-9) $4.95 Introduction to Symbolic Logic and Its Applications, Rudolf Carnap. (60453-5) $5.95 Introduction to the T heory of Fourier’s Series and Integrals, H.S. Carslaw. (60048-3) $7.50 Conformal Mapping on Riemann Surfaces, Harvey Cohn. (64025-6) $7.50 Elementary Statistics, Frederick E. Croxton. (60506-X) $6.95 Foundations of Mathematical Logic, Haskell B. Curry. (63462-0) $7.95 Introduction to N onlinear D ifferential and Integral Equations, H.T. Davis. (60971-5) $9.95 Foundations of Modern Analysis, Avner Friedman. (64062-0) $5.50 Curvature and Homology, Samuel I. Goldberg. (64314-X) $6.95 Differential G eometry, Heinrich W. Guggenheimer. (63433-7) $6.95 Ordinary D ifferential Equations, E.L. Ince. (60349-0) $10.95 Lie Algebras, Nathan Jacobson. (63832-4) $7.00 Elements of the T heory of Functions, Konrad Knopp. (60154-4) $3.95 T heory of Functions, Part I, Konrad Knopp. (60156-0) $3.50 T heory of Functions, Part II, Konrad Knopp. (60157-9) $3.95 (continued on back flap) An Introduction to LINEAR ALGEBRA AND TENSORS M. A. AKIVIS V V GOLDBERG Revised English Edition Translated and Edited by Richard A. Silverman DOVER PUBLICATIONS, INC. N EW YORK Copyright © 1972 by Richard A. Silverman. All rights reserved under Pan American and International Copyright Conventions. Published in Canada by General Publishing Com­ pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC2H 7EG. This Dover edition, first published in 1977, is an unabridged and corrected republication of the revised English edition published by Prentice-Hall, Inc., Englewood Cliffs, N J . , in 1972 under the title Intro­ ductory Linear Algebra. International Standard Book Number: 0-486-63545-7 Library of Congress Catalog Card N um ber: 77-78589 Manufactured in the United States of America Dover Publications, Inc. 31 East 2nd Street Mineola, N.Y. 11501 CONTENTS Editor’s Preface, vii LINEAR SPACES, Page 1. 1. Basic Concepts, 1. 2. Linear Dependence, 4. 3. Dimension and Bases, 8. 4. Orthonormal Bases. The Scalar Product, 12. 5. The Vector Product. Triple Products, 16. 6. Basis Transformations. Tensor Calculus, 22. 7. Topics in Analytic Geometry, 29. MULTILINEAR FORMS AND TENSORS, Page 38. 8. Linear Forms, 38. 9. Bilinear Forms, 41. 10. Multilinear Forms. General Definition of a Tensor, 44. 11. Algebraic Operations on Tensors, 50. 12. Symmetric and Antisymmetric Tensors, 55. LINEAR TRANSFORMATIONS, Page 64. 13. Basic Concepts, 64. 14. The Matrix of a Linear Transformation and Its Determinant, 68. 15. Linear Transformations and Bilinear Forms, 78. 16. Multiplication of Linear Transformations and Matrices, 87. 17. Inverse Transformations and Matrices, 94. 18. The Group of Linear Transformations and Its Subgroups, 98. V Vi CONTENTS FURTHER TOPICS, Page 107. 19. Eigenvectors and Eigenvalues, 107. 20. The Case of Distinct Eigenvalues, 117. 21. Matrix Polynomials and the Hamilton-Cayley Theorem, 121. 22. Eigenvectors of a Symmetric Transformation, 124. 23. Diagonalization of a Symmetric Transformation, 127. 24. Reduction of a Quadratic Form to Canonical Form, 133. 25. Representation of a Nonsingular Transformation, 138. SELECTED HINTS AND ANSWERS, Page 144. BIBLIOGRAPHY, Page 161. INDEX, Page 163. EDITOR’S PREFACE The present book, stemming from the first four chap­ ters of the authors’ Tensor Calculus (Moscow, 1969), constitutes a lucid and completely elementary introduction to linear algebra. The treatment is virtually self-contained. In fact, the mathematical background assumed on the part of the reader hardly exceeds a smattering of calculus and a casual acquaintance with determinants. A special merit of the book, reflecting its lineage, is its free use of tensor notation, in particular the Einstein summation convention. Each of the 25 sections is equipped with a problem set, leading to a total of over 250 problems. Hints and answers to most of these problems can be found at the end of the book. As usual, I have felt free to introduce a number of pedagogical and mathematical improvements that occurred to me in the course of the translation. R. a. s. v ii 1 LINEAR SPACES 1. Basic Concepts In studying analytic geometry, the reader has undoubtedly already encountered the concept of a free vector, i.e., a directed line segment which can be shifted in space parallel to its original direction. Such vectors are usually denoted by boldface Roman letters like a, b , . . . , x, y , . . . It can be assumed for simplicity that the vectors all have the same initial point, which we denote by the letter O and call the origin o f coordinates. Two operations on vectors are defined in analytic geometry: a) Any two vectors x and y can be added (in that order), giving the sum x + y; b) Any vector x and (real) number a can be multiplied, giving the product A-x or simply Ax. The set of all spatial vectors is closed with respect to these two operations, in the sense that the sum of two vectors and the product of a vector with a number are themselves both vectors. The operations of addition of vectors x, y, z , . . . and multiplication of vectors by real numbers A, p , . . . have the following properties: 1) x + y = y + x; 2) (x + y) + z = x + (y + z); 3) There exists a zero vector 0 such that x + 0 = x; 4) Every vector x has a negative (vector) y = —x such that x + y = 0; 5) 1 • x = x; 6) X(px) = (A/*)x; 1 2 LINEAR SPACES CHAP. 1 7) W + M)x = Ax + //x; 8) A(x + y) = Ax + Ay. However, operations of addition and multiplication by numbers can be defined for sets of elements other than the set of spatial vectors, such that the sets are closed with respect to the operations and the operations satisfy the properties l)-8) just listed. Any such set of elements is called a linear space (or vector space), conventionally denoted by the letter L. The elements of a vector space L are often called vectors, by analogy with the case of ordinary vectors. Example 1. The set of all vectors lying on a given straight line / forms a linear space, since the sum of two such vectors and the product of such a vector with a real number is again a vector lying on /, while properties l)-8) are easily verified. This linear space will be denoted by L x.t Example 2. The set of all vectors lying in a given plane is also closed with respect to addition and multiplication by real numbers, and clearly satisfies properties l)-8). Hence this set is again a linear space, which we denote by L2. Example 5. Of course, the set of all spatial vectors is also a linear space, denoted by L 3. Example 4. The set of all vectors lying in the xy-plane whose initial points coincide with the origin of coordinates and whose end points lie in the first quadrant is not a linear space, since it is not closed with respect to multipli­ cation by real numbers. In fact, the vector Ax does not belong to the first quadrant if A < 0. Example 5. Let L„ be the set of all ordered w-tuples X = (Xj, x 2, . . . , x„), y = (y u y 2, . . . , y „),. . . of real numbers x 19. . . , y„ , . . . with addition of elements and multiplication of an element by a real number A defined by X + y = (*1 + X2 + ^2» • • • > Xn + y*)> . . Ax = (A*!, Ax2, . . . , Xx„). Then Ln is a linear space, since Ln is closed with respect to the operations (1) which are easily seen to satisfy properties l)-8). For example, the zero element in Ln is the vector 0 = (0, 0 , . . . , 0), while the negative of the vector x is just - x = (-X j, - x 2, . . . , Example 6. As is easily verified, the set of all polynomials t Concerning the meaning of the subscript here and below, see Sec. 3.

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