An Introduction to Linear Algebra and Tensors

194 Pages·2010·9.1 MB·English

by  M. A. Akivis

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(continued on back flap)
An Introduction to
LINEAR ALGEBRA
AND TEN SO RS
M. A. AKIVIS
V V GOLDBERG
Revised English Edition
Translated and Edited by
Richard A. Silverman
DOVER PUBLICATIONS, INC.
NEW 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 Number: 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.
vii

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 of 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 Lx.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 L3.
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, x2,..., x„), y = (yu y2,..., y„),. . .
of real numbers x19..., 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 = (-Xj, -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.