Lectures in Abstract Algebra: II. Linear Algebra

290 Pages·1953·7.12 MB·English

by  Nathan Jacobson (auth.)

Graduate Texts in Mathematics 31
Editorial Board: F. W. Gehring
P. R. Halmos (Managing Editor)
C. C. Moore
Nathan Jacobson
Lectures in
Abstract Algebra
II. Linear Algebra
Springer Science+Business Media, LLC
Nathan Jacobson
Yale University
Department of Mathematics
New Haven, Connecticut 06520
Managing Editor
P. R. Halmos
Indiana University
Department of Mathematics
Swain Hali East
Bloomington, Indiana 47401
Editors
F. W. Gehring c. C. Moore
University of Michigan University of California at Berkeley
Department of Mathematics Department of Mathematics
Ann Arbor, Michigan 48104 Berkeley, California 94720
AMS Subject Classification
15-01
Library of Congress Cataloging in Publication Data
Jacobson, Nathan, 1910-
Lectures in abstract algebra.
(Graduate texts in mathematics; 31 )
Reprint of the 1951-1964 ed. published by Van Nostrand,
New York in The University series in higher mathematics,
M. A. Stone, L. Nirenberg, and S. S. Chem, eds.
Inc1udes index.
CONTENTS: ---2. Linear
algebra.
1. Algebra, Abstract. 1. Title. II. Series.
QAI62.J3 1975 512'.02 75-15564
Ali rights reserved.
No part of this book may be translated or reproduced in any
form without written permission from Springer-Verlag.
© 1953 N. Jacobson
Urspriinglich erschienen bei N. Jacobson 1953
Softcover reprint of the hardcover 1s t edition 1953
ISBN 978-1-4684-7055-0 ISBN 978-1-4684-7053-6 (eBook)
DOI 10.1007/978-1-4684-7053-6
TO
MICHAEL
PREFACE
The present volume is the second in the author's series of
three dealing with abstract algebra. For an understanding of
this volume a certain familiarity with the basic concepts treated
in Volume I: groups, rings, fields, homomorphisms, is presup
posed. However, we have tried to make this account of linear
algebra independent of a detailed knowledge of our first volume.
References to specific results are given occasionally but some of
the fundamental concepts needed have been treated again. In
short, it is hoped that this volume can be read with complete
understanding by any student who is mathematically sufficiently
mature and who has a familiarity with the standard notions of
modern algebra.
Our point of view in the present volume is basically the abstract
conceptual one. However, from time to time we have deviated
somewhat from this. Occasionally formal calculational methods
yield sharper results. Moreover, the results of linear algebra are
not an end in themselves but are essential tools for use in other
branches of mathematics and its applications. It is therefore
useful to have at hand methods which are constructive and which
can be applied in numerical problems. These methods sometimes
necessitate a somewhat lengthier discussion but we have felt that
their presentation is justified on the grounds indicated. A stu
dent well versed in abstract algebra will undoubtedly observe
short cuts. Some of these have been indicated in footnotes.
We have included a large number of exercises in the text.
Many of these are simple numerical illustrations of the theory.
Others should be difficult enough to test the better students. At
any rate a diligent study of these is essential for a thorough un
derstanding of the text.
vii
V111 PREFACE
At various stages in the writing of this book I have benefited
from the advice and criticism of many friends. Thanks are par
ticularly due to A. H. Clifford, to G. Hochschild, and to I. Kap
lansky for suggestions on earlier versions of the text. Also I am
greatly indebted to W. H. Mills, Jr. for painstaking help with the
proofs and for last minute suggestions for improvements of the
text.
J.
N.
New Haven, Conn.
September, 1952
CONTENTS
CHAPTER I: FINITE DIMENSIONAL VECTOR SPACES
SECTION PAGE
1. Abstract vector spaces 3
2. Right vector spaces 6
3. o-modules. . . . . . 7
4. Linear dependence . . 9
5. Invariance of dimensionality 13
6. Bases and matrices. . . . . 15
7. Applications to matrix theory 18
8. Rank of a set of vectors 22
9. Factor spaces . . . . . . . 25
10. Algebra of subspaces . . . . 25
11. Independent subspaces, direct sums 28
CHAPTER II: LINEAR TRANSFORMATIONS
1. Definition and examples . . . . . . 31
2. Compositions of linear transformations 33
3. The matrix of a linear transformation 36
4. Compositions of matrices . . . . . . 38
5. Change of basis. Equivalence and similarity of matrices. 41
6. Rank space and null space of a linear transformation 44
7. Systems of linear equations . . . . . . . . 47
8. Linear transformations in right vector spaces . . . 49
9. Linear functions . . . . . . . . . . . . . . . . 51
10. Duality between a finite dimensional space and its conjugate
space. . . . . . . . . . . . . 53
11. Transpose of a linear transformation 56
12. Matrices of the transpose 58
13. Projections . . . . . . . . . . . 59
CHAPTER III: THE THEORY OF A SINGLE LINEAR TRANSFORMATION
1. The minimum polynomial of a linear transformation 63
2. Cyclic subspaces. . . . .. ....... . 66
ix
x CONTENTS
SECTION PAGE
3. Existence of a vector whose order is the minimum polynomial 67
4. Cyclic linear transformations . . . . . . . . . . . . 69
5. The cp[AJ-module determined by a linear transformation. 74
6. Finitely generated o-modules, 0, a principal ideal domain 76
7. Normalization of the generators of ~ and of ~ .... 78
8. Equivalence of matrices with elements in a principal ideal
domain. . . . . . . . . . . . .. ....... 79
9. Structure of finitely generated o-modules . 85
10. Invariance theorems . . . . . . . 88
11. Decomposition of a vector space relative to a linear trans-
forma tion. . . . . . . . . . . . . . . 92
12. The characteristic and minimum polynomials 98
13. Direct proof of Theorem 13 . . 100
14. Formal properties of the trace and the characteristic poly
nomial . . . . . . . . . . . . . . . . . . . . . . . 103
15. The ring of o-endomorphisms of a cyclic o-module . . . . . 106
16. Determination of the ring of o-endomorphisms of a finitely
generated o-module, 0 principal . . . . . . . . . . . . 108
17. The linear transformations which commute with a given lin-
ear transformation . . 110
18. The center of the ring ~ 113
CHAPTER IV: SETS OF LINEAR TRANSFORMATIONS
1. Invariant subspaces 115
2. Induced linear transformations 117
3. Composition series. . 120
4. Decomposability. . . . 122
5. Complete reducibility 124
6. Relation to the theory of operator groups and the theory of
modules . . . . . . . . . . . . . . . . . . . 126
7. Reducibility, decomposability, complete reducibility for a
single linear transformation . . . . . . . . . . . . . . 128
8. The primary components of a space relative to a linear trans-
formation. . . . . . . . . . . . . . 130
9. Sets of commutative linear transformations . . . . . . . . 132
CHAPTER v: BILINEAR FORMS
1. Bilinear forms . . . . . . 137
2. Matrices of a bilinear form 138
CONTENTS Xl
SECTION PAGE
3. Non-degenerate forms . . . . . . . . . . . . . . . . . 140
4. Transpose of a linear transformation relative to a pair of bi
linear forms. . . . . . . . . . . . . . . . . . . . . 142
5. Another relation between linear transformations and bilinear
forms ........ 145
6. Scalar products . . . . . . . . . . 147
7. Hermitian scalar products ..... 150
8. Matrices of hermitian scalar products 152
9. Symmetric and hermitian scalar products over special divi-
sion rings ..... . 154
10. Alternate scalar products . . . . . 159
11. Wi tt's theorem . . . . . . . . . 162
12. Non-alternate skew-symmetric forms 170
CHAPTER VI: EUCLIDEAN AND UNITARY SPACES
1. Cartesian bases . . . . . . . . . . . . 173
2. Linear transformations and scalar products . . . . . 176
3. Orthogonal complete reducibility ......... 177
4. Symmetric, skew and orthogonal linear transformations. 178
5. Canonical matrices for symmetric and skew linear transfor-
mations ..................... 179
6. Commutative symmetric and skew linear transformations. 182
7. Normal and orthogonal linear transformations. . . . 184
8. Semi-definite transformations . . . . . . . . . . . 186
9. Polar factorization of an arbitrary linear transformation 188
10. Unitary geometry . . . . . . . . . . . 190
11. Analytic functions oflinear transformations. . . . . 194
CHAPTER VII: PRODUCTS OF VECTOR SPA9ES
1. Product groups of vector spaces . . . . 199
2. Direct products of linear transformations 202
3. Two-sided vector spaces . . . . . . . 204
4. The Kronecker product. . . . . . . . 208
5. Kronecker products of linear transformations and of matrices 211
6. Tensor spaces . . . . . . . . . . . 213
7. Symmetry classes of tensors. . . . . . . 217
8. Extension of the field of a vector space . . 221
9. A theorem on similarity of sets of matrices 222
XlI CONTENTS
SECTION PAGE
10. Alternative definition of an algebra. Kronecker product of
algebras . . . . . . . . . . . 225
CHAPTER VIII: THE RING OF LINEAR TRANSFORMATIONS
1. Simplicity of 2 227
2. Operator methods 229
3. The left ideals of 2 230
4. Right ideals. . . 232
5. Isomorphisms of rings of linear transformations 233
CHAPTER IX: INFINITE DIMENSIONAL VECTOR SPACES
1. Existence of a basis 239
2. Invariance of dimensionality 240
3. Subspaces. . . . . . . . . 242
4. Linear transformations and matrices . 243
5. Dimensionality of the conjugate space 244
6. Finite topology for linear transformations 248
7. Total subspaces of ffi* . . . . . . . . . 251
8. Dual spaces. Kronecker products . . 253
9. Two-sided ideals in the ring of linear transformations. 256
10. Dense rings of linear transformations. . 259
11. Isomorphism theorems . . . . . . . . . 264
12. Anti-automorphisms and scalar products . 267
13. Schur's lemma. A general density theorem 271
14. Irreducible algebras of linear transformations 274
Index ................ . 277