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Schaum's Outline of Linear Algebra (6e)
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Linear Algebra
Sixth Edition
Seymour Lipschutz, PhD
Temple University
Marc Lars Lipson, PhD
University of Virginia
Schaum’s Outline Series
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Copyright © 2018 by McGraw-Hill Education. All rights reserved. Except as permitted under the United States
Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means,
or stored in a database or retrieval system, without the prior written permission of the publisher.
ISBN: 978-1-26-001145-6
MHID: 1-26-001145-3
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MHID: 1-26-001144-5.
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SEYMOUR LIPSCHUTZ is on the faculty of Temple University and formally taught at the Polytechnic In-
stitute of Brooklyn. He received his PhD in 1960 at Courant Institute of Mathematical Sciences of New York
University. He is one of Schaum’s most prolic authors. In particular, he has written, among others, Beginning
Linear Algebra, Probability, Discrete Mathematics, Set Theory, Finite Mathematics, and General Topology.
MARC LARS LIPSON is on the faculty of the University of Virginia and formerly taught at the University
of Georgia, he received his PhD in nance in 1994 from the University of Michigan. He is also the coauthor of
Discrete Mathematics and Probability with Seymour Lipschutz.
TERMS OF USE
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of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store
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Black plate (3,1)
Preface
Linear algebra has in recent years become an essential part of the mathematical background required by
mathematicians and mathematics teachers, engineers, computer scientists, physicists, economists, and statis-
ticians, among others. This requirement reflects the importance and wide applications of the subject matter.
This book is designed for use as a textbook for a formal course in linear algebra or as a supplement to all
current standard texts. It aims to present an introduction to linear algebra which will be found helpful to all
readers regardless of their fields of specification. More material has been included than can be covered in most
first courses. This has been done to make the book more flexible, to provide a useful book of reference, and to
stimulate further interest in the subject.
Each chapter begins with clear statements of pertinent definitions, principles, and theorems together with
illustrative and other descri ptive material. This is followed by graded sets of solved and supplementary
problems. The solved problems serve to illustrate and amplify the theory, and to provide the repetition of basic
principles so vital to effective learning. Numerous proofs, especially those of all essential theorems, are
included among the solve d problems. The supplementary problems serve as a complete review of the material
of each chapter.
The first three chapters treat vectors in Euclidean space, matrix algebra, and systems of linear equations.
These chapters provide the motivation and basic computational tools for the abstract investigations of vector
spaces and linear mappings which follow. After chapters on inner product spaces and orthogonality and on
determinants, there is a detailed discussion of eigenvalues and eigenvectors giving conditions for representing a
linear operat or by a diagonal matrix. This naturally leads to the study of various canonical forms, specifically,
the triangular, Jordan, and rational canonical forms. Later chapters cover linear functions and the dual space V*,
and bilinear, quadratic, and Hermitian forms. The last chapter treats linear operators on inner product spaces.
The main changes in the sixth edition are that some parts in Appendix D have been added to the main part of
the text, that is, Chapter Four and Chapter Eight. There are also many additional solved and supplementary
problems.
Finally, we wish to thank the staff of the McGraw-Hill Schaum’s Outline Series, especially Diane Grayson,
for their unfailing cooperation.
S
EYMOUR LIPSCHUTZ
MARC LARS LIPSON
iii
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Black plate (4,1)
List of Symbols
A ¼½a
ij
, matrix, 27
A ¼½
a
ij
, conjugate matrix, 38
jAj, determinant, 266, 270
A*, adjoint, 379
A
H
, conjugate transpose, 38
A
T
, transpose, 33
A
þ
, Moore–Penrose inverse, 420
A
ij
, minor, 271
AðI; JÞ, minor, 275
AðVÞ, linear operators, 176
adj A, adjoint (classical), 273
A B, row equivalence, 72
A ’ B, congruence, 362
C, complex numbers, 11
C
n
, complex n-space, 13
C½a; b, continuous functions, 230
Cð f Þ, companion matrix, 306
colsp ðAÞ, column space, 120
dðu; vÞ, distance, 5, 243
diagða
11
; ...; a
nn
Þ, diagonal matrix, 35
diagðA
11
; ...; A
nn
Þ, block diagonal, 40
detðAÞ, determinant, 270
dim V, dimension, 124
fe
1
; ...; e
n
g, usual basis, 125
E
k
, projections, 386
f : A ! B, mapping, 166
FðXÞ, function space, 114
G F, composition, 175
HomðV; UÞ, homomorphisms, 176
i, j, k,9
I
n
, identity matrix, 33
Im F, image, 171
JðlÞ, Jordan block, 331
K, field of scalars, 112
Ker F, kernel, 171
mðtÞ, minimal polynomial, 305
M
m;n
; m n matrices, 114
n-space, 5, 13, 229, 242
P(t), polynomials, 114
P
n
ðtÞ; polynomials, 114
projðu; vÞ, projection, 6, 236
projðu; VÞ, projection, 237
Q, rational numbers, 11
R, real numbers, 1
R
n
, real n-space, 2
rowsp ðAÞ, row-space, 120
S
?
, orthogonal complement, 233
sgn s, sign, parity, 269
spanðSÞ, linear span, 119
trðAÞ, trace, 33
½T
S
, matrix representation, 197
T*, adjoint, 379
T-invariant, 329
T
t
, transpose, 353
kuk, norm, 5, 13, 229, 243
½u
S
, coordinate vector, 130
u v , dot product, 4, 13
hu; vi, inner product, 228, 240
u v , cross product, 10
u v , tensor product, 398
u ^ v , exterior product, 403
u v , direct sum, 129, 329
V ffi U, isomorphism, 132, 171
V W, tensor product, 398
V*, dual space, 351
V**, second dual space, 352
V
r
V, exterior product, 403
W
0
, annihilator, 353
z, complex conjugate, 12
Zðv; TÞ, T-cyclic subspace, 332
d
ij
, Kronecker delta, 37
DðtÞ, characteristic polynomial, 296
l, eigenvalue, 298
P
, summation symbol, 29
iv
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