Linear Algebra: Challenging Problems for Students (Johns Hopkins Studies in the Mathematical Sciences)

266 Pages·2009·5.96 MB·English

by  Fuzhen Zhang

lgeb _a
Linear
I· l '1. II F Z Ill \ \ c
2ND ED,ITIONI
LINEAR ALGEBRA
Johns Hopkins Studies in the
Mathematical Sciences
in association wth the
Department of Mathematical Sciences
The Johns Hopkins University
LINEAR ALGEBRA
Challenging Problems
for Students
Second Edition
Fuzhen Zhang
The Johns Hopkins University Press
Baltimore
© 1996, 2009 The Johns Hopkins Univet·sity Press
All rights reserved. Published 2009
Pnnted in tho United States of America on acid-froo papor
987654321
The .Johns Hopkins University Pre~s
2715 North Charles Street
llaltimore, Maryland 21218-4363
www.pressjhu edu
Library of Congress Control Number: 2008936105
A catalog record for this book is available fJ'Om the Btitlsh Library.
Special disc-ounts are m·ailable for bulk purchases o{this book.
For more information, please l:onlad Spel"illl Sales at 410-516-6936 or
[email protected] 11.edu.
The Johns Hopkins University Press uses environmentally fiiendly book
materials, mcludmg recycled text paper that is composed of at least 30
percent post-consumer waste, whenever possible. All of our book papers
are acid-free, and our jackets and covers are pnnt.ed on pape1 with recycled
content.
To the memory of my grandfather and parents
This page intentionally left blank
Contents
Preface to the 2nd Edition ............................................ ix
Preface ................................................................ xi
Frequently Used Notation and Terminology .......................... xiii
Frequently Used Theorems ............................................ xv
Chapter 1 Vector Spaces .................................... 1
Definitions and Facts ..................................... 1
Vector space · Vector spaces lR'\ C'\ 1Pn[x], and C[a, b]
Matrices Mmxn(1F) ·Column space 1m. A· Null space
Ker A · Linear independence · Basis · Dimension · Span
Sum · Direct sum · Dimension identity · Subspace
Chapter 1 Problems ...................................... 9
Chapter 2 Determinants, Inverses and Rank of Matrices,
and Systems of Linear Equations ................ 21
Definitions and Facts .................................... 21
Determinant · Elementary operations · Inverse · Minor
Cofactor · Adjoint matrix · Rank · Vandermonde deter
minant · Determinants of partitioned matrices · Linear
equation system · Solution space · Cramer's rule
Chapter 2 Problems ..................................... 26
Chapter 3 Matrix Similarity, Eigenvalues, Eigenvectors,
and Linear 'Iransform.ations ••••••••••••••.•••••• 45
Definitions and Facts .................................... 45
Similarity · Diagonalization · Eigenvalue · Eigenvector
Eigenspace · Characteristic polynomial · Trace · Trian
gulariza.tion · Jordan canonical form · Singular value
Singular value decomposition · Linear transformation
Matrix representation · Commutator [A, B] · Image
1m A · Kernel Ker A · Invariant subspaces
Chapter 3 Problems ..................................... 51
vii
viii CONTENTS
Chapter 4 Special Matrices ...•....•................•....•. 75
Definitions and Facts .................................... 75
Hermitian matrix · Skew-Hermitian matrix · Positive
semidefinite matrix · Square root · Trace inequalities
Hadamard determinantal inequality· Spectral decom
position · Hadamard product · Unitary matrix· Real
orthogonal matrix · Normal matrix · Involution A2 = I
Nilpotent matrix Am = 0 · Idempotent matrix A2 = A
Permutation matrix
Chapter 4 Problems ..................................... 77
Chapter 5 Inner Product Spaces .•....................•... 103
Definitions and Facts ................................... 103
Inner product · Vector norm · Distance · Cauchy
Schwarz inequality · Orthogonality · Field of values
Orthogonal complement · Orthonormal basis · Adjoint
transformation · Orthogonal transformation · Dual
space · Projection
Chapter 5 Problems .................................... 107
Hints and Answers for Chapter 1 ...•......•............... 121
Hints and Answers for Chapter 2 ........•••......•••••.••• 133
Hints and Answers for Chapter 3 .•••••.••••.••••.••••••••• 153
Hints and Answers for Chapter 4 .•...........••..•••.•.... 185
Hints and Answers for Chapter 5 ..........•............... 224
Notation ............................................................ 239
Main R.eferences ..................................................... 241
Index ................................................................ 243
Preface to the 2nd Edition
This is the second, revised, and expanded edition of the linear algebra
problem book Linear Algebra: Challenging Problems for Students. The first
edition of the book, containing 200 problems, was published in 1996. In
addition to about 200 new problems in this edition, each chapter starts with
definitions and facts that lay out the foundations and groundwork for the
chapter, followed by carefully selected problems. Some of the new problems
are straightforward; some are pretty hard. The main theorems frequently
needed for solving these problems are listed on page xv.
My goal has remained the same as in the first edition: to provide a book
of interesting and challenging problems on linear algebra and matrix theory
for upper-division undergraduates and graduate students in mathematics,
statistics, engineering, and related fields. Through working and practicing
on the problems in the book, students can learn and master the basic
concepts, skills, and techniques in linear algebra and matrix theory.
During the past ten years or so, I served as a collaborating editor for
American Mathematical Monthly problem section, Wisociate editor for the
International Linear Algebra Society Bulletin IMAGE Problem Corner, and
editor for several other mathematical journals, from which some problems in
the new edition have originated. I have also benefited from the math confer
ences I regularly attend; they are the International Linear Algebra Society
(ILAS) Conferences, Workshops on Numerical Ranges and Numerical Radii,
R. C. Thompson (formerly Southern California) Matrix Meetings, and the
International Workshops on Matrix Analysis and Applications. For exam
ple, I learned Problem 4.21 from M.-D. Choi at the !LAS Shanghai Meeting
in 2007; Problem 4.97 was a recent submission to IMAGE by G. Goodman
and R. Hom; some problems were collected during tea breaks.
I am indebted to many colleagues and friends who helped with the re
vision; in particular, I thank Jane Day for her numerous comments and
suggestions on this version. I also thank Nova Southeastern University
(NSU) and the Farquhar College of Arts and Sciences (FCAS) of the uni
versity for their support through various funds, including the President's
Faculty Research and Development Grants (Awards), FCAS Minigrants,
and FCAS Faculty Development FUnds.
Readers are welcome to communicate with me at [email protected].
ix