Solutions Manual for Lang’s Linear Algebra

192 Pages·1996·2.94 MB·English

by  Rami Shakarchi (auth.)

Solutions Manual for Lang's
Linear Algebra
Springer Science+Business Media, LLC
Rami Shakarchi
Solutions Manual for Lang's
Linear Algebra
Springer
Rami Shakarchi
P.O. Box 205612 Yale Station
New Haven, CT 06520
USA
Mathematies Subjeet Classifieation (1991): 15-01
Printed on acid-free paper.
© 1996 Springer Scienee+Business Media New York
Origina11y published by Springer-Verlag New York Ine. in 1996
Ali rights reserved. This work may not be translated or copied in whole or in part without the written
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and Merchandise Marks Act, may according1y be used freely by anyone.
Produetion managed by Lesley Poliner; manufaeturing supervised by Joe Quatela.
Carnera-ready eopy prepared by the author.
987654321
ISBN 978-0-387-94760-0 ISBN 978-1-4612-0755-9 (eBook) SPIN 10536794
DOI 10.1007/978-1-4612-0755-9
To my parents
Preface
The present volume contains all the exercises and their solutions of Lang's'
Linear Algebra. Solving problems being an essential part of the learning
process, my goal is to provide those learning and teaching linear algebra with a
large number of worked out exercises. Lang's textbook covers all the topics in
linear algebra that are usually taught at the undergraduate level: vector spaces,
matrices and linear maps including eigenvectors and eigenvalues, determinants,
diagonalization of symmetric and hermitian maps, unitary maps and matrices,
triangulation, Jordan canonical form, and convex sets. Therefore this solutions
manual can be helpful to anyone learning or teaching linear algebra at the college
level.
As the understanding of the first chapters is essential to the comprehension
of the later, more involved chapters, I encourage the reader to work through all of
the problems of Chapters I, II, III and IV. Often earlier exercises are useful in
solving later problems. (For example, Exercise 35, §3 of Chapter II shows that
a strictly upper triangular matrix is nilpotent and this result is then used in
Exercise 7, §1 of Chapter X.) To make the solutions concise, I have included
only the necessary arguments; the reader may have to fill in the details to get
complete proofs.
Finally, I thank Serge Lang for giving me the opportunity to work on this
solutions manual, and I also thank my brother Karim and Steve Miller for their
helpful comments and their support.
Rami Shakarchi
Yale, 1996
I thank Rami Shakarchi very much for having prepared this answer book.
Serge Lang
Yale, 1996
Contents
CHAPTER I
Vector Spaces 1
§ 1. Definitions 1
§2. Bases 5
§4. Sums and Direct Sums 10
CHAPTER II
Matrices 12
§ 1. The Space of Matrices 12
§2. Linear Equations 17
§3. Multiplication of Matrices 18
CHAPTER III
Linear Mappings 34
§ 1. Mappings 34
§2. Linear Mappings 37
§3. The Kernel and Image of a Linear Map 42
§4. Composition and Inverse of Linear Mappings 50
§5. Geometric Applications 55
CHAPTER IV
Linear Maps and Matrices 57
§ 1. The Linear Map Associated with a Matrix 57
§2. The Matrix Associated with a Linear Map 57
§3. Bases, Matrices and Linear Map 60
CHAPTER V
Scalar Products and Orthogonality 65
§ 1. Scalar Products 65
§2. Orthogonal bases, Positive Definite Case 66
§3. Application to Linear Equations; the Rank 71
x CONTENTS
§4. Bilinear Map and Matrices 75
§5. General Orthogonal Bases 78
§6. The Dual Space and Scalar Products 79
§7. Quadratic Fonns 81
§8. Sylvester's Theorem 82
CHAPTER VI
Determinants 85
§2. Existence of Determinants 85
§3. Additional Properties of Detenninants 85
§4. Cramer's rule 91
§5. Triangulation of a Matrix by Column Operations 91
§6. Pennutations 93
§7. Expansion Fonnula and Uniqueness of Detenninants 95
§8. Inverse of a Matrix 98
§9. The Rank of Matrix and Subdetenninants 99
CHAPTER VII
Symmetric, Hermitian and Unitary Operators 101
§ 1. Symmetric Operators 101
§2. Hermitian Operators 107
§3. Unitary Operators 110
CHAPTERVm
Eigenvectors and Eigenvalues 117
§ 1. Eigenvectors and Eigenvalues 117
§2. The Characteristic Polynomial 121
§3. Eigenvalues and Eigenvectors of Symmetric Matrices 135
§4. Diagonalization of a Symmetric Linear Map 136
§5. The Hennitian Case 148
CHAPTER IX
Polynomials and Matrices 154
§2. Polynomials of Matrices and Linear Maps 154
CHAPTER X
Triangulation of Matrices and Linear Maps 156
§ 1. Existence of Triangulation 156
§3. Diagonalization of Unitary Maps 159
CONTENTS Xl
CHAPTER XI
Polynomials and Primary Decomposition 160
§1. The Euclidean Algorithm 160
§2. Greatest Common Divisor 161
§3. Unique Factorization 162
§4. Application to the Decomposition of a Vector Space 169
§5. Schur's Lemma 170
§6. The Jordan Normal Form 173
CHAPTER XII
Convex Sets 176
§4. The ~ein-Milman Theorem 176
APPENDIX
Complex Numbers 179
CHAPTER I
Vector Spaces
I, §1 Definitions
1. Let V be a vector space. Using the properties VS 1 through VS 8, show
that if c is a number then cO = O.
SOLUTION. We have cO=c(O+O)=cO+cO, but we also have
cO = O+cO, hence
cO+cO= O+cO.
Adding (-cO) to both sides shows that cO = O.
2. Let c be a number "# 0, and v an element of V. Prove that if cv = 0, then
v=O.
SOLUTION. Exercise 1 implies, O=(l/c)O=(l/c)(cv)=(c/c)v=lv=v.
3. In the vector space of functions, what is the function satisfying the condi
tion VS 2?
SOLUTION. The zero function, namely, f(x) = 0 for all x plays the role
of the identity.
4. Let V be a vector space and v, w two elements of V. If v + w = 0, show
that w = -v.
SOLUTION. We have -w = -w+O = -w+(v+ w) = v+ w-w = v.
5. Let V be a vector space, and v, w be two elements of V such that
v + w = v. Show that w = O.
SOLUTION. We know that v + 0 = v so v + 0 = v + w. Adding -v to
both sides shows that 0 = w.