(Graduate texts in mathematics 135) steven roman (auth ) advanced linear algebra springer new york (1992)

Graduate Texts in Mathematics

S. Axler

135

Editorial Board
F.W. Gehring K.A. Ribet

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Graduate Texts in Mathematics
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32
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TAKEUTI/ZARING.Introduction to
Axiomatic Set Theory. 2nd ed.
OXTOBY.Measure and Category. 2nd ed.
SCHAEFER.Topological Vector Spaces.
2nd ed.
HILTON/STAMMBACH.A Course in
Homological Algebra. 2nd ed.
MAC LANE. Categories for the Working
Mathematician. 2nd ed.
HUGHES/PIPER.Projective Planes.
J.-P. SERRE.A Course in Arithmetic.
TAKEUTI/ZARING.Axiomatic Set Theory.
HUMPHREYS.Introduction to Lie Algebras

and Representation Theory.
COHEN.A Course in Simple Homotopy
Theory.
CONWAY.Functions of One Complex
Variable I. 2nd ed.
BEALS.Advanced Mathematical Analysis.
ANDERSON/FULLER.Rings and Categories
of Modules. 2nd ed.
GOLUBITSKY/GUILLEMIN.Stable Mappings
and Their Singularities.
BERBERIAN.Lectures in Functional
Analysis and Operator Theory.
WrNTER.The Structure of Fields.
ROSENBLATT.Random Processes. 2nd ed.
HALMOS.Measure Theory.
HALMOS.A Hilbert Space Problem Book.
2nd ed.
HUSEMOLLER.Fibre Bundles. 3rd ed.
HUMPHREYS.Linear Algebraic Groups.
BARNES/]VIACK.An Algebraic Introduction
to Mathematical Logic.
GREUB.Linear Algebra. 4th ed.
HOLMES.Geometric Functional Analysis
and Its Applications.
HEWITT/STROMBERG.Real and Abstract
Analysis.
MANES.Algebraic Theories.
KELLEY.General Topology.
ZARISKI/SAMUEL.Commutative Algebra.
Vol.I.

ZAPdSrO/SAMUEL.Commutative Algebra.
Vol.II.
JACOBSON.Lectures in Abstract Algebra I.
Basic Concepts.
JACOBSON.Lectures in Abstract Algebra II.
Linear Algebra.
JACOBSON.Lectures in Abstract Algebra
III. Theory of Fields and Galois Theory.
HIRSCH.Differential Topology.

34
35
36
37
38
39
40
41

42
43
44
45
46
47
48
49
50
51
52

53
54
55

56
57
58
59
60
61
62
63

SPITZER.Principles of Random Walk.
2nd ed.
ALEXANDE~ERMER.Several Complex
Variables and Banach Algebras. 3rd ed.
KELLEY/NAMIOKAet al. Linear
Topological Spaces.
MONK.Mathematical Logic.
GRAUERT/FRITZSCHE.Several Complex
Variables.
ARVESON.An Invitation to C*-Algebras.
KEMENY/SNELL/KNAPP.Denumerable
Markov Chains. 2nd ed.
APOSTOL.Modular Functions and Dirichlet
Series in Number Theory.
2nd ed.
J.-P. SERRE.Linear Representations of
Finite Groups.

GILLMAN/JERISON.Rings of Continuous
Functions.
KENDIG.Elementary Algebraic Geometry.
LOEVE.Probability Theory I. 4th ed.
LoI~vE.Probability Theory II. 4th ed.
MOISE.Geometric Topology in
Dimensions 2 and 3.
SACHS/~qVu.General Relativity for
Mathematicians.
GRUENBERG/~vVEIR.Linear Geometry.
2nd ed.
EDWARDS.Fermat's Last Theorem.
KLINGENBERG.A Course in Differential
Geometry.
HARTSHORNE.Algebraic Geometry.
MANN.A Course in Mathematical Logic.
GRAVEK/V~ATKINS.Combinatorics with
Emphasis on the Theory of Graphs.
BROWN/PEARCY.Introduction to Operator
Theory I: Elements of Functional
Analysis.
MASSEY.Algebraic Topology: An
Introduction.
CROWELL/Fox.Introduction to Knot
Theory.
KOBLITZ.p-adic Numbers, p-adic Analysis,
and Zeta-Functions. 2nd ed.
LANG.Cyclotomic Fields.
ARNOLD.Mathematical Methods in
Classical Mechanics. 2nd ed.

WHITEHEAD.Elements of Homotopy
Theory.
KARGAPOLOV/]~ERLZJAKOV.Fundamentals
of the Theory of Groups.
BOLLOBAS.Graph Theory.

(continued after index)

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Steven Roman

Advanced Linear Algebra
Second Edition

___ S p r i n g e r

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Steven Roman
University of California, Irvine
Irvine, California 92697-3875
USA


Editorial Board."
S. Axler
Mathematics Department

San Francisco State
University
San Francisco, CA 94132
USA


EW. Gehring
Mathematics Department
East Hall
University of Michigan
Ann Arbor, MI 48109
USA


K.A. Ribet
Mathematics D e p a r t m e n t
University of California,
Berkeley
Berkeley, CA 94720-3840
USA


Mathematics Subject Classification (2000): 15-xx
Library of Congress Cataloging-in-Publication Data
Roman, Steven.
Advanced linear algebra / Steven Roman.--2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-387-24766-1 (acid-free paper)
1. Algebras, Linear. I. Title.

QA184.2.R66 2005
512'.5-- dc22
2005040244
ISBN 0-387-24766-1

Printed on acid-free paper.

© 2005 Steven Roman
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+ Business Media, LLC, 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter
developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or
not they are subject to proprietary rights.
Printed in the United States of America.
98765432
springer.com

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To Donna
and to my poker buddies
Rachelle, Carol and Dan

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Preface to the Second Edition

Let me begin by thanking the readers of the first edition for their many helpful
comments and suggestions. The second edition represents a major change from
the first edition. Indeed, one might say that it is a totally new book, with the
exception of the general range of topics covered.
The text has been completely rewritten. I hope that an additional 12 years and
roughly 20 books worth of experience has enabled me to improve the quality of
my exposition. Also, the exercise sets have been completely rewritten.
The second edition contains two new chapters: a chapter on convexity,
separation and positive solutions to linear systems (Chapter 15) and a chapter on
the QR decomposition, singular values and pseudoinverses (Chapter 17). The
treatments of tensor products and the umbral calculus have been greatly
expanded and I have included discussions of determinants (in the chapter on
tensor products), the complexification of a real vector space, Schur's lemma and
Geršgorin disks.
Steven Roman

Irvine, California February 2005

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Preface to the First Edition

This book is a thorough introduction to linear algebra, for the graduate or
advanced undergraduate student. Prerequisites are limited to a knowledge of the
basic properties of matrices and determinants. However, since we cover the
basics of vector spaces and linear transformations rather rapidly, a prior course

in linear algebra (even at the sophomore level), along with a certain measure of
“mathematical maturity,” is highly desirable.
Chapter 0 contains a summary of certain topics in modern algebra that are
required for the sequel. This chapter should be skimmed quickly and then used
primarily as a reference. Chapters 1–3 contain a discussion of the basic
properties of vector spaces and linear transformations.
Chapter 4 is devoted to a discussion of modules, emphasizing a comparison
between the properties of modules and those of vector spaces. Chapter 5
provides more on modules. The main goals of this chapter are to prove that any
two bases of a free module have the same cardinality and to introduce
noetherian modules. However, the instructor may simply skim over this chapter,
omitting all proofs. Chapter 6 is devoted to the theory of modules over a
principal ideal domain, establishing the cyclic decomposition theorem for
finitely generated modules. This theorem is the key to the structure theorems for
finite-dimensional linear operators, discussed in Chapters 7 and 8.
Chapter 9 is devoted to real and complex inner product spaces. The emphasis
here is on the finite-dimensional case, in order to arrive as quickly as possible at
the finite-dimensional spectral theorem for normal operators, in Chapter 10.
However, we have endeavored to state as many results as is convenient for
vector spaces of arbitrary dimension.
The second part of the book consists of a collection of independent topics, with
the one exception that Chapter 13 requires Chapter 12. Chapter 11 is on metric
vector spaces, where we describe the structure of symplectic and orthogonal
geometries over various base fields. Chapter 12 contains enough material on
metric spaces to allow a unified treatment of topological issues for the basic

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x

Preface

Hilbert space theory of Chapter 13. The rather lengthy proof that every metric
space can be embedded in its completion may be omitted.
Chapter 14 contains a brief introduction to tensor products. In order to motivate
the universal property of tensor products, without getting too involved in
categorical terminology, we first treat both free vector spaces and the familiar
direct sum, in a universal way. Chapter 15 [Chapter 16 in the second edition] is
on affine geometry, emphasizing algebraic, rather than geometric, concepts.
The final chapter provides an introduction to a relatively new subject, called the
umbral calculus. This is an algebraic theory used to study certain types of
polynomial functions that play an important role in applied mathematics. We
give only a brief introduction to the subject c emphasizing the algebraic
aspects, rather than the applications. This is the first time that this subject has
appeared in a true textbook.
One final comment. Unless otherwise mentioned, omission of a proof in the text
is a tacit suggestion that the reader attempt to supply one.
Steven Roman

Irvine, California

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Contents

Preface to the Second Edition, vii
Preface to the First Edition, ix

Preliminaries, 1
Part 1 Preliminaries, 1
Part 2 Algebraic Structures, 16

Part I—Basic Linear Algebra, 31
1

Vector Spaces, 33
Vector Spaces, 33
Subspaces, 35
Direct Sums, 38
Spanning Sets and Linear Independence, 41
The Dimension of a Vector Space, 44
Ordered Bases and Coordinate Matrices, 47
The Row and Column Spaces of a Matrix, 48
The Complexification of a Real Vector Space, 49
Exercises, 51

2

Linear Transformations, 55
Linear Transformations, 55
Isomorphisms, 57
The Kernel and Image of a Linear Transformation, 57
Linear Transformations from -  to -  , 59
The Rank Plus Nullity Theorem, 59
Change of Basis Matrices, 60
The Matrix of a Linear Transformation, 61
Change of Bases for Linear Transformations, 63
Equivalence of Matrices, 64

Similarity of Matrices, 65
Similarity of Operators, 66
Invariant Subspaces and Reducing Pairs, 68

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xii

Contents

Topological Vector Spaces, 68
Linear Operators on = d , 71
Exercises, 72
3

The Isomorphism Theorems, 75
Quotient Spaces, 75
The Universal Property of Quotients and
the First Isomorphism Theorem, 77
Quotient Spaces, Complements and Codimension, 79
Additional Isomorphism Theorems, 80
Linear Functionals, 82
Dual Bases, 83
Reflexivity, 84
Annihilators, 86
Operator Adjoints, 88
Exercises, 90

4

Modules I: Basic Properties, 93
Modules, 93
Motivation, 93
Submodules, 95
Spanning Sets, 96
Linear Independence, 98
Torsion Elements, 99
Annihilators, 99
Free Modules, 99
Homomorphisms, 100
Quotient Modules, 101
The Correspondence and Isomorphism Theorems, 102
Direct Sums and Direct Summands, 102
Modules Are Not As Nice As Vector Spaces, 106
Exercises, 106

5

Modules II: Free and Noetherian Modules, 109
The Rank of a Free Module, 109
Free Modules and Epimorphisms, 114
Noetherian Modules, 115
The Hilbert Basis Theorem, 118
Exercises, 119

6

Modules over a Principal Ideal Domain, 121
Annihilators and Orders, 121

Cyclic Modules, 122
Free Modules over a Principal Ideal Domain, 123
Torsion-Free and Free Modules, 125

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Contents

Prelude to Decomposition: Cyclic Modules, 126
The First Decomposition, 127
A Look Ahead, 127
The Primary Decomposition, 128
The Cyclic Decomposition of a Primary Module, 130
The Primary Cyclic Decomposition Theorem, 134
The Invariant Factor Decomposition, 135
Exercises, 138
7

The Structure of a Linear Operator, 141
A Brief Review, 141
The Module Associated with a Linear Operator, 142
Orders and the Minimal Polynomial, 144
Cyclic Submodules and Cyclic Subspaces, 145
Summary, 147
The Decomposition of = , 147
The Rational Canonical Form, 148
Exercises, 151

8

Eigenvalues and Eigenvectors, 153
The Characteristic Polynomial of an Operator, 153
Eigenvalues and Eigenvectors, 155
Geometric and Algebraic Multiplicities, 157
The Jordan Canonical Form, 158
Triangularizability and Schur's Lemma, 160
Diagonalizable Operators, 165
Projections, 166
The Algebra of Projections, 167
Resolutions of the Identity, 170
Spectral Resolutions, 172
Projections and Invariance, 173
Exercises, 174

9

Real and Complex Inner Product Spaces , 181
Norm and Distance, 183
Isometries, 186
Orthogonality, 187
Orthogonal and Orthonormal Sets, 188
The Projection Theorem and Best Approximations, 192
Orthogonal Direct Sums, 194
The Riesz Representation Theorem, 195
Exercises, 196

10

Structure Theory for Normal Operators, 201

The Adjoint of a Linear Operator, 201

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xiii


xiv

Contents

Unitary Diagonalizability, 204
Normal Operators, 205
Special Types of Normal Operators, 207
Self-Adjoint Operators, 208
Unitary Operators and Isometries, 210
The Structure of Normal Operators, 215
Matrix Versions, 222
Orthogonal Projections, 223
Orthogonal Resolutions of the Identity, 226
The Spectral Theorem, 227
Spectral Resolutions and Functional Calculus, 228
Positive Operators, 230
The Polar Decomposition of an Operator, 232
Exercises, 234

Part II—Topics, 235
11

Metric Vector Spaces: The Theory of Bilinear Forms, 239

Symmetric, Skew-Symmetric and Alternate Forms, 239
The Matrix of a Bilinear Form, 242
Quadratic Forms, 244
Orthogonality, 245
Linear Functionals, 248
Orthogonal Complements and Orthogonal Direct Sums, 249
Isometries, 252
Hyperbolic Spaces, 253
Nonsingular Completions of a Subspace, 254
The Witt Theorems: A Preview, 256
The Classification Problem for Metric Vector Spaces, 257
Symplectic Geometry, 258
The Structure of Orthogonal Geometries: Orthogonal Bases, 264
The Classification of Orthogonal Geometries:
Canonical Forms, 266
The Orthogonal Group, 272
The Witt's Theorems for Orthogonal Geometries, 275
Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 277
Exercises, 279

12

Metric Spaces, 283
The Definition, 283
Open and Closed Sets, 286
Convergence in a Metric Space, 287
The Closure of a Set, 288

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Contents

Dense Subsets, 290
Continuity, 292
Completeness, 293
Isometries, 297
The Completion of a Metric Space, 298
Exercises, 303
13

Hilbert Spaces, 307
A Brief Review, 307
Hilbert Spaces, 308
Infinite Series, 312
An Approximation Problem, 313
Hilbert Bases, 317
Fourier Expansions, 318
A Characterization of Hilbert Bases, 328
Hilbert Dimension, 328
A Characterization of Hilbert Spaces, 329
The Riesz Representation Theorem, 331
Exercises, 334

14

Tensor Products, 337
Universality, 337
Bilinear Maps, 341
Tensor Products, 343

When Is a Tensor Product Zero? 348
Coordinate Matrices and Rank, 350
Characterizing Vectors in a Tensor Product, 354
Defining Linear Transformations on a Tensor Product, 355
The Tensor Product of Linear Transformations, 357
Change of Base Field, 359
Multilinear Maps and Iterated Tensor Products, 363
Tensor Spaces, 366
Special Multilinear Maps, 371
Graded Algebras, 372
The Symmetric Tensor Algebra, 374
The Antisymmetric Tensor Algebra:
The Exterior Product Space, 380
The Determinant, 387
Exercises, 391

15

Positive Solutions to Linear Systems:
Convexity and Separation 395
Convex, Closed and Compact Sets, 398
Convex Hulls, 399

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xv


xvi

Contents

Linear and Affine Hyperplanes, 400
Separation, 402
Exercises, 407
16

Affine Geometry, 409
Affine Geometry, 409
Affine Combinations, 41
Affine Hulls, 412
The Lattice of Flats, 413
Affine Independence, 416
Affine Transformations, 417
Projective Geometry, 419
Exercises, 423

17

Operator Factorizations: QR and Singular Value, 425
The QR Decomposition, 425
Singular Values, 428
The Moore–Penrose Generalized Inverse, 430
Least Squares Approximation, 433
Exercises, 434

18

The Umbral Calculus, 437
Formal Power Series, 437

The Umbral Algebra, 439
Formal Power Series as Linear Operators, 443
Sheffer Sequences, 446
Examples of Sheffer Sequences, 454
Umbral Operators and Umbral Shifts, 456
Continuous Operators on the Umbral Algebra, 458
Operator Adjoints, 459
Umbral Operators and Automorphisms
of the Umbral Algebra, 460
Umbral Shifts and Derivations of the Umbral Algebra, 465
The Transfer Formulas, 470
A Final Remark, 471
Exercises, 472

References, 473
Index, 475

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Chapter 0

Preliminaries

In this chapter, we briefly discuss some topics that are needed for the sequel.
This chapter should be skimmed quickly and used primarily as a reference.

Part 1 Preliminaries
Multisets
The following simple concept is much more useful than its infrequent

appearance would indicate.
Definition Let : be a nonempty set. A multiset 4 with underlying set : is a
set of ordered pairs
4 ~ ¸²  Á  ³ “



 :Á   {b Á



£



for  Ê ạ

where {b ~ á  ạ. The number  is referred to as the multiplicity of the
elements  in 4 . If the underlying set of a multiset is finite, we say that the
multiset is finite. The size of a finite multiset 4 is the sum of the multiplicities
of all of its elements. …
For example, 4 ~ ¸²Á ³Á ²Á ³Á ²Á ³¹ is a multiset with underlying set
: ~ á  ạ. The elements  has multiplicity . One often writes out the
elements of a multiset according to multiplicities, as in 4 ~ ¸Á Á Á Á Á ¹ .
Of course, two mutlisets are equal if their underlying sets are equal and if the
multiplicity of each element in the comon underlying set is the same in both
multisets.

Matrices
The set of  d  matrices with entries in a field - is denoted by CÁ ²- ³ or

by CÁ when the field does not require mention. The set CÁ ²< ³ is denoted
by C ²- ³ or C À If (  C, the ²Á ³-th entry of ( will be denoted by (Á .
The identity matrix of size  d  is denoted by 0 . The elements of the base

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2

Advanced Linear Algebra

field - are called scalars. We expect that the reader is familiar with the basic
properties of matrices, including matrix addition and multiplication.
The main diagonal of an  d  matrix ( is the sequence of entries
(Á Á (Á (
where  ~ miná ạ.
Definition The transpose of (  CÁ is the matrix (! defined by
²(! ³Á ~ (Á
A matrix ( is symmetric if ( ~ (! and skew-symmetric if (! ~ c(. …
Theorem 0.1 (Properties of the transpose) Let (, )  CÁ . Then
1) ²(! ³! ~ (
2) ²( b )³! ~ (! b ) !
3) ²(³! ~ (! for all   4) ²()³! ~ ) ! (! provided that the product () is defined
5) det²(! ³ ~ det²(³. …

Partitioning and Matrix Multiplication
Let 4 be a matrix of size  d . If ) á ạ and * á ạ then
the submatrix 4 ) *à is the matrix obtained from 4 by keeping only the
rows with index in ) and the columns with index in * . Thus, all other rows and
columns are discarded and 4 ´)Á *µ has size () ( d (* (.

Suppose that 4  CÁ and 5  CÁ . Let
1) F ~ ¸) Á à Á ) ¹ be a partition of ¸Á à Á ¹
2) G ~ ¸* Á Ã Á * ạ be a partition of á ạ
3) H ~ á+ + ạ be a partition of á ạ
(Partitions are defined formally later in this chapter.) Then it is a very useful fact
that matrix multiplication can be performed at the block level as well as at the
entry level. In particular, we have
´4 5 µ´) Á + µ ~  4 ´) Á * µ5 ´* Á + µ
*  G

When the partitions in question contain only single-element blocks, this is
precisely the usual formula for matrix multiplication


´4 5 µÁ ~ 4Á 5Á
~

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Preliminaries

3

Block Matrices
It will be convenient to introduce the notational device of a block matrix. If )Á
are matrices of the appropriate sizes then by the block matrix
4~

v )Á

Å
w )Á

)Á
Å
)Á

Ä

)Á y
Å
Ä )Á zblock

we mean the matrix whose upper left submatrix is )Á , and so on. Thus, the
)Á 's are submatrices of 4 and not entries. A square matrix of the form
v )
x 
4 ~x
Å
w 

 Ä  y
Ỉ Ỉ Å {
{
Ỉ Æ 
Ä  ) zblock

where each ) is square and  is a zero submatrix, is said to be a block
diagonal matrix.

Elementary Row Operations
Recall that there are three types of elementary row operations. Type 1
operations consist of multiplying a row of ( by a nonzero scalar. Type 2
operations consist of interchanging two rows of (. Type 3 operations consist of
adding a scalar multiple of one row of ( to another row of (.
If we perform an elementary operation of type  to an identity matrix 0 , the
result is called an elementary matrix of type  . It is easy to see that all
elementary matrices are invertible.
In order to perform an elementary row operation on (  CÁ we can perform
that operation on the identity 0 , to obtain an elementary matrix , and then take
the product ,(. Note that multiplying on the right by , has the effect of
performing column operations.
Definition A matrix 9 is said to be in reduced row echelon form if
1) All rows consisting only of 's appear at the bottom of the matrix.
2) In any nonzero row, the first nonzero entry is a . This entry is called a
leading entry.
3) For any two consecutive rows, the leading entry of the lower row is to the
right of the leading entry of the upper row.
4) Any column that contains a leading entry has 's in all other positions. …
Here are the basic facts concerning reduced row echelon form.

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4

Advanced Linear Algebra

Theorem 0.2 Matrices (Á )  CÁ are row equivalent, denoted by ( — ) ,
if either one can be obtained from the other by a series of elementary row

operations.
1) Row equivalence is an equivalence relation. That is,
a) ( — (
b) ( — ) ¬ ) — (
c) ( — ) , ) — * ¬ ( — * .
2) A matrix ( is row equivalent to one and only one matrix 9 that is in
reduced row echelon form. The matrix 9 is called the reduced row
echelon form of (. Furthermore,
( ~ , Ä, 9
where , are the elementary matrices required to reduce ( to reduced row
echelon form.
3) ( is invertible if and only if its reduced row echelon form is an identity
matrix. Hence, a matrix is invertible if and only if it is the product of
elementary matrices. …
The following definition is probably well known to the reader.
Definition A square matrix is upper triangular if all of its entries below the
main diagonal are . Similarly, a square matrix is lower triangular if all of its
entries above the main diagonal are . A square matrix is diagonal if all of its
entries off the main diagonal are . …

Determinants
We assume that the reader is familiar with the following basic properties of
determinants.
Theorem 0.3 Let (  CÁ ²- ³. Then det²(³ is an element of - . Furthermore,
1) For any )  C ²- ³,
det²()³ ~ det²(³det²)³
2) ( is nonsingular (invertible) if and only if det²(³ £ .
3) The determinant of an upper triangular or lower triangular matrix is the
product of the entries on its main diagonal.
4) If a square matrix 4 has the block diagonal form

v )
x 
4 ~x
Å
w 

 Ä  y
Ỉ Ỉ Å {
{
Ỉ Ỉ 
Ä  ) zblock

then det²4 ³ ~  det²) ³. …

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Preliminaries

5

Polynomials
The set of all polynomials in the variable % with coefficients from a field - is
denoted by - ´%µ. If ²%³  - ´%µ, we say that ²%³ is a polynomial over - . If
²%³ ~  b  % b Ä b  %
is a polynomial with  £  then  is called the leading coefficient of ²%³
and the degree of ²%³ is , written deg ²%³ ~ . For convenience, the degree
of the zero polynomial is cB. A polynomial is monic if its leading coefficient
is .
Theorem 0.4 (Division algorithm) Let  ²%³Á ²%³  - ´%µ where deg ²%³ € .

Then there exist unique polynomials ²%³Á ²%³  - ´%µ for which
 ²%³ ~ ²%³²%³ b ²%³
where ²%³ ~  or   deg ²%³  deg ²%³. …
If ²%³ divides ²%³, that is, if there exists a polynomial  ²%³ for which
²%³ ~  ²%³²%³
then we write ²%³ “ ²%³.
Theorem 0.5 Let  ²%³Á ²%³  - ´%µ. The greatest common divisor of  ²%³ and
²%³, denoted by gcd² ²%³Á ²%³³, is the unique monic polynomial ²%³ over for which
1) ²%³ “  ²%³ and ²%³ “ ²%³
2) if ²%³ “  ²%³ and ²%³ “ ²%³ then ²%³ “ ²%³.
Furthermore, there exist polynomials ²%³ and ²%³ over - for which
gcd² ²%³Á ²%³³ ~ ²%³ ²%³ b ²%³²%³

…

Definition The polynomials  ²%³Á ²%³  - ´%µ are relatively prime if
gcd² ²%³Á ²%³³ ~ . In particular,  ²%³ and ²%³ are relatively prime if and
only if there exist polynomials ²%³ and ²%³ over - for which
²%³ ²%³ b ²%³²%³ ~ 

…

Definition A nonconstant polynomial  ²%³  - ´%µ is irreducible if whenever
 ²%³ ~ ²%³²%³ then one of ²%³ and ²%³ must be constant. …
The following two theorems support the view that irreducible polynomials
behave like prime numbers.
Theorem 0.6 A nonconstant polynomial  ²%³ is irreducible if and only if it has
the property that whenever  ²%³ “ ²%³²%³ then either  ²%³ “ ²%³ or
 ²%³ “ ²%³. …

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6

Advanced Linear Algebra

Theorem 0.7 Every nonconstant polynomial in - ´%µ can be written as a product
of irreducible polynomials. Moreover, this expression is unique up to order of
the factors and multiplication by a scalar. …

Functions
To set our notation, we should make a few comments about functions.
Definition Let  Â : Ư ; be a function from a set : to a set ; .
1) The domain of  is the set : .
2) The image or range of  is the set im ~ á :ạ.
3)  is injective (one-to-one), or an injection, if % £ & ¬  ²%³ £  ²&³.
4)  is surjective (onto ; ), or a surjection, if im² ³ ~ ; .
5)  is bijective, or a bijection, if it is both injective and surjective.
6) Assuming that   ; , the support of  is
supp² ³ ~ ¸  :  Ê ạ



If  Â : ¦ ; is injective then its inverse  c ¢ im² ³ ¦ : exists and is welldefined as a function on im² ³.
It will be convenient to apply  to subsets of : and ; . In particular, if ? ‹ :
and if @ ‹ ; , we set
 ? ~ á % % ?ạ
and
 c @ ³ ~ ¸  : “  ² ³  @ ¹

Note that the latter is defined even if  is not injective.
Let   : Ư ; . If ( ‹ : , the restriction of  to ( is the function  O(  ( Ư ;
defined by
 O( ²³ ~  ²³
for all   (. Clearly, the restriction of an injective map is injective.

Equivalence Relations
The concept of an equivalence relation plays a major role in the study of
matrices and linear transformations.
Definition Let : be a nonempty set. A binary relation — on : is called an
equivalence relation on : if it satisfies the following conditions:

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Preliminaries

7

1) (Reflexivity)
—
for all   : .
2) (Symmetry)
—¬—
for all Á   : .
3) (Transitivity)
 — Á  —  ¬  — 
for all Á Á   : . …
Definition Let — be an equivalence relation on : . For   : , the set of all
elements equivalent to  is denoted by

à ~ á : “  — ¹
and called the equivalence class of . …
Theorem 0.8 Let — be an equivalence relation on : . Then
1)   ´µ ¯   ´µ ¯ ´µ ~ ´µ
2) For any Á   : , we have either ´µ ~ ´µ or ´µ q ´µ ~ J. …
Definition A partition of a nonempty set : is a collection á( ( ạ of
nonempty subsets of : , called the blocks of the partition, for which
1) ( q ( ~ J for all  £ 
2) : ~ ( r Ä r ( . …
The following theorem sheds considerable light on the concept of an
equivalence relation.
Theorem 0.9
1) Let — be an equivalence relation on : . Then the set of distinct equivalence
classes with respect to — are the blocks of a partition of : .
2) Conversely, if F is a partition of : , the binary relation — defined by
 —  if  and  lie in the same block of F
is an equivalence relation on : , whose equivalence classes are the blocks
of F .
This establishes a one-to-one correspondence between equivalence relations on
: and partitions of : . …
The most important problem related to equivalence relations is that of finding an
efficient way to determine when two elements are equivalent. Unfortunately, in

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8

Advanced Linear Algebra

most cases, the definition does not provide an efficient test for equivalence and
so we are led to the following concepts.
Definition Let — be an equivalence relation on : . A function   : Ư ; , where
; is any set, is called an invariant of — if it is constant on the equivalence
classes of — , that is,
 —  ¬  ²³ ~  ²³
and a complete invariant if it is constant and distinct on the equivalence
classes of — , that is,
 —  ¯  ²³ ~  ²³
A collection ¸ Á Ã Á  ¹ of invariants is called a complete system of
invariants if
 —  ¯  ²³ ~  ²³ for all  ~ Á à Á 

…

Definition Let — be an equivalence relation on : . A subset * ‹ : is said to be
a set of canonical forms (or just a canonical form) for — if for every  : ,
there is exactly one   * such that  — . Put another way, each equivalence
class under — contains exactly one member of * . …
Example 0.1 Define a binary relation — on - ´%µ by letting ²%³ — ²%³ if and
only if ²%³ ~ ²%³ for some nonzero constant   - . This is easily seen to be
an equivalence relation. The function that assigns to each polynomial its degree
is an invariant, since
²%³ — ²%³ ¬ deg²²%³³ ~ deg²²%³³
However, it is not a complete invariant, since there are inequivalent polynomials
with the same degree. The set of all monic polynomials is a set of canonical
forms for this equivalence relation. …
Example 0.2 We have remarked that row equivalence is an equivalence relation
on CÁ ²- ³. Moreover, the subset of reduced row echelon form matrices is a
set of canonical forms for row equivalence, since every matrix is row equivalent

to a unique matrix in reduced row echelon form. …
Example 0.3 Two matrices (, )  C ²- ³ are row equivalent if and only if
there is an invertible matrix 7 such that ( ~ 7 ) . Similarly, ( and ) are
column equivalent, that is, ( can be reduced to ) using elementary column
operations if and only if there exists an invertible matrix 8 such that ( ~ )8.
Two matrices ( and ) are said to be equivalent if there exist invertible
matrices 7 and 8 for which

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Preliminaries

9

( ~ 7 )8
Put another way, ( and ) are equivalent if ( can be reduced to ) by
performing a series of elementary row and/or column operations. (The use of the
term equivalent is unfortunate, since it applies to all equivalence relations, not
just this one. However, the terminology is standard, so we use it here.)
It is not hard to see that an  d  matrix 9 that is in both reduced row echelon
form and reduced column echelon form must have the block form
1 ~ >

0
cÁ

Ác
cÁc ?block

We leave it to the reader to show that every matrix ( in C is equivalent to
exactly one matrix of the form 1 and so the set of these matrices is a set of
canonical forms for equivalence. Moreover, the function  defined by
 ²(³ ~  , where ( — 1 , is a complete invariant for equivalence.
Since the rank of 1 is  and since neither row nor column operations affect the
rank, we deduce that the rank of ( is  . Hence, rank is a complete invariant for
equivalence. In other words, two matrices are equivalent if and only if they have
the same rank. …
Example 0.4 Two matrices (, )  C ²- ³ are said to be similar if there exists
an invertible matrix 7 such that
( ~ 7 )7 c
Similarity is easily seen to be an equivalence relation on C . As we will learn,
two matrices are similar if and only if they represent the same linear operators
on a given -dimensional vector space = . Hence, similarity is extremely
important for studying the structure of linear operators. One of the main goals of
this book is to develop canonical forms for similarity.
We leave it to the reader to show that the determinant function and the trace
function are invariants for similarity. However, these two invariants do not, in
general, form a complete system of invariants. …
Example 0.5 Two matrices (, )  C ²- ³ are said to be congruent if there
exists an invertible matrix 7 for which
( ~ 7 )7 !
where 7 ! is the transpose of 7 . This relation is easily seen to be an equivalence
relation and we will devote some effort to finding canonical forms for
congruence. For some base fields - (such as s, d or a finite field), this is
relatively easy to do, but for other base fields (such as r), it is extremely
difficult. …

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10

Advanced Linear Algebra

Zorn's Lemma
In order to show that any vector space has a basis, we require a result known as
Zorn's lemma. To state this lemma, we need some preliminary definitions.
Definition A partially ordered set is a pair ²7 Á  ³ where 7 is a nonempty set
and  is a binary relation called a partial order, read “less than or equal to,”
with the following properties:
1) (Reflexivity) For all   7 ,

2) (Antisymmetry) For all Á   7 ,
   and    implies  ~ 
3) (Transitivity) For all Á Á   7 ,
   and    implies   
Partially ordered sets are also called posets. …
It is customary to use a phrase such as “Let 7 be a partially ordered set” when
the partial order is understood. Here are some key terms related to partially
ordered sets.
Definition Let 7 be a partially ordered set.
1) A maximal element is an element   7 with the property that there is no
larger element in 7 , that is
  7Á   ¬  ~ 
2) A minimal element is an element   7 with the property that there is no
smaller element in 7 , that is
  7Á   ¬  ~ 
3) Let Á   7 . Then "  7 is an upper bound for  and  if
  " and   "

The unique smallest upper bound for  and , if it exists, is called the least
upper bound of  and  and is denoted by lubá ạ.
4) Let   7 . Then M  7 is a lower bound for  and  if
M   and M  
The unique largest lower bound for  and , if it exists, is called the
greatest lower bound of  and  and is denoted by glbá ạ.

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