Unformatted text preview: Undergraduate Texts in Mathematics
Editors S.Axler
FWo Gehring
K.A. Ribet Springer Science+Business Media, LLC Undergraduate Texts in Mathematics
Anglin: Mathematics: A Concise History
and Philosophy.
Readings in Mathematics.
Anglin/Lambek: The Heritage of
Thales.
Readings in Mathematics.
Apostol: Introduction to Analytic
Number Theory. Second edition.
Armstrong: Basic Topology.
Armstrong: Groups and Symmetry.
Axler: Linear Algebra Done Right.
Second edition.
Beardon: Limits: A New Approach to
Real Analysis.
Bak/Newman: Complex Analysis.
Second edition.
Banchoff/Wermer: Linear Algebra
Through Geometry. Second edition.
Berberian: A First Course in Real
Analysis.
Bix: Comics and Cubics: A Concrete
Introduction to Algebraic Curves.
Readings in Mathematics.
Bremaud: An Introduction to
Probabilistic Modeling.
Bressoud: Factorization and Primality
Testing.
Bressoud: Second Year Calculus.
Readings in Mathematics.
Brickman: Mathematical Introduction
to Linear Programming and Game
Theory.
Browder: Mathematical Analysis:
An Introduction.
Buskes/van Rooij: Topological Spaces:
From Distance to Neighborhood.
Cederberg: A Course in Modem
Geometries.
Childs: A Concrete Introduction to
Higher Algebra. Second edition.
Chung: Elementary Prob ability Theory
with Stochastic Processes. Third
edition.
Cox/Little/O'Shea: Ideals, Varieties,
and Algorithms. Second edition.
Croom: Basic Concepts of Algebraic
Topology.
Curtis: Linear Algebra: An Introductory
Approach. Fourth edition. DevIin: The Joy of Sets: Fundamentals
of Contemporary Set Theory.
Second edition.
Dixmier: General Topology.
Driver: Why Math?
Ebbinghaus/Flum/Thomas:
Mathematical Logic. Second edition.
Edgar: Measure, Topology, and Fractal
Geometry.
Elaydi: Introduction to Difference
Equations.
Exner: An Accompaniment to Higher
Mathematics.
Fine/Rosenberger: The Fundamental
Theory of Algebra.
Fischer: Intermediate Real Analysis.
Flanigan/Kazdan: Ca1culus Two: Linear
and Nonlinear Functions. Second
edition.
Fleming: Functions of Several Variables.
Second edition.
Foulds: Combinatorial Optimization for
Undergraduates.
Foulds: Optimization Techniques: An
Introduction.
FrankIin: Methods of Mathematical
Economics.
Gordon: Discrete Probability.
Hairer/Wanner: Analysis by Its History.
Readings in Mathematics.
Halmos: FiniteDimensional Vector
Spaces. Second edition.
Halmos: Naive Set Theory.
Hämmerlin/Hoffmann: Numerical
Mathematics.
Readings in Mathematics.
Hijab: Introduction to Calculus and
Classical Analysis.
Hilton/Holton/Pedersen: Mathematical
Reflections: In a Room with Many
Mirrors.
Iooss/Joseph: Elementary Stability
and Bifurcation Theory. Second
edition.
Isaac: The Pleasures of Probability.
Readings in Mathematics.
(continued after index) LarrySmith Linear Algebra
Third Edition With 23 Illustrations , Springer LarrySmith
Mathematisches Institut
Universität Göttingen
Bunsenstrasse 35
Gottingen, D37073
Germany
Editorial Board S.Axler
Mathematics
Department
San Francisco State
University
San Francisco, CA 94132
USA FW. Gehring
Mathematics
Department
EastHall
University ofMichigan
AnnArbor, MI 48109
USA K.A. Ribet Department of
Mathematics
Universi ty of California
at Berkeley
Berkeley, CA 947203840
USA Mathematics Subject Classification (1991): 1501 LibraryofCongress CataloginginPublication Data
Smith, Larry.
Linear Algebra / Larry Smith.  3rd. ed.
cm.  (Undergraduate texts in mathematics)
p.
Inc1udes bibliographical references and index.
ISBN 9781461272380
ISBN 9781461216704 (eBook)
DOI 10.1007/9781461216704
1. Algebras, Linear. 1. Title. II. Series.
QA184.S63 1998
512 .5dc21
9816278 3rd Printed on acidfree paper.
© 1978, 1984, 1998 Springer Science+Business Media New York
Originally published by SpringerVerlag New York, Ine. in 1998
Softcover reprint ofthe hardcover 3rd edition 1998
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Typeset by the author using f..S TEX. 987654321
ISBN 9781461272380 Fri,\ ·ch Wilhelm/.B~~sel (17841846) (KOnigsberg, Prussia)
;~t; Lewis C~Jlj(l832;}898) (Oxford, England)
1\~stin Cauc9~{(if7891857)(Paris, France)
Artliu.fCayley ('t~21,:r1895) (Cambridge, England)
Ga~~~l C~~erh70tt1752)(Switzerland)
Rent~~~cifrtes (1596l'~650) (Paris, France)
Euclid of Alexaftdria (365 B~Qt:lBC) (Alexandria, Asia Minor)
Joseph Fouri~J;;~:lf6s.:..1830)(Paris, France)
Abraha4j,~
. ". (16§,'l,7""1744) (London, England)
Jcergen Pede~ ram (185Q+l91 ,Copenhagen, Denmark)
William Rowan Hamilton.~~~~5) (Dublin, Ireland)
Charles Hermi
.221901} (Paris, France)
Camille JQJ!~'
8381922) (Paris, France)
Joseph Louis',~grange(1~~.~J,~~~) (Turin, Italy)
Adrien Marie Legenare (17&~1~~~) (mUlo se, Paris, France,
Berlit1\ GeJ:itany)
MarcAntoine des Chenes Par~e'V81 (17551
) (Paris, France)
. ,Hungary)
Friedrich Riesz (18801956) (.0
1.0. Rodrigues (born at the dohcentury) (Paris, France)
P.F. Sarms (born a
d of thQ.,"leth century)
(Perpignan, trasbourg, Ftahce
Erhard Schmidt (18761959) (Ber .
'~y)
Hermann Amandus Schwarz (184
. (Halle, G6ttingen,
Berlin, Ge
y)
James Joseph Sylvester (18141J)7) (UniveI:§jtyl){Vrrginia,
Charlottesville, VA, Johns HopkinaiUnivers~~8altimo MD) Preface This text was originally written for a one semester course in linear algebra at the (U.S.) sophomore undergraduate level, preferably directly
following a one variable calculus course, so that linear algebra could
be used in a course on multidimensional calculus and/or differential
equations. Students at this level generally have had little contact with
complex numbers or abstract mathematics, so the book deals almost exclusively with real finitedimensional vector spaces, but in a setting and
formulation that permits easy generalization to abstract vector spaces.
The parallel complex theory is developed in part in the exercises.
The goal of the first two editions was the principal axis theorem for
real symmetric linear transformations. Twenty years of teaching in
Germany, where linear algebra is a one year course taken in the first
year of study at the university, has modified that goal. The principal
axis theorem becomes the first of two goals, and to be achieved as
originally planned in one semester, a more or less direct path is followed
to its proof. As a consequence there are many subjects that are not
developed, and this is intentional: this is only an introduction to linear
algebra. As compensation, a wide selection ofexamples of vector spaces
and linear transformations is presented, to serve as a testing ground
for the theory. Students with a need to learn more linear algebra
can do so in a course in abstract algebra, which is the appropriate
setting. Through this book they will be taken on an excursion to the
algebraic/analytic zoo, and introduced to some of the animals for the
first time. Further excursions can teach them more about the curious
habits of some of these remarkable creatures. In the second edition of the book I added, among other things, a safari
into the wilderness of canonical forms, where the hardy student could vii viii Preface pursue the Jordan form, which has become the second goal ofthis book,
with the tools developed in the preceding chapters. In this edition
I have added the tip of the iceberg of invariant theory to show that
linear algebra alone is not capable of solving these canonical forms
problems, even in the simplest case of 2 x 2 complex matrices.
Gottingen, Germany, February 1998 Larry Smitfi Contents Preface vii 1. Vectors in the Plane and in Space
1.1 First Steps
1.2 Exercises 1
1
12 2. Vector Spaces 15 2.1 Axioms for Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Cartesian (or Euclidean) Spaces.......................
2.3 Some Rules for Vector Algebra
2.4 Exercises 15
18
21
22 3. Examples of Vector Spaces 25 4. Subspaces 35 5. Linear Independence and Dependence 47 3.1 Three Basic Examples
3.2 Further Examples of Vector Spaces...................
3.3 Exercises
. . . . . . . . .. 4.1 Basic Properties of Vector Subspaces
4.2 Examples of Subspaces
4.3 Exercises 25
27
30 35
41
42 5.1 Basic Definitions and Examples....................... 47
5.2 Properties of Independent and Dependent Sets
50
5.3 Exercises
53
ix x Contents 6. FiniteDimensional Vector Spaces and Bases 57
6.1 FiniteDimensional Vector Spaces.....................
6.2 Properties of Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
6.3 Using Bases...........................................
6.4 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. The Elements of Vector Spaces:
A Summing Up 57
61
65
71 75 7.1 Numerical Examples.................................. 75
7.2 Exercises
82 8. Linear Transformations 8.1 Definition of Linear Transformations
8.2 Examples of Linear Transformations. . . . . . . . . . . . . . . . ..
8.3 Properties of Linear Transformations
8.4 Images and Kernels of Linear Transformations
8.5 Some Fundamental Constructions
8.6 Isomorphism of Vector Spaces
8.7 Exercises 9. Linear Transformations:
Examples and Applications
9.1 Numerical Examples
9.2 Some Applications
9.3 Exercises 85 85
89
91
94
98
102
109 113 113
123
124 10. Linear Transformations and Matrices 129 11. Representing Linear Transformations
by Matrices 159 12. More on Representing
Linear Transformations by Matrices 185 10.1 Linear Transformations and Matrices in m.3
10.2 Some Numerical Examples
10.3 Matrices and Their Algebra
10.4 Special Types of Matrices
10.5 Exercises 129
134
136
141
151 11.1 Representing a Linear Transformation by a Matrix .. 159
11.2 Basic Theorems
165
174
11.3 Change of Bases
11.4 Exercises
178 12.1 Projections 185 Contents xi 191
12.2 Nilpotent Transfonnations
193
12.3 Cyclic Transfonnations
12.4 Exercises .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 195 13. Systems of Linear Equations 199 14. The Elements of Eigenvalue and
Eigenvector Theory 227 15. Inner Product Spaces 267 16. The Spectral Theorem and Quadratic
Forms 307 17. Jordan Canonical Form 343 13.1 Existence Theorems
13.2 Reduction to Echelon Fonn
13.3 The Simplex Method
13.4 Exercises 14.1 The Rank of an Endomorphism
14.2 Eigenvalues and Eigenvectors
14.3 Detenninants
14.4 The Characteristic Polynomial
14.5 Diagonalization Theorems
14.6 Exercises 15.1 Scalar Products
15.2 Inner Product Spaces
15.3 Isometries
15.4 The Riesz Representation Theorem
15.5 Legendre Polynomials
15.6 Exercises 16.1 SelfAdjoint Transfonnations
16.2 The Spectral Theorem
16.3 The Principal Axis Theorem for Quadratic Fonns
16.4 A Proof of the Spectral Theorem in the General Case
16.5 Exercises 17.1 Invariant Subspaces
17.2 Nilpotent Transfonnations
17.3 The Jordan Nonnal Fonn
17.4 Square Roots
17.5 The HamiltonCayley Theorem
17.6 Inverses
17.7 Exercises 201
209
216
224 227
230
238
245
253
260 268
274
288
291
298
301 308
316
324
335
338 345
350
357
372
374
376
377 xii Contents 18. Application to Differential Equations 381 19. The Similarity Problem 405 18.1 Linear Differential Systems: Basic Definitions
18.2 Diagonalizable Systems
18.3 Application of Jordan Form
18.4 Exercises 19.1 The Fundamental Problem of Linear Algebra
19.2 A Bit of Invariant Theory
19.3 Exercises 381
386
395
402 405
406
409 A. Multilinear Algebra and Determinants 411 B. Complex Numbers 433 A.1 Multilinear Forms
411
415
A.2 Determinants
A.3 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 428 B.1 The Complex Numbers
B.2 Exercises Font Usage
Notations
Index 433
441 443
445
447 Chapter 1
Vectors in the Plane and in Space In physics certain quantities such as force, displacement, velocity, and
acceleration possess both a magnitude and a direction and they are
most usually represented geometrically by drawing an arrow with the
magnitude and direction of the quantity in question. Physicists refer
to the arrow as a vector, and call the quantity so represented a vector
quantity. In the study of the calculus the student has no doubt also
encountered what are called vectors, particularly in connection with
the study of lines and planes and the differential geometry of space
curves. We begin by reviewing these ideas and codifying the algebra of
vectors.
1.1 First Steps Vectors as they appear in physics and the study of curves and surfaces
can be described as ordered pairs of points (P, Q) which we call the
vector from P to Q and often denote by PQ. This is substantially
the same as the physics definition, since all it amounts to is a technical
description of the word "arrow". P is called the initial point and Q
the terminal point.
For our purposes it will be convenient to regard two vectors as being
equal if they have the same magnitude and the same direction. In
other words, we will regard PQ and RS as determining the same vector
ifRS results by moving PQ parallel to itself.
N.B. Vectors that conform to this definition are called free vectors, since
we are "free to pick" their initial point. Not all "vectors" that occur in
nature conform to this convention. If the vector quantity depends not
1
L. Smith, Linear Algebra
© Springer Science+Business Media New York 1998 2 1. Vectors in the Plane and in Space only on its direction and magnitude but as well on its initial point it is
called a bound vector. For Example, torque is a bound vector. In the
force vector diagram represented by Figure 1.1.1 PQ does not have the
same effect as RS in pivoting a bar. In this book we will consider only
free vectors.
p R Q e 11 ' s Figure 1.1.1
With this convention of equality of vectors in mind it is clear that if
we fix a point 0 in space called the origin, then we may regard all
our vectors as having their initial point at O. The vector OP will very
>
often be abbreviated to P if the point 0 which serves as the origin of
>
all vectors is clear from context. The vector P is called the position
vector of the point P relative to the origin O.
In physics vector quantities such as force vectors are often added together to obtain a resultant force vector. This process may be described
as follows. Suppose an origin 0 has been fixed. Given vectors P and ij
their sum is defined by the Figure 1.1.2. That is, draw the parallelogram determined by the three points P, 0 and Q. Let R be the fourth
>
>
>
vertex and set P + Q = R.
P o b?7
Q R Q Figure 1.1.2
The following basic rules of vector algebra may be easily verified by
elementary Euclidean geometry.
~ ~ 7 ~ (1) P+Q=Q+P.
~7 7777 (2) (P + Q) + R = P + (Q + R).
(3) P+ 0 =P = 0 + P. It is also possible to define the operation of multiplying a vector by a
number. Suppose we are given a vector P and a number a. If a> 0, we 1.1 First Steps
~ 3 ~ let aP be the vector with the same direction as P only a times as long
(see Figure 1.1.3).
1.5P o
Figure 1.1.3
~ If a < 0, we set aP equal to the vector of magnitude a times the magnitude ofP but having direction opposite to that ofP (see Figure 1.1.4).
~ ~ Figure 1.1.4
If a =0 we set aP equal to O. It is then easy to show that vector algebra
satisfies the following additional rules.
(4) P + (lP) =O.
~ ~ ~ ~ ~ = aP + aQ.
(a + b)P = aP+ bP.
~ ~ ~ (5) a(P + Q)
(6) ~ ~ ~ ~ ~ ~ (7) (a b)P = a(bP).
7 (8) OP = 0, 1P =P.
7 7 7 Note that Rule 6 involves two types of addition, namely addition of
numbers and addition of vectors.
Vectors are particularly useful in studying lines and planes in space.
Suppose that an origin 0 has been fixed and L is the line through the
two points P and Q as in Figure 1.1.5. Suppose that R is any other
point on L. Consider the position vector R. Since the two points P, Q
completely determine the line L, it is quite reasonable to look for some
~ 4 1. Vectors in the Plane and in Space
7 7 7 relation among the vectors P, Q, and R.
L o.,.:::... ~ Figure 1.1.5
One such relation is provided by Figure 1.1.6.
L o Figure 1.1.6
Observe that 7 7 7 S+P=Q. 7 7 Therefore, if we write P for (I)P, we see that
7 7 7 S=QP. 1.1 First Steps 5 Notice that there is a number t such that
> > T= tS.
Moreover,
> > > R=P+T,
and hence we find
~ R =P+ t(QP).
7 77 Equation (*) is called the vector equation of the line L. To make
practical computations with this equation it is convenient to introduce
in addition to the origin 0 a Cartesian coordinate system as in Figure
1.1.7. Every point P then has coordinates (x, Y, z), and if we have two
points P and Q with coordinates (xp, yp, zp) and (xQ, YQ, zQ) then it is
> '> quite easy to check that the vector P + Q is the position vector of the
point with components (xp + xQ> yp + YQ, Zp + zQ).
z y x Figure 1.1.7
'> Likewise, for a number a the vector aP is the position vector of the
point with coordinates (axp, ayp, azp). Thus we find by considering
the coordinates of the points represented equation (*) that (x, Y, z) lies
on the line L through P, Q if and only if there is a number t such that
x = Xp + t(xQ xp), Y = yp+t(YQYp),
Z = Zp + t(zQ  zp). EXAMPLE 1: Does the point (1, 2, 3) lie on the line passing through
the points (4, 4, 4) and (1, 0, I)? 6 1. Vectors in the Plane and in Space SOLUTION: Let L be the line through P = (4, 4, 4) and Q = (1, 0, 1).
Then the points of L must satisfy the equations x = 4+ t(14) = 43t, =4+ t(04) = 44t,
z =4+ t(14) =43t, y where t is a number. Let us check whether this is possible: namely,
does there exist a number t such that 1 =43t,
2 = 44t,
3=43t. The first equation gives 3 = 3t t = 1. Putting this in the last equation gives 3 = 43 = 1,
which is impossible. Therefore (1,2,3) does not lie on the line through
(4,4,4) and (1, 0, 1).
EXAMPLE 2: Let L 1 be the line through the points (1, 0, 1) and
(1,1, 1). Let L 2 be the line through the points (0,1,0) and (1,2,1). Determine whether the lines L 1 and L 2 intersect. If so find their point
of intersection.
SOLUTION: The equations of L 1 are
x = 1 + t1(1 1) = 1, y = 0 + t1(1 0) = t1, z = 1 + t1(1 1) = 1.
The equations of L 2 are =0 + (1 0)t2 = t2,
y = 1 + (2  1)t2 = 1 + t2,
z =0 + (1 0)t2 =t2·
x Ifa point lies on both of these lines we must have
1 = t2, =1 + t2,
1 = t2. t1 1.1 First Steps 7 Therefore t2 = 1 and tl = 2. Hence (1,2,1) is the only point these lines
have in common.
EXAMPLE 3: Determine whether the lines L 1 and L 2 with equations
X L1 { y z = 1 3t,
= 1 + 3t,
= t, =23t,
=4+3t,
z = 1 + t, X L2 { y have a point in common.
SOLUTION: If a point (x, y, z) lies on both lines, it must satisfy
both sets of equations, so there is a number tl such that = 1 3tt,
Y = 1 + 3tt,
x and a number t2 with x = 23t2,
Y =4+ 3t2, Z = 1 + t2, and the answer to the problem is reduced to determining whether in fact
two such numbers can be found, that is if the simultaneous equations =2  3t2,
1 + 3tl =4 + 3t2,
tl = 1 + t2,
1  3tl have any solutions. Writing these equations in the more usual form
they become
3 =3tl3t2, =3tl + 3t2,
1 =tl + t2. 3 By dividing the first equation by 3, the second by 3, and multiplying
the third by 1 we get
1 = tl  t2,
1 = tl  t2, 1 = tl  t2, 8 1. Vectors in the Plane and in Space giving as the only requirement on tl and t2 that = 1 + t2...
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