Linear algebra (3rd ed)

Students at this level generally have had little contact withcomplex numbers or abstract mathematics, so the book deals almost ex-clusively with real finite-dimensional vector spaces, bu

Undergraduate Texts in Mathematics

Editors

S.Axler

FWo Gehring K.A Ribet

Springer Science+Business Media, LLC

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

Apostol: Introduction to Analytic

Number Theory Second edition

Armstrong: Basic Topology

Armstrong: Groups and Symmetry

Axler: Linear Algebra Done Right

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

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

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

Mathematics Subject Classification (1991): 15-01

LibraryofCongress Cataloging-in-Publication Data

K.A Ribet Department of Mathematics Universi ty of California

at Berkeley Berkeley, CA 94720-3840 USA

Linear Algebra / Larry Smith - 3rd ed

p cm - (Undergraduate texts in mathematics)

Inc1udes bibliographical references and index

ISBN 978-1-4612-7238-0 ISBN 978-1-4612-1670-4 (eBook)

DOI 10.1007/978-1-4612-1670-4

1 Algebras, Linear 1 Title II Series

QA184.S63 1998

Printed on acid-free paper

© 1978, 1984, 1998 Springer Science+Business Media New York

Originally published by Springer-Verlag New York, Ine in 1998

Softcover reprint ofthe hardcover 3rd edition 1998

All rights reserved This work may not be translated or copied in whole or in part wi thout thewrittenpermissionofthepublisher Springer Scienee+Business Media, LLC,

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as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone

Production managed by Anthony Guardiola; manufacturing supervised by JacquiAshri Typeset by the author using f S TEX

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ISBN 978-1-4612-7238-0

Fri,\ ·chWilhelm/.B~~sel(1784 1846) (KOnigsberg, Prussia)

;~t; Lewis C~Jlj(l832-;}898) (Oxford, England)

1\~stin Cauc9~{(if789 1857)(Paris, France)

Artliu.fCayley('t~21,:r1895)(Cambridge, England)

Ga~~~l C~~erh70tt1752)(Switzerland)

Rent~~~cifrtes (1596l'~650)(Paris, France)

Euclid of Alexaftdria (365B~Qt:lBC)(Alexandria, Asia Minor)

JosephFouri~J;;~:lf6s.: 1830)(Paris, France)

Abraha4j,~ " (16§,'l,7""1744) (London, England)JcergenPede~ ram (185Q+l91 ,Copenhagen, Denmark)William RowanHamilton.~~~~5)(Dublin, Ireland)Charles Hermi 22-1901} (Paris, France)

CamilleJQJ!~' 838-1922) (Paris, France)

JosephLouis',~grange(1~~.~J,~~~)(Turin, Italy)

Adrien Marie Legenare(17&~1~~~)(mUlo se, Paris, France,

Berlit1\ GeJ:itany)Marc-Antoine des ChenesPar~e'V81(1755-1 ) (Paris, France)Friedrich Riesz (1880-1956)(.0 ,Hungary)

1.0 Rodrigues (born at the doh-century) (Paris, France)

P.F Sarms (born a d of thQ.,"leth -century)

(Perpignan, trasbourg, FtahceErhard Schmidt (1876-1959) (Ber '~y)

Hermann Amandus Schwarz (184 (Halle, G6ttingen,

Berlin, Ge y)James Joseph Sylvester (1814 1J)7) (UniveI:§jtyl){Vrrginia,Charlottesville, VA, JohnsHopkinaiUnivers~~8altimo MD)

This text was originally written for a one semester course in linear gebra at the (U.S.) sophomore undergraduate level, preferably directlyfollowing a one variable calculus course, so that linear algebra could

al-be used in a course on multidimensional calculus and/or differentialequations Students at this level generally have had little contact withcomplex numbers or abstract mathematics, so the book deals almost ex-clusively with real finite-dimensional vector spaces, but in a setting andformulation 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 forreal symmetric linear transformations Twenty years of teaching inGermany, where linear algebra is a one year course taken in the firstyear of study at the university, has modified that goal The principalaxis theorem becomes the first of two goals, and to be achieved asoriginally planned in one semester, a more or less direct path is followed

to its proof As a consequence there are many subjects that are notdeveloped, and this is intentional: this is only an introduction to linearalgebra As compensation, a wide selection ofexamples ofvector spacesand linear transformations is presented, to serve as a testing groundfor the theory Students with a need to learn more linear algebracan do so in a course in abstract algebra, which is the appropriatesetting Through this book they will be taken on an excursion to thealgebraic/analytic zoo, and introduced to some of the animals for thefirst time Further excursions can teach them more about the curioushabits of some of these remarkable creatures

Inthe second edition of the book I added, among other things, a safariinto the wilderness of canonical forms, where the hardy student could

vii

Gottingen, Germany, February 1998 Larry Smitfi

3.2 Further Examples of Vector Spaces 27

5 Linear Independence and Dependence 47

5.1 Basic Definitions and Examples 475.2 Properties of Independent and Dependent Sets 50

ix

x Contents

6 Finite-Dimensional Vector Spaces and Bases 57

6.1 Finite-Dimensional Vector Spaces 57

6.2 Properties of Bases 61

6.3 Using Bases 65

6.4 Exercises 71

7 The Elements of Vector Spaces: A Summing Up 75 7.1 Numerical Examples 75

7.2 Exercises 82 8 Linear Transformations 85 8.1 Definition of Linear Transformations 85 8.2 Examples of Linear Transformations 89

8.3 Properties of Linear Transformations 91 8.4 Images and Kernels of Linear Transformations 94 8.5 Some Fundamental Constructions 98 8.6 Isomorphism of Vector Spaces 102 8.7 Exercises 109 9 Linear Transformations: Examples and Applications 113 9.1 Numerical Examples 113 9.2 Some Applications 123 9.3 Exercises 124 10 Linear Transformations and Matrices 129 10.1 Linear Transformations and Matrices inm.3 129 10.2 Some Numerical Examples 134 10.3 Matrices and Their Algebra 136 10.4 Special Types of Matrices 141 10.5 Exercises 151 11 Representing Linear Transformations by Matrices 159 11.1 Representing a Linear Transformation by a Matrix 159

12 More on Representing

Linear Transformations by Matrices 185

14 The Elements of Eigenvalue and

16.3 The Principal Axis Theorem for Quadratic Fonns 32416.4 A Proof of the Spectral Theorem in the General Case 335

xii Contents

18 Application to Differential Equations 381

18.1 Linear Differential Systems: Basic Definitions 381

19.1 The Fundamental Problem of Linear Algebra 405

A Multilinear Algebra and Determinants 411

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 themagnitude 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 withthe study of lines and planes and the differential geometry of spacecurves We begin by reviewing these ideas and codifying the algebra ofvectors

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 Qand often denote byPQ. This is substantiallythe same as the physics definition, since all it amounts to is a technicaldescription 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 beingequal if they have the same magnitude and the same direction Inother words, we will regardPQand RS as determining the same vector

ifRS results by movingPQparallel 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 innature conform to this convention Ifthe 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 iscalled a bound vector. For Example, torque is a bound vector In theforce vector diagram represented by Figure 1.1.1PQdoes not have thesame effect asRSin pivoting a bar In this book we will consider onlyfree vectors

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 allour vectors as having their initial point at O The vectorOPwill very

to-as follows Suppose an origin 0 hto-as been fixed Given vectorsPandij

their sum is defined by the Figure 1.1.2 That is, draw the gram determined by the three pointsP, 0 and Q Let R be the fourth

point on L Consider the position vectorR Since the two pointsP, Q

completely determine the lineL,it is quite reasonable to look for some

4 1 Vectors in the Plane and in Space

1.1 First Steps 5Notice that there is a numbertsuch that

on the lineL throughP,Qif and only if there is a numbertsuch that

6 1 Vectors in the Plane and in Space

SOLUTION: LetL be the line throughP =(4, 4, 4) and Q=(1,0, 1).Then the points ofL must satisfy the equations

EXAMPLE2: 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 linesL 1andL 2intersect If so find their point

1=t2.

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

8 1 Vectors in the Plane and in Space

giving as the only requirement ontland t2that

tl=1+t2.

What does this mean? Itmeans that no matter what value oft2 wechoose, there is a value oftl, namelytl=1+t2,that satisfies equations(~). By varying the valuesoft2we get all the points on the lineL 2. Foreach such value oft2the fact that there is a (corresponding) value oftl

solving equations(~)shows that every point of the line L2lies on theline L l Therefore these lines must be the same!

The lesson to be learned from this Example is that the equations of aline are not unique This should be geometrically clear since we onlyused two points of the line to determine the equations, and there aremany such possible pairs of points

EXAMPLE 4: Determine if the lines L l and L2with equations

have a point in common

SOLUTION:As in Example 3, our task is to determine whether theset of simultaneous equations

Adding the first two equations gives

sotlmust equal 1 Putting this into the last equation, we get

1.1 First Steps 9

sot2must equal 2 But substituting these values oftlandt2into either

of the first two equations leads to a contradiction, namely

-1=-1+2=1,-1=-1 - 2=-3

Therefore, no values oftl and t2can simultaneously satisfy equations(1E1E), so the lines have no point in common

In Chapter 13 we will take up the study of solving simultaneous linearequations in detail There we will explain various techniques and

"tests" that will make the problems encountered in Examples 3 and 4routine

Suppose thatP,Q, and R are three noncollinear points, i.e., the points

P, Qand R do not all lie on the same line Then they determine aunique plane ll Ifwe introduce a fixed origin 0 then it is possible

to derive a vector equation satisfied by the points ofll ConsideringFigure 1.1.8 shows that

+ ~ -7 -7 -7 -7A-Q=s(P-Q)+ t(R-Q);

that is,

-7 -7 -7 -7 -7 -7

A =s{P - Q)+t{R - Q)+Q

Equation(1E1E)is called thevector equationofthe planell Compare it

to the vector equation ofa line Note the presence ofthe two parameters

sand tinstead of the single parametert.

II

Figure 1.1.8

10 1 Vectors in the Plane and in Space

If we now introduce a coordinate system and pass to components inequation(~~),we obtain

where we may take (or twice these values, or -7 times, etc.)

a =(YR - YQ)(zp - zQ) - (ZR - zQ}(YP - YQ),

b=(ZR - zQ}(xp - xQ) - (XR - xQ}(zp - zQ),

C=(XR - xQ}(YP - YQ) - (YR - YQ}(xp - xQ),

d =-{axQ+bYQ+cZQ}.

Equation(++)is also called the equation ofthe plane II

EXAMPLE 5: Find the equation of the plane through the points

(1,0, I), (0, 1,0), (1, 1, I)

Determine whether the point (0, 0, O) lies in this plane

SOLUTION:We know that the equation has the form

ax+by +cz+d =0,and all we must do is find suitable values for a, b,c,d. (Rememberthey are not unique) We must have

a+c + d =0,

b+d =0,

a+b+c+d =0,since the points (1, 0, 1), (0, 1, O), and (1,1, I) lie in this plane Thus

a+C=0, d =0, b =0, a =-c.

So the plane has the equation

x-z =0,and (0, 0, O) lies in it

for suitable numbersa, b, c, u, v, w. Since such points must lie in bothplanes we have

a+ut-(c+wt)=O,

a+ut +b+vt+c+wt+1=0,for all values oft. Putt =o. Then

a-c=0,

a+b +c+1=o.

The first equation yieldsa=c Combining this with the second equationand settingb =1yields 2a+2=o. Hence a =-1=c Next putt =1.Then

O=a+ut-(c+wt)=-I+u+l-w, 0= a +ut+b+vt+c+wt+1

12 1 Vectors in the Plane and in Space

1.2 Exercises

1 On the dedication page of this book are the names, and where theyworked, ofmathematicians who are connected with the history oflinearalgebra and whose names appear elsewhere in this book Find out whothey were: this means that you may have to go to the library and learnhow to look up subjects from mathematics and history Do it

2 Suppose that an origin0 and a coordinate system have been fixed.Let P be a point Define vectors E;, E;, and E; by requiring thatthey be the position vectors of the points (1,0,0), (0,1,0), and (0,0,1),respectively Let the coordinates ofP be(xp, yp,zp). Show that

~ ~ + +

P =xpE1+ypE2+zpEs

The vectors xpE- - -1, ypE2, zpEs

(2) Find an equation for a lineL 2with no points in common with

TI (Le., L 2 should be parallel toTI).

(3) Find an equation for a lineL 3lying in the plane TI

8 Let P =(Xl, Yl, ZI), Q =(X2, Y2, Z2) be two points Show that themidpoint of the line segment joining them has coordinates

1.2 Exercises 13

10 Find the equation of the line through the origin bisecting the angleformed by AOB, where A = (1, 0, 0), B = (0, 0,1)

+ :;;:;-;t, -+

11 Verify that vectors PQ andltbrepresent the same vector T, where

P = (0, 1, 1), Q = (1, 3, 4), R = (1, 0, -1), S = (2, 2, 2) Find the coordinates

(6) T = aP + bQ + cR, where a, b, c are given constants.

16 In each of(IH7) below find a vector equation ofthe line satisfyingthe given conditions:

(1) Passing through the point P = (-2,1) and having slope ~

(2) Passing through the point (0, 3) and parallel to the x-axis.(3) The tangent line to y =x 2at (2, 4)

(4) The line parallel to the line of (c) passing through the origin.(5) The line passing through points (1, 0, 1) and (1, 1, 1)

(6) The line passing through the origin and the midpoint of the

line segment PQ, where P = (2' 2,0), Q = (0, 0, 1)

(7) The line in thexy-plane passing through (1, 1, 0) and(0,1, 0)

17 In each of(1H7) determine a vector equation ofthe plane satisfyingthe given conditions:

(1) The plane determined by (0, 0), (1, 0) and (1,1)

(2) The plane determined by (0, 0, 1), (1, 0, 1), and (1,1, 1)

are squares and rectangles, but any four sided figure with straight sides is a

quadrangle; the angles do not haveto be right angles, and the sides do not allhave to have the same length

14 1.Vectors in the Plane and in Space

(3) The plane determined by (1,0,0), (0, 1,0), and (1, 1, 1) Does

the origin lie on this plane?

(4) The plane parallel tothe xy-plane and containing the point

(1,1,1)

(5) The plane through the origin and containing the pointsP =

(1,0,0), Q =(0, 1,0)

(6) The plane through three pointsA, B,C, where A=(1, 0, 1), B=

(-1,2,3),and C =(2, 6,1) Does the origin lie on this plane?(7) The plane parallel to yz-plane passing through the point

(1, 1, 1).

18 Let a be a real number Determine the equation of a line L inthe plane, passing through the origin, such that the area of the figurebounded byL and the two lines

x+y - a =0, x =0

Chapter 2

Vector Spaces

In the previous chapter we reviewed the intuitive notions connectedwith vectors in space and their algebra We derived vector equationsfor lines and planes and saw how once a coordinate system was chosenthese vector equations lead to the familiar equations of analytic geom-etry However, particularly in application to physics, it is often veryimportant to know the relation between the equations for the sameplane (or line) in different coordinate systems This leads us to the no-tion of acoordinate transformation. The appropriate domain in which

to study such transformations is that of the abstract vector spaces to

be introduced in this chapter

2.1 Axioms for Vector Spaces

The following definition lays the basis for our study of vector spaces

DEFINITION: Avector space is a set 0/, whose elements are called vectors, together with two operations The first operation, calledvec-

tor addition, assignstoeach pairofvectorsAandBa vectordenoted

byA+B, called their sum Thesecondoperation, calledscalar

mul-tiplication, assigns to each vectorA and each number1 r a vector

denoted byrA The two operations are requiredtohave the following properties:

AxIOM1 A+B=B+A for each pairofvectorsAandB

(commu-tative lawofvector addition).

AxIOM2 (A+B)+C=A+(B+C) for each tripleofvectorsA, B,

and C(associative lawofvector addition).

1 Unless stated to the contrary the wordnumberwill mean a real number

15

L Smith, Linear Algebra

© Springer Science+Business Media New York 1998

16 2 Vector Spaces

AxIOM3 There isa unique vector0,called thezero vector, such

thatA + 0 = A foreveryvector A

AxIOM 4.For eachvector A there correspondsa unique vector-A

such thatA+(-A)=O

AxIOM5 r(A +B) = r A + r B for each realnumber r andeach pair

ofvectors A and B

AxIOM 6 (r +s)A= rA +sAforeach pairofrealnumbers rand

s andeachvector A

AxIOM 7.(rs)A= r(sA)foreach pairr,s ofrealnumbers andeach

vectorA

AxIOM8 For each vector A, 1A = A

The development of linear algebra in this book uses the axiomatic method. That is, vector, vector addition, and scalar multiplica- tion constitute the basic terms of the theory They are not defined,but rather our study of linear algebra will be based on the proper-ties of these terms as specified by the preceding eight axioms In theaxiomatic treatment, what vectors, vector addition, and scalar multi-plication are is immaterial; rather what is important is the propertiesthat these quantities have as consequences of the axioms Thus in ourdevelopment of the theory we may not use properties of vectors thatare not stated in, or are consequences of, the preceding axioms Wemay use any properties of vectors, etc that are stated in the axioms:for example, that the vector 0 is unique, or that A = 1A for any vector

A On the other hand, we maynotuse that a vector is an arrow with

a specified head and tail

The advantage of the axiomatic approach is that results so obtainedwill apply to any special case or example that we wish to consider Theconverse is definitely false, i.e., results obtained in a special case neednot apply to all Presently we will see a large number of examples ofvector spaces Let us begin with some elementary consequences of theaxioms

PROPOSITION 2.1.1: OA=O

PROOF:We have

A= 1A=(l+O)A= 1A+OA=A+OA

by using Axioms 8, 6, and 8 again Tothe equation

A=A+OA

we apply Axiom 1, getting

A=OA+A

2.1Axioms for Vector Spaces 17

Ifwe apply Axiom 4, we obtain by Axiom 2,

0= A+ (-A) = (OA+A) + (-A) = OA+ (A+ (-A»

=OA+O=OA

where the last equality is by Axiom 3 That is ,

O=OA,

which is the desired conclusion 0

NOTATION: Wewill reserve boldface capita11ettersforvectorsand

small italic lettersfor numbers(whicharealso calledscalars) 2

The proof of Proposition2.1.1was given in considerable detail to trate how results are deduced by the axiomatic method In the sequel

illus-we will not be so detailed in our proofs, leaving to the reader the task

of providing as much detail as is felt necessary

PROPOSITION2.1.2: (-1)A =-A

PROOF: By Proposition2.1.1we have

o= OA = (1 - 1)A = 1A + (-1)A = A + (-1)A

Add-Ato both sides, giving

-A = -A + (A + (-1)A) = (-A + A) + (-1)A

= (A-A) + (-1)A = 0 + (-1)A=(-1)A+ 0

=(-1)A,

as required 0

PROPOSITION2.1.3: 0+A =A

PROOF: Exercise 0

These formal deductions may seem like a sterile intellectual

exercise-an indication of the absurdity of too much reliexercise-ance on abstraction exercise-andformalism Quite to the contrary, however, they illustrate the advan-tages of the abstract formulation of a mathematical theory For ifthe basic terms are not defined, the possibility is opened of assigning

to them content in new and unforeseen ways Ifin this way the ioms become true statements about the meanings assigned to the basic

ax-2Actually the font used for numbers is called\mit inis'lEXand not \it. Seethe list offonts used at the end of the book

18 2 Vector Spaces

terms vector, vector addition, andscalar multiplication, then we haveconstructed a model for the abstract theory: and in this model all thetheorems, propositions, etc that we deduced from the axioms are alsotrue statements about the assigned meanings ofvector, vector addition,

andscalar multiplication.

2.2 Cartesian (or Euclidean) Spaces

One of the standard models of a vector space is the generalization ofthe familiar Euclidean plane usually encountered in a calculus course.Before plunging into a discussion of these examples, we pause to makesome comments on the axioms Note that Axioms 3 and 4 are of a

"different character" than the others Apart from Axioms 3 and 4,verifying that an axiom holds in an example usually may be reduced

to some property of real numbers Axioms 3 and 4 are different inthat they assert the existence of things and their uniqueness In otherwords, one will have to specify which vector in the example underdiscussion is the zero vector, and then verify that it has the propertystated in the axiom and that it is the only vector in the example withthis property

Here is one standard model of a vector space

DEFIN1TION: Letk be a positive integer TheCartesian k-space,

denoted by lRk

, is the set of all k-tuples (ab a2, , , ak) ofk real

numbers, together with the two operations(vectoraddition)

and(scalar multiplication)

If(al,"" ak)is a vector in lRk

,then the number ai is called the i-thcomponent of the vector (aI, , ak). We can think of lRI as beingjust lR by with its usual addition and multiplication by ignoring theparentheses ( ) around the numbers

THEOREM2.2.1: For each positive integer k,lRk

2.2 Cartesian (or Euclidean) Spaces 19

of A and B, we shall mean the vector (al + bl , , ak +bk)jthat is, wedefine

A+B=(al+bI, , ak+bk).

Likewise, for a real numberr and a k-tuple A we define

Axioms 1-8 for a vector space then become statements about k-tuples,

and we must verify that they are true statements.

PROOF OF 2.2.1 : We will verify the axioms in turn

AxIOM1 Let A = (al, , ak), B =(bI, , b k ). Then

A+O=(al+O, , ak+O)=(al, , ak)=A.

Moreover ifB = (b l , , b k )is any vector such that

A+B =A,then

and therefore

al + bl = al ~ bl = 0,

ak+bk = ak => bk OJi.e., B=O Thus 0 is the unique vector with the property that A + 0 = A,and Axiom 3 holds

20 2 Vector Spaces

AxIOM4 Let A=(at, , ak)and set -A=(-at, , -ak).Then

A+(-A)=(al, , ak)+(-al, , -ak)

=(al-at, , ak-ak)=(O, , O)=O~

Moreover, ifC=(ct, , Ck)is any vector such that

A+C =0,then

and therefore

al +CI = 0 ~ CI = -aI,

a2+C2 = 0 ~ C2 = -a2,

ak+Ck = 0 ~ Ck = -ak ;

Le., C =-A Thus -A is the unique vector with the property that

A+(-A)=0 and Axiom 4 holds

AxIOM5 Let r be a real number and let A=(at, , ak) and B=

(b l , •.• , b k )be vectors Then

rCA + B) =real+b l , , ak +bk )=(r(al+b l ), , r(ak +bk»

= (ral+rbt, , rak +rb k )=(rat, , rak)+(rb l , , rbk)

=r(al, , ak)+r(bt, , bk)=rA+rB,

so that Axiom 5 is satisfied

AxIOM6 Let r,sbe numbers and A = (at, , ak). Then

(r +s)A= «r +s)al, , (r +s)ak)

= (ral+sat, , rak +sak)

= (raI, , rak)+(sal,.'" sak)

=r(at, , ak)+s(at, , ak)

=rA+sA,

so Axiom 6 holds

AxIOM7 Let r,sbe numbers and A = (at, , ak). Then

(rs)A=(rsat, , rsak)=r(saI, ,sak)

= r(s(at, , ak»=r(s(A»,

so Axiom 7 holds

AxIOM8 Instant

Therefore,lR k is a vector space 0

2.3 Some Rules for Vector Algebra 212.3 Some Rules for Vector Algebra

In the chapters that follow we will introduce many additional examples

of vector spaces 'Ib close this chapter let us indicate some further ementary consequences of the axioms: the generalized associative andcommutative laws, which allow us to drop parentheses from formulaswhen they serve no useful purpose (such as grouping terms together toemphasize the result of a computation not shown in the text)

el-PRoPOSITION2.3.1 (Generalized Associative Law): Letn~ 3 be an

integer. Thenanytwowaysofassociatinga sum

ofthe order in which thesumis taken.

It will be convenient to introduce some notations for describing therelationships between vectors and vector spaces, or more generallybetween sets and their elements

NOTATION: Wewillusethe symboleas anabbreviation for "isan

element of." Thusx eS shouldberead as, x isanelementofthe set

S The symbolc isanabbreviation for "is contained in."ThusS c T should beread as, the set S is contained in the set T. IfS c T and

S :f: T then we usethe symbol c or~ toindicate this, and write for example S ~ T in this case.3

IfSand T, aresets then the collectionofelements contained in either set is denotedbyS u T Thus xeS u T is equivalenttoxeSorx e T The collectionofall elements commontoboth sets is denotedbyS n T Thus xeS nTis equivalenttoxeS and x e T The set S u T is called theunionofS and T, and S nTis theintersectionofSand T. We

denoteby0 the empty set, i.e., the set withnoelements.

The axioms for a vector space that we have given are the axioms for

a realvector space, that is, a vector space whose scalars are thereal numbers, which we have denoted by JR It is also possible, and of-ten important, to study vector spaces whose scalars are the complex

3There are many other variations of these symbols, such as; e.g.,~, *,etc

22 2 Vector Spaces

numbers,which we denote bycr:. A vector space with complex scalars

is called a complex vector space The axioms for a complex tor space are exactly the same as for a real vector space except thatthe numbers (= scalars) are to be complex The generic example of

vec-a complex vector spvec-ace is the complex Cvec-artesivec-an spvec-acecek of k-tuplesA= (al, , ak)ofcomplexnumbers, where for vectors A =(al, , ak)

and B = (bb"" b k ) and for scalars r Ece, vector addition and scalarmultiplication are given by

A+B=(al+bl , , ak+bk)

and

In the next few chapters we will study only real vector spaces, ing where necessary the modifications required in the complex case.2.4 Exercises

indicat-1 Let A = {(2a, a) Ia Em}, B = {( b, b) Ib Em}. Find A u Band

6 Show that A n (B n C)=(A n B)n (A n C) and A n (B u C)=

(An B)u (An C), whereA, B,C are sets

7 LetA ={x Em I IxI>I}, B = {X Em I-2<x <3} FindA u B

andAnB.

8 Let 'l! be a vector space Prove each of the following statements:

(a) IfA E'l!anda is a number, thenaA= °if and only ifa =0 or

A =0, or both

(b) IfA E'l!andais a number, thenaA=A if and only ifa=1or

A =0, or both

9 Let 'l! be the set of all ordered pairs of real numbers (a, b). Define

an addition for the elements of 'l! by the rule

(a, b)$(c, d)=(a+c, b+d),

2.4 Exercises 23and a multiplication of elements of0/by numbers by the rule

r· (e, d) = (re, 0).

Is 0/with these two operations a vector space? Justify your answer

10 Let 0/ and 'W be vector spaces Denote by 0/ EB'W the set of allordered pairs (A, B), where A Eo/and BE'W. Define an addition forthe elements of0/EB'Wby the rule

(A, B)+(C, D)=(A+C, B +D),and a multiplication of elements of0/ by numbers by the rule

@ For any two pointsp, q Ell the distance from p to q is thesame as the distance fromT(p) toT(q).

Introduce Cartesian coordinates in II and show:

(a) IfT is a translation, then for everyp =(x, y) Ell,T(p) =(x +

lI,x+l2), whereT(O,0)=(It,l2) T is said to be the tionbyl =(It,l2).

transla-(b) Ifl =(It,l2) Ell, show that

T(p)=(x+lt,y+l2), p=(x,y)Ell

defines a translation ofll

If T,S are translations, define their sumT EB S by

(TEBS)(p) = T(S(p)).

(c) Show thatT EB S is again a translation

IfT is the translation byl = (It,l2)and a is a number, define a T to

be the translation by(alt, al2)'

(d) Show that the set of translations with the addition EB andscalar multiplication is a vector space

12 Show that if a vector space contains two elements, then it containsinfinitely many

13 LetA,B be elements of the vector space0/ Show that there exists

a unique element X of0/such that A+X=B

Chapter 3

Examples of Vector Spaces

Before embarking on our study of the elementary properties of vectorspaces and their linear subspaces in the succeeding chapters, let uscollect a list of examples of vector spaces Of basic importance arethe three examples rn. k , Pn(rn.), and Fun(S) described in Section 3.1.They illustrate three of the viewpoints that have contributed to thedevelopment oflinear algebra and its applications: geometry, analysis,and combinatorics

3.1 Three Basic Examples

We have already encountered the Cartesian k-spacern k in Section 2.2,and so for the sake of completeness let us begin by listing this example:EXAMPLE 1: rn. k •

The first new example in this chapter is primarily designed to destroythe beliefthat a vector is a quantity with both direction and magnitudeand to give meaning to the phrase in our comments on axiomatics in

Chapter 2 that "the possibility is opened ofassigning to them (the axioms

of a vector space) content in new and unforeseen ways."

EXAMPLE 2: Pn(rn.).

The vectors inP n(rn.)are polynomials

p(x)= ao+aIX+ +amx m

of degree less than or equal ton,that is,m:5:n. Addition ofvectors isto

be ordinary addition ofpolynomials, and multiplication ofa polynomial

25

L Smith, Linear Algebra

© Springer Science+Business Media New York 1998

26 3 Examples of Vector Spaces

by a number will be the ordinary product of a polynomial by a number.With these interpretations of the basic terms, namely

vector +-+polynomial of degree~n,

vector additon +-+addition of polynomials,

scalar multiplication +-+multiplication of a polynomial

by a number,

we obtain an example of a vector space 1bverify that Pn(IR) is deed a vector space we must check that the eight declarative sentences

in-obtained from these interpretations of the basic terms vector, vector

addition, scalar multiplication, are true sentences This is a forward deduction from the (assumed) properties of real numbers fol-lowing the pattern of Proposition 2.2.1 and will be left to the diligentreader

straight-Note that in this example it is very difficult to say what direction orlength a vector has

EXAMPLE3: Let S be a nonempty set and Fun(S) the set of allfunctionsf :S - IR.Iff,g EFun(S), define their sum

(rf)(s) =r(f(s)),

for all s ES.

Equipped with these definitions of vector addition and scalar cation the set Fun(S) becomes a vector space The zero vector ofFun(S)

multipli-is the function

O:S-IRdefined by

O(s)=0

for all s ES; that is, 0 is the constant function that takes the value 0 for all s ES The negative off EFun(S) is the function

-f : S -IRdefined by

(-f)(s)=-f(s).

3.2 Further Examples of Vector Spaces 27

Itis routine to verify that the axioms of a real vector space are satisfiedfor Fun(S)

This completes the list ofbasic examples The exercises at the end ofthechapter introduce further examples and develop some of the propertiesoflRk

,Pn(lR),and Fun(S)

3.2 Further Examples of Vector Spaces

We collect in this section some further examples of vector spaces thatindicate only a small portion of the vector spaces encountered in prac-tice

EXAMPLE 1: (Cas a real vector space

Recall that (C denotes the complex numbers A complex number1

looks like

e=a+bi,

where a, b are real numbers, andi is a number such thati2 =-1 Thenumberais called the real part of e and the numberb the imaginarypart of e Addition of complex numbers is componentwise, i.e.,

e' +e" =(a' +a")+(b' +b")i

for complex numbers e' =a' +b'i and e" =a" +b"i Multiplication isdetermined by the rule

e'e" =(a' a" - b' b")+Ca'b"+a" b')i

The vectors in our vector space will be complex numbers Addition ofvectors is to be the ordinary addition of complex numbers, and scalarmultiplication the familiar process of multiplying a complex number

by a real number With these interpretations of the basic terms,

vector ~complex numbervector addition ~addition of complex numbers

scalar multiplication ~multiplication of a complex number by a

num-of complex numbers using ideas developed later in this book see Appendix B,

so you may want to come back to this example after further study of this book

28 3 Examples of Vector Spaces

Note that in this example we are not using all of the structure that

we have, for, we may in effect, multiply two complex numbers, that

is, in this example we may multiply two vectors, something it is notalways possible to do in a vector space This is a possibility2 worthy offurther study, and we will do just that when we study spaces of lineartransformations and linear algebras

REMARK: The product of two polynomials of degree at mostn willhave degree at most2n, but in general will not be of degree n,so youcannot multiply elements ofP nem.)in any obvious way

Examples 2 and 3 in Section 3.1 are very important, and they areprototypes for many others of the same type These examples arecharacterized by the fact that their "vectors" are actually functions ofsome type or other Here are some related examples

EXAMPLE2: FunG::(S)

We can make the set of allcomplex-valued functions on a nonemptyset S, Le., FunG::(S)= {f :S~ C}, into a complex vector space bysetting

(f+g)(s)=f(s)+g(s), (cf)(s) =c(f(s»

for allf,g EFunG::(S), s ES, and C EC

in this example is:

vector < >polynomial,vector addition < >addition of polynomials,

scalar multiplication < >multiplication of a polynomial

by a number

2The multiplication of complex numbers enjoys several important properties,among which are:

(1) the commutative law for multiplication c' c" =c" c' holds;

(2) for every nonzero complex number c there is an inverse c-1such thatcc-1 =1, and hence

(3) the cancellation law if c' c" =0 and c' -f.0 then c" =0 holds

The onlyfinite-dimensional(a term we will define and study in Chapter 6)real vector space with these properties is in fact the complex numbers