Linear Algebra

Linear

Algebra

Stephen H. FriedbergArnold J. lnselLawrenceE. SpenceIllinois State University

Prentice Hall, Englewood Cliffs, New Jersey 07632

Library of Congress Cataloging-m-Publication Data

Friedberg, Stephen H.Linear algebra/Stephen H. Friedberg, Arnold J. Insel, Lawrence

E. Spcnce.\342\200\2242nd ed.

p. cm.

Includes indexes.ISBN 0-13-537102-31.Algebras, Linear. I. Insel, Arnold J. II. Spcnce,Lawrence

E III. Title.QA184.F75 1989 88-28568512\\5\342\200\224dc!9 CIP

Editorial/production supervisionand interior design: Kathleen M, Lafferty

Coverdesign:Wanda LubelskaManufacturing buyer: Paula MassenaroCover illustration: Based on a

lithographby Victor Vasarely,

\302\2511989, 1979 by Prentice-Hail, Inc.A Paramount Communications CompanyEnglewoodCliffs,New Jersey 07632

All rights reserved. No part of this bookmay be reproduced, in any form or by any means,without permission in writing from the publisher.

Printedin the United States of America

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Contents

Preface

Vector Spaces1.1 Introduction 11.2 Vector Spaces 6

1.3 Subspaces 141.4 Linear Combinationsand Systems of Linear

Equations 21

1.5 Linear Dependence and Linear Independence 311.6 Bases and Dimension 351.7* Maximal Linearly Independent Subsets 49

Index ofDefinitionsfor Chapter 1 52

Linear Transformations and Matrices

2.1 LinearTransformations,NullSpaces,and Ranges2.2 The Matrix Representation of a Linear

Transformation 652.3 Compositionof Linear Transformations and Matrix

Multiplication 72

2.4 Invertibility and Isomorphisms 842.5 The Change of Coordinate Matrix 932.6* Dual Spaces 101

\342\231\246Sections denoted by an asterisk are optional.

viii

2.7* Homogeneous LinearDifferentialEquationswith Constant Coefficients 108

Index of Definitions for Chapter 2 125

3 ElementaryMatrix Operationsand Systems of Linear Equations 127

3.1 ElementaryMatrixOperationsand ElementaryMatrices 128

3.2 The Rank of a Matrix and MatrixInverses 1323.3 - Systems of Linear Equations\342\200\224Theoretical Aspects 1473.4 Systems of Linear Equations\342\200\224Computational

Aspects 160

.\342\200\242Index of Definitions for Chapter 3 170

4 Determinants 171

4.1 Determinants of Order 2 1714.2 Determinants of Order n 1824.3 \342\200\236Properties of Determinants 1904.4 . Summary\342\200\224Important Facts about Determinants 205


5 Diagonalization 2145.1 Eigenvalues and Eigenvectors 2145.2 Diagonalizability 2315.3* Matrix Limitsand Markov Chains5.4 Invariant Subspaces

and the Cayley-Hamilton TheoremIndex of Definitionsfor Chapter 5

6 Inner Product Spaces 295

6.1 InnerProductsand Norms 2956.2 The Gram-Schmidt Orthogonalization Process

and Orthogonal Complements 3046.3 The Adjoint of a Linear Operator 314

252

280293

Contents ix

6.4 Normal and Self-Adjoint Operators 3256.5 Unitary and Orthogonal Operators

andTheir Matrices 3336.6 Orthogonal Projections and the Spectral Theorem 3486.1 Bilinear and Quadratic Forms 3556.8* Einstein's Special Theory of Relativity 38563* Conditioning and the Rayleigh Quotient 3986.10* The Geometry of OrthogonalOperators 406


Canonical Forms 416

7.1 GeneralizedEigenvectors 4167.2 Jordan Canonical Form 4307.3 The Minimal Polynomial 4517.4* RationalCanonical Form 459


Appendices 481

A Sets 481

B Functions 483

C Fields 484D ComplexNumbers 488E Polynomials 492

Answers to Selected Exercises 501

List of Frequently Used Symbols 518

Index of Theorems 519

Index 522

Preface

The languageand concepts of matrix theory and, more generally, of linearalgebra have comeinto widespread usage in the social and natural sciences,computer science,and statistics.In addition,linear algebra continues to be ofgreat importance in modern treatmentsof geometry and analysis.

The primary purpose of Linear Algebra, Second Edition,is to present acareful treatment of the principal topics of linear algebraand to illustratethepower of the subject through a variety of applications. Althoughthe only formalprerequisite for the book is a one-year course in calculus,the material inChapters 6 and 7 requires the mathematical sophisticationof typical collegejuniors and seniors (who may or may not have had some previous exposure tothe subject).

The book is organized to permita number of different courses (rangingfrom three to six semester hours in length) to be taught from it. The corematerial (vector spaces,linear transformations and matrices, systems of linearequations, determinants, and diagonalization) is foundin Chapters 1 through 5.The remaining chapters, treating inner product spacesandcanonicalforms, arecompletely independent and may be studied in any order. In addition,throughout the book are a variety of applications to such areas as differentialequations, economics, geometry, and physics. These applications are not centralto the mathematical development, however, and may be excluded at thediscretionof the instructor. '

We have attempted to make it possible for many of theimportanttopicsoflinear algebra to be covered in a one-semester course. This goalhas led us todevelop the major topics with fewer unnecessary preliminaries than in atraditional approach. (Our treatment of the Jordan canonical form, for instance,doesnot

require any theory of polynomials.) The resulting economy permits usto covermostof the book (omitting many of the optional sections and a detaileddiscussionofdeterminants) in a one-semester four-hour course for students whohave had somepriorexposure to linear algebra.

Chapter 1 of the book presents the basic theory of finite-dimensionalvector spaces\342\200\224subspaces, linear combinations, linear dependence and inde-

xi

xii Preface

pendence, bases, and dimension. The chapter concludes with an optionalsectionin which we prove the existence of a basis in infinite-dimensionalvectorspaces.

Linear transformations and their relationship to matrices are the subjectof Chapter2. We discuss there the null space and range of a linear

transformation, matrix representations of a transformation, isomorphisms, andchange of coordinates.Optionalsectionson dual spaces and homogeneouslinear differential equations end the chapter.

The applicationsof vector space theory and linear transformations tosystems of linearequationsare found in Chapter 3. We have chosen to defer thisimportantsubjectso that it can be presented as a consequence of the precedingmaterial.This

approach allows the familiar topic of linear systems to illuminatethe abstracttheory and permits us to avoid messy matrix computations in thepresentationof Chapters 1 and 2. There will be occasional examples in thesechapters,however, where we shall want to solve systems of linear equations.(Ofcourse,these examples will not be a part of the theoretical development.)Thenecessary background is contained in Section 1.4.

Determinants, the subject of Chapter4, are of much less importance thanthey once were. In a shortcoursewe prefer to treat determinants lightly so thatmore time may be devoted to the material in Chapters 5 through 7.Consequently we have presented two alternatives in Chapter 4\342\200\224a completedevelopment of the theory (Sections 4.1 through 4.3) and a summaryof the importantfacts that are needed for the remaining chapters (Section4.4).

\342\200\242Chapter 5 discusses eigenvalues, eigenvectors, and diagonalization. One of

the mostimportantapplicationsof this material occurs in computing matrixlimits. We h&ve therefore included an optional section on matrix limits andMarkovchainsin this chapter even though the most general statement of someof the results

requires a knowledge of the Jordan canonical form. Section 5.4containsmaterialon invariantsubspacesand the Cayley-Hamilton theorem.

Inner product spaces are the subject ofChapter6.The basic mathematicaltheory (inner products; the Gram-Schmidt process; orthogonal complements;adjointtransformations;normal,self-adjoint, orthogonal, and unitaryoperators; orthogonal projections; and the spectral theorem) is contained in Sections6.1 through 6.7. Sections 6.8, 6.9, and 6.10 contain diverse applicationsof therich inner product space structure.

Canonical forms are treated in Chapter7.Sections7.1and 7.2 develop theJordan form, Section 7.3 presents the minimal polynomial,and Section 7.4discusses the rational form.

There are five appendices. The first four, which discuss sets, functions,fields, and complex numbers, respectively, are intendedto review basic ideasused throughout the book. Appendix E on polynomialsis used primarily inChapters 5 and 7, especially in Section 7.4. We prefernot to discusstheappendices independently but rather to refer to them as the needarises.

The following diagram illustrates the dependencies among the variouschapters.

Preface xiii

Chapter 1

''

Chapter 2

1'Chapter 3

1 '

Sections 4.1-4.3or Section 4.4

* '

Sections 5.1 and 5.2

1^\"\"^\"\"^^

Section 5.4

i '

Chapter 7

One final word is required about our notation.Sections denoted by anasterisk (*) are optional and may be omittedas the instructor sees fit. Anexercise denoted by the daggersymbol(f) is not optional, however\342\200\224we use thissymbol to identify an exercise that will be cited at some later point of the text.

Specialthanks are due to Kathleen M. Lafferty for her cooperation andfinework during the production process.

DIFFERENCES BETWEEN THE FIRST AND SECOND EDITIONS

In the 10 years that have elapsed sincethe first edition, we have discovered moreefficient ways to present a numberofkey topics. Most of the material that; reliedon direct sumsin the first edition has been rewritten, and the results about directsumsare contained in optional subsections or relegated to the exercises.Also,anumber of exercises and examples have been added to reflect the changesmadein the existing topics and the additional new material.

In Chapter 1manyproofsare more readable without the dependence ondirect sums. Section 1.3no longercontainsany results about direct sums exceptin the exercises. Section 1.4now employsGaussianelimination.

In Chapter 2 the proof of the dimension theorem in Section2.1no longeruses direct sums. In Section 2.3 the proof ofTheorem2.15,and in Section 2.4 theproof of Theorem 2.21,havebeenimproved.Also,exercises on quotient spaces

xiv Preface

have been added to Section2.4.The material on the Change of coordinate matrixin Section2.5hasbeen rewritten. In particular, Theorem 2.24 is less general, butmore understandable,*the general case of Theorem 2.24 is included in theexercises.

In Chapter3 systems of equations are now solved by the more efficientGaussianeliminationmethod, rather than by the Gauss-Jordan technique usedin the firstedition. '

In Chapter 4a considerably simplified proof of Cramer's rule is presentedthat no longer depends on the classical adjoint. In fact, theclassicaladjointnowappears as a simple exercise that exploits the new developmentofCramer'srule.

In Chapter 5 the proofs of Theorems 5.10 and 5.12in Section5.2have beensimplified. Theorem 5.14 characterizes diagonalizability without the earlierdependenceon \"direct sums. Direct sums are now treated as an optionalsubsection.Gerschgorin'sdisk theorem is now contained in Section 5.3, andSection 5.4 combinesthe Cayley-Hamiltontheorem and invariant subspaces.The minimal polynomial is now found in Section7.3.

Chapters 6 and 7 in the first edition have been interchangedin the secondedition. This alteration not only reflects the authors' tastes, but those of anumber of our readers.

In Chapter 6 the materialin Section6.2 has been rewritten to improve theclarity and flowof thetopics.The results needed for least squares now appear inSection 6.2 and no longer rely on direct sums. Least squares and minimalsolutions are introducedearlierandare now contained in Section 6.3. In Section6.4 the addition of Schur'stheorem allows a completely new and improvedapproach to normal and self-adjointoperatorswhich no longer relies on resultsconcerning T*-invariant subspaces. These resultsare now exercises. Section 6.5has been rewritten to take advantageof the new approach in Section 6.4. A newsubsection of Section 6.5 treatsrigidmotions in the plane. Sylvester's law ofinertia is now includedwith the materialon bilinear forms in Section 6.6. Forreasons of continuity, the optionalsectionson the special theory of relativity,conditioning, and the geometry of orthogonaloperatorsare now located at theend of the chapter.

In Chapter 7 the proofs in Section 7.1 have been improved. In Section7.2the proof of the Jordan canonical form has been simplifiedand no longerusesdirect sums; which are dealt with in a subsection. Section7.3 now treats theminimal polynomial. In Section 7.4 the proof of therationalcanonical form hasbeen simplified and no longer uses either direct sumsor quotientspaces.

Stephen if. Friedberg

Arnold J. lnselLawrence E. Spence

Vector Spaces

1.1 INTRODUCTION

Many familiar physical notions, such as forces, velocities,1 and accelerations,involvebotha magnitude (the amount of the force, velocity, or acceleration) anda direction.Any such entity involving both magnitude and direction is called avector.Vectors are represented by arrows in which the length of the arrowdenotes the magnitude of the vector and the direction of the arrow representsthe direction of the vector. In most physical situations involvingvectors,onlythe magnitude and direction of the vector are significant; consequently,weregard vectors with the same magnitude and direction as beingequalirrespective of their positions. In this section the geometry of vectorsis discussed.Thisgeometry is derived from physical experiments that test the manner in whichtwo vectors interact.

Familiar situations suggest that when two vectorsact simultaneously at apoint, the magnitude of the resultant vector (thevectorobtainedby adding thetwo original vectors) need not be the sumof the magnitudes of the original two.For example, a swimmer swimmingupstreamat the rate of 2 miles per houragainst a current of 1mileper hour does not progress at the rate of 3 milesperhour. For in this instance the motions of the swimmerand current oppose eachother, and the rate of progress of the swimmeris only 1 mile per hour upstream.If, however, the swimmer is movingdownstream (with the current), then his orher rate of progressis 3 milesper hour downstream.

Experiments show that vectors add according to the followingparallelogram law (see Figure 1.1),

Parallelogram Law for Vector Addition. Thesumof two vectors x and ythat act at the same pointP is the vector in the parallelogram having x and y asadjacent sides that is represented by the diagonal beginningat P.

fThe word velocity is being used here in its scientificsense\342\200\224as an entity havingboth magnitude and direction. The magnitudeof a velocity (without regard for thedirection of motion) is calledits speed.

1

2Chap. 1 Vector Spaces

Figure 1.1

Since opposite sides ofa parallelogramareparalleland of equal length, theendpoint Q of the arrow representingx + y can also be obtained by allowing xto act at P and then allowing y to act at the endpoint of x Similarly,theendpointof the vector x + y can be obtained'by firstpermittingj;to actat P andthen allowing x to4act at the endpoint of j;.Thustwo vectors x and y that bothact at a pointP may be added \"tail-to-head\"; that is, either xory may be appliedat P anda vector having the same magnitude and direction as the other may beappliedto the endpoint of the first. If this is done, the endpointof the secondvector is the endpoint of x 4- y.

The additionof vectors can be described algebraically with the use ofanalytic geometry.In the plane containing x and y, introduce a coordinatesystem with P at the origin. Let (als a2) denote the endpoint of x and (bu b2)denote the endpoint of y. Then as Figure 1.2(a)shows,theendpointQ of x 4- y is(ai + bx,

Sec. 1.1 Introduction 3

vector x is representedby an arrow, then for any real number t ^ 0 thevectortxwill be represented by an arrow having the same direction as the arrowrepresenting x but having length t times the length of thearrow representing x Ift < 0, the vector tx will be representedby an arrow having the oppositedirection as x and havinglength|t| times the length of the arrow representing x.Two nonzero vectorsx andy are called parallel if y = tx for some nonzerorealnumber t. (Thus nonzero vectors having the same direction or oppositedirectionsare parallel.)

To describe scalar multiplication algebraically, again introduce acoordinate system into a plane containing the vector x so that x emanates from theorigin. If the endpoint of x has coordinates(au a^), then the coordinates of theendpoint of tx are easilyshownto be (tax, ta2) [see Figure 1.2(b)].

The algebraic descriptions ofvectoradditionand scalar multiplication forvectors in a plane yield the followingpropertiesfor arbitrary vectors x, y, and zand arbitrary real numbersa andb:

1.x 4- y = y 4- x.2. (x + y) + z = x + (y + z).3. Thereexists a vector denoted 0 such that x + 0 = x for each vector x.4. For each vector x there is a vector y such that x + y = 0.5. lx = x6. (ab)x = a(bx).7. a(x + y) = ax 4- ay.8. (a + b)x = ax-\\-bx.

Argumentssimilarto those given above show that these eight properties,as well as the geometric interpretations of vector addition and scalarmultiplication, are true also for vectors acting in space rather than in a plane. We willuse these results to write equations of linesand planesin space.

Consider first the equation of a line in space that passesthrough twodistinct points P and Q. Let O denote theoriginofa coordinate system in space,and let u and v denote the vectors that begin at O and end at P and g,respectively. If w denotes the vector beginning at P and endingat g, then t'tail-to-head\" addition shows that u + w = v, and hence w = v \342\200\224us where \342\200\224udenotes the vector (\342\200\224l)u. (See Figure 1.3, in which quadrilateral OPQR is aparallelogram.)Since a scalar multiple of w is parallel to w but possibly of adifferent length than ws any point on the line joining P and Qmay beobtainedasthe endpoint of a vector that begins at P and has the form tw for some realnumber t. Conversely, the endpoint of every vector of the form tw that begins atP lieson the line joining P and Q, Thus an equation of the linethrough P and Qis x = u + tw = u + t(v \342\200\224u), where t is a real number and x denotesan arbitrarypoint on the line. Notice also that the endpoint R of the vector v \342\200\224u in Figure1.3 has coordinates equal to the differenceof the coordinatesof Q and P.

Chap. 1 Vector Spaces

Figure 1.3

Example 1We will find the equation of the line through the points P and Q havingcoordinates (\342\200\2242,0,1)and (4,5,3), respectively. The endpoint R of thevector emanatingfrom the origin and having the same direction as thevector beginning at P and terminating at Q has coordinates(4,5,3) \342\200\224(\342\200\224X0,1) = (6, 5,2). Hence the desired equation is

\342\226\240

; x= (-2, 0,1) +\302\242(6,5, 2). I

Now let P, g, and R denote any threenoncollinearpointsin space. Thesepoints determine a unique plane, whose equation can be found by use of ourprevious observations about vectors.Letu and v denote the vectors beginning atP and ending #t Q and R, respectively.Observe that any point in the planecontaining P, Q;and R is the endpoint S of a vector x beginningat P andhavingthe form txu + t2v for some real nunfbers tx and t2. The endpoint of t^u will bethe point of intersectionof the line through P and Q with the line through Sparallelto the line through P and R (see Figure 1.4), A similar procedure willlocate the endpoint of t2v. Moreover, for any real numbers tl and t2i txu + t2v isa vectorlyingin the plane containing P, Q, and R. It follows that an equationofthe plane containing P, g, and R is

x = P + txu + t2V,where tl and t2 are arbitrary reaL numbers and x denotes an arbitrary point inthe plane. \342\200\242

*- RFigure 1.4

Sec. 1.1 Introduction 5

Example 2Let P, g, and R be the points having coordinates (1,0,2), ( \342\200\2243, \342\200\2242,4), and(1,8, \342\200\2245), respectively. The endpoint of the vector emanating from the originand having the same length and direction as the vector beginning at P andterminating at Q is

(-3,-2,4)-(1,0,2) = (-4,-2,2).Similarly,the endpointof the vector emanating from the origin and having thesame lengthanddirectionas the vector beginning at P and terminating at R is(1,8, \342\200\2245)- (1,0,2) = (0,8, \342\200\2247). Hence the equation of the plane containing thethree givenpointsis

^ = (1,0,2)+ ^(-4,-2,2) + ^(0,8,-7). 1Any mathematical structure possessing the eight properties on page 3 is

calleda \"vectorspace.\" In the next section we formally define a vectorspaceand

consider many examples of vector spaces other than the ones mentionedabove.

EXERCISES

1. Determine if the vectors emanating from the origin and terminating at thefollowing pairs of points are parallel.(a) (3, 1, 2) and (6,4, 2)(b) .(-3, 1,7) and (9, -3, -21)(c) (5, -6, 7) and (-5, 6, -7)(d) (2,0, -5) and (5,0, -2)

2. Find the equations of the lines through the following pairs ofpointsinspace.(a) (3, -2, 4) and (-5, 7,1)(b) (2,4,0) and(-3, -6,0)(c) (3, 7, 2) and (3, 7, -8)(d) (-2, -1,5) and (3, 9, 7)

3. Find the equations of the planescontainingthe following points in space.(a) (2, -5, -1), (0,4, 6), and (-3, 7, 1)(b) (3, -6, 7), (-2, 0, -4), and (5, -9, -2)(c) (-8, 2,0),(1,3,0),and (6, -5,0)(d) (1,1, 1),(5, 5, 5), and (-6, 4, 2)

4. What are the coordinates of the vector 0 in theEuclideanplane that satisfiescondition 3 on page 3? Prove that this choiceof coordinatesdoessatisfycondition 3.

5. Prove that if the vector x emanatesfrom the origin of the Euclidean plane andterminates at the point with coordinates (al3a2), then the vector tx thatemanates from the originterminates at the point with coordinates (tau ta2).

6 Chap.1 Vector Spaces

6. Show that the midpoint of the line segmentjoining thepoints(a, b) and (c, d)is ((a + c)/2s (b + d)/2).

7. Prove that the diagonals of a parallelogram bisect each other.

1.2 VECTOR SPACES

Because such diverse entities as the forces acting in a planeand the polynomialswith real number coefficients both permit natural definitionsof additionandscalar multiplication that possess properties 1 through 8 on page 3,it is naturalto abstract these properties in the following definition.

Definition. A vector space (or linear space) V over a field* F consistsofaset on which two operations (called addition and scalar multiplication,respectively) are defined so that for each pair of elementsx, y in V there is a uniqueelement x 4- y; in V, and for each element a in F and eachelementx in V there is aunique element ax in V, such that thefollowing conditions hold:

(VS 1)_ For all x, y in V, x + y = y + x (commutativityof addition).,(VS 2) For all x, y, z- in V, (x + y) + z = x + (y + z) (associativity of

addition).(VS 3) There exists an elementin V denoted by 0 such that x + 0 = x for

eachx in V.(VS 4) For each element x in V there exists an element y in V such that

x\"+y = 0.(VS 5) For each element x m,V, lx = x.(VS6) For each pair a, b of elements in F and each elementx in V,

(ab)x = a(bx).(VS 7) For each element a in F and each pair of elements xs y in V,

a(x + y) = ax + ay.(VS 8) For each pair a, b of elements in F and each element x in V,

(a + b)x= ax+ bx.The elements x + y and ax are calledthesumo/ x and y and the product of a andx, respectively.

Theelements of the field F are called scalars and the elementsof the vectorspace V are called vectors. The reader should not confusethis use of the word\"vector\" with the physical entity discussed in Section1.1;the word \"vector\" isnow being used to describe any element of a..vector space.

A vector space-will frequently be discussed in the text withoutexplicitlymentioning its field of scalars. The reader is cautioned to remember,however,

fSee Appendix C. With few exceptions, however, the reader may interpret the word\"field\" to mean \"field of real numbers\"(which we denote by R) or \"field of complexnumbers\"(which we denote by C).

Sec. 1.2 VectorSpaces 7

thatevery vector space will be regarded as a vector space overa given field, which

will be denoted by F.In the remainder of thissectionwe introduce several important examples

of vector spaces that will be studied throughout the text. Observe that indescribing a vector space it is necessary to specify not only the vectors but alsothe operationsof additionandscalarmultiplication.

An object of the form (al3... ,a\342\200\236),where the entries at are elements of afield F, is calledan n-tuple with entries from F. Two n-tuples (a1?..., an) and(bu..., bn) are defined to be equal if and only if a-x = bt for i = 1,2,..., n.Example1Theset of all n-tuples with entries from a field F formsa vector space, which wedenote by Fn, under the operations of coordinatewiseadditionandmultiplication; that is, if x = (au...,an)efn,y = (bli...ibn)e Fns and c 6F, then

x + y = (fli + bx,..., an + bn) and ex = {cax,..., can).For example, in R4,

(3, -2, 0, 5) + (-1, 1, 4, 2)= (2, -1, 4, 7)and

-5(1, -2,0,3) = (-5,10,0,-15).Elements of Fn will often be written as column vectors:

rather thanas row vectors (au..., an). Since a 1-tuple with entry fromF may beregarded as an element of F, we will write F ratherthan F1for the vector spaceof 1-tuples from F. 1

An m x n matrix with entries from a field F is arectangular array of the

form

fl21 \302\25322 \342\200\242\"\302\2532n

\\f*ml am2' ' '

amnjwhere each entry au(l < i < m, 1 < j < n) is an element of F. The entries&iu&iii \342\200\242-\342\226\240>ain of the matrix above compose the \\th row of the matrix and willoften be regarded as a row vector in Fn, whereas the entriesaip a2jJ...,amJcompose the jth column of the matrix and will often be regarded as a column

8 Chap. 1 VectorSpaces

vector in Fm. The m x n matrix having each entry equal to zeroiscalled the zeromatrix and is denoted by 0.

In this book we denote matrices by capital italic letters (e.g., A, B, and C),andwe denote the entry of a matrix A that lies in row i and column j by Au.Inaddition,if the number of rows and columns of a matrix are equal,the matrix iscalled square.

Two m x n matrices A and B are defined to be equal if and only if all theircorrespondingentriesareequal,that is, if and only if Au = Btj for 1 < i < m and1 < j < n.

Example 2Theset of all m x n matrices with entries from a fieldF isa vector space, whichwe denote by Mmxn(F), under the followingoperationsofadditionand scalarmultiplication: For A, B e Mmxn(F) and c eF,

(A + B)u = Au + By and (cA)i}= cAu.For instance,

and

in M2x3(R). 1

Example 3

Let S be any nonemptysetandF be any field, and let ^(5, F) denote the set ofallfunctions from S into F. Two elements f and g in J*\"(S, F) are defined to beequal if f(s) = g(s)for eachseS. The set ^(S,F) is a vector space under theoperationsofadditionand scalar multiplication defined for f, g e ^(S, F) andc eF by

\342\226\240 W + gKs) = m + g(s) and (cf)(s)= c[/M]for each seS. Note that these are the familiar operationsofadditionand scalarmultiplication for functions as used in algebra and calculus. 1

A polynomial with coefficients from a field F is an expressionof the formf(x) = anxn + fln-iX\"\"1 + \342\226\240\342\200\242\342\200\242+ axx + a0,

where n is a nonnegative integerand an,,.., a0 are elements of F, If f(x) = 0,that is,ifan = \342\200\242\342\200\242\342\200\242= a0 = 0, then f(x) is called thezeropolynomial and the degreeof f(x) is said to be \342\200\2241; otherwise, the degree of a polynomial is defined to be the

Sec.1.2 Vector Spaces 9

largest exponent of x that appears in the representationf(x) = anxn + fln-xx\"-1 + \342\200\242\342\200\242\342\200\242+ a0

with a nonzero coefficient. Note that the polynomialsof degree zero are of theform f(x) = c for some nonzeroscalarc.Two polynomials f(x) and g{x) areequal if and only if they have the same-degree and their coefficients of likepowers of x are equal.

When F is a field containing an infinite number ofelements,we will usuallyregard a polynomial with coefficients from F as a functionfrom F into F, In thiscase the value of the function

f(x)= anxn + fln-xx\"-1 + \342\200\242\342\200\242'\342\200\242+ fl0at c e F is the scalar

/(c) = ancn + fln_1cn-1+--- + a0-Here either of the notations f or f(x) will be used for the polynomial function

f(x) = anxn + fln-xx\"-1 + .. \342\200\242+ a0.Example 4

The set of all polynomialswith coefficientsfroma field F is a vector space, whichwe denote by P(F)Sunder the following operations: For

f(x) = anxn + an_lxn-1 + ---+^and

.g(x)

= bux* + h-iX?-1 + \342\200\242-+ b0

in P(F) and c 6 F,

if + fiOM = (A\302\273+ bu)xn + (fln-x + dn-Jx\"-1 + -\342\226\240\342\200\242+ (flo .+ b0)and

(cfXx) = canxn + cfl\342\200\236_!*\"-l + \342\200\242\342\226\240\342\200\242+ ca0. 1

We will see in Exercise21ofSection2.4that the vector space defined in, theexample below is essentiallythesameas P(F).Example 5

Let F be any field. A sequence in F is a function a from the positiveintegersintoF. As usual, the sequence a such that a(n) = an will be denoted by {an}. Let Vconsist of all sequences{an}inF that have only a finite number of nonzero termsan. If {an}and-{fcn}are sequences in V and teF, then {an} + {bn}is thatsequence {c\342\200\236}in V such that cn = an+ bn (n = 1,2,...) and t{an} is that sequence{dn}in V such that dn = tan (\302\253= 1,2,...). Under these operations V is a vectorspace. 1

10 Chap. 1 Vector Spaces

Our next two examples contain sets on which an additionand scalarmultiplication are defined, but which are not vector spaces.

Example6Let S = {(al3 a2): al,a2s R}. For (au a2\\ (bu b2) e S and ce R, define

(fli\302\273

Sec. 1.2 Vector Spaces 11

Proposition 1.2. In any vector spaceV the following statements are true:(a) Ox = Ofor eachx e V.(b) (\342\200\224a)x

=

\342\200\224(ax)=

a(\342\200\224x) for each a 6 F and each x 6 V.(c) a0= Ofor each a e F.Proo/. (a) By (VS 8), (VS1),and (VS 3) it follows that

Ox + Ox = (0 + 0)x= Ox = 0 + Ox.Hence Ox = 0 by Proposition 1.1.

(b) The element\342\200\224

(ax) is the unique element of V such thatax 4- [\342\200\224(ax)] = 0. Hence if ax 4- (\342\200\224a)x = 0, Corollary 2 of Proposition 1.1implies that (-a)x = -(ax). But by (VS 8),

ax + (-a)x = [a + (-a)]x = Ox = 0by (a). Thus (-a)x = -(ax). In particular,(-l)x = -x. So by (VS 6)

a(-x) = a[(-l)x] = [a(-l)]x = (-a)x.The proof of (c) is similar to the proofof(a). 1

EXERCISES

1.Label the follpwing statements as being true or false.(a) Everyvectorspacecontains a zero vector.(b) A vector space may have more than one zero vector.(c) In any vector space ax = bx impliesthat a = b.(d) In any vector space ax = ay implies that x = y.(e) An element of Fn may be regarded as an elementof Mnxl(F).(f) An m x n matrix has m columns and n rows.(g) In P(F) only polynomials of the same degree may be added.(h) If f andg are polynomials of degree n, then f + g is a polynomial-of

degree n.

(i) If f is a polynomial of degreen and c isa nonzero scalar, then cf is apolynomial of degree n.

(j) A nonzero element of F may be considered to be an elementof P(F)having degree zero.

(k) Two functions in 1F(S7F) are equalif and only if they have the samevalues at each element of 5.

2. Write the zero vector of M3x4(F).3. If

what are M13,M2iiandM22?


4. Perform the operations indicated.(a)

(b)

(c)

(d)

(e) (2x4 -:7x3 + 4x + 3)+ (8x3 + 2x2 - 6x + 7)(f) (-3x3*+.7x2 + 8x- 6) + (2x3 - 8x + 10)(g) 5(2x7- 6x4 + Sx2 - 3x)(h) 3(x5 - 2x3+ 4x+ 2)

Exercises 5 and 6 show why the definitions of matrix additionand scalarmultiplication (as defined in Example 2) are the appropriate ones.

5. RichardGard(Effects of Beaver on Trout in Sagehen Creek, California, J.WildlifeManagement,25,221-242) reports the following number of trouthaving crossed beaver dams inSagehenCreek:,

Brook trout

Rainbow trout

Brown trout *

Upstream

Fall

8

33

Crossings

Spring

300

Summer

i

0

0

Downstream Crossings

Brook troutRainbowtroutBrown trout

Fall

9

3i

Spring

i0i

Summer

400

Record the upstreamand downstream crossings as data in two 3x3matrices and

verify that the sum of these matrices gives the total numberof

Sec.1.2 Vector Spaces 13

crossings (both upstream and downstream) categorized by trout speciesandseason.

6. At the end of May a furniture store had the followinginventory:

Early Mediter-

American Spanish ranean Danish

Living room suites 4 2 i 3Bedroom suites 5 i i 4

Dining room suites 3 12 6

Record these data as a 3 x 4 matrix M.Topreparefor its June sale, the storedecided to double its inventoryon each of the items above. Assuming thatnone of the presentstockissolduntil the additional furniture arrives, verifythat'the inventory on hand aftertheorderis.filled is described by the matrix2M. If the inventory at the endof June is described by the matrix

A =

interpret 2M\342\200\224 A. How many suites were sold during the June sale?7. LetS= {0,1}and F = R, the field of real numbers. In ^(5, R), show that

f = g and f + g = h, where f(x) = 2x + 1, g(x) = 1 + Ax \342\200\2242x2, andh{x) = 5x+l.

8. In any vectorspaceV, show that (a + b)(x + y) = ax + ay + bx+ by forany xjeV and any a,beF.

9. Prove Corollaries1 and 2 to Proposition 1.1 and Proposition 1.2(c).10. Let V denote the set of all difFerentiable real-valued functions defined on the

realline.Prove that V is a vector space under the operations ofadditionandscalar

multiplication defined in Example 3.

11. Let V = {0}consistofa single vector 0, and define 0 + 0 = 0 and cO = 0 foreach c in F. Prove that V is a vector space over F. (V is called the zerovectorspace.)

12.A real^valued function defined on the real line is called an even function iff(\342\200\224x)

= f[x) for each real number x Prove that the set of even functionsdefined on the real line with the operations of addition and scalarmultiplication defined in Example 3 is a vector space.

13. Let V denote the set of ordered pairs of real numbers.If (^, a2) and (bi, b2)are elements of V and c is an elementofF, define

(fli, a2) + Q>iyb2) = (fli + bl9 a2b2) and c(au a2) = (cau'a2).Is V a vector space under these operations?Justifyyour answer.


14. Let V = {(al,...,an): aseC for i = 1,2,...,n}.Is V a vector space over thefield of real numbers with the operationsof coordinatewise addition andmultiplication?

15. Let V = {(ax,..., an):a{6 R for i = 1,2,..., n}. Is V a vector space over thefield of complexnumberswith the operations of coordinatewise additionand multiplication?

16. Let V denote the set of all m x n matrices with realnumberentries, and let Fbe the field of rational numbers. Is V a vector space over F under the usualdefinitions of matrixadditionand scalar multiplication?

17. Let V = {(al9 a2): al9 a2 e jR}. For (au a2), (bl9 b2) e V and csR, define(\302\253i> a2) + {bx, b2) = {ax +bl9a2 + b2)

and

[(0,0) ifc = 0c(aua2)= If a2\\

c&u\342\200\224 if c #0.

Is V.a vector'space under these operations? Justifyyour answer.18. Let V = {(al9 a2): al,a2E C}. For (al9a2),(bl9 b2) 6 V and c 6 C, define

(flu ai) + 0>ub2) = (fli + 2bl9 a2 + 3b2) and c[au a2)= (cau ca2).Is V a vectpr space under these operations? Justify your answer.

19. Let V = {(a\\, a2): al,a2s F}, where F is an arbitrary field. Define additionof elements of V coordinatewise,- and for c 6 F and (ax, a2) e V, define

c{au fl2) = (^1.0).Is V a vector space under these operations? Justify your answer.

20. How many elements are there in the vector space Mmxn(Z2)?

1.3 SUBSPACES

In the study of any algebraic structure it is of interest to examinesubsetsthatpossessthe same structure as the set under consideration. The appropriatenotion of substructure for vector spaces is introduced in this section.

Definition. A subset W of a vector space V over a field F is calledasubspace of\\fif\\N is a vector space over F under the operationsofadditionandscalar multiplication defined on V.

In any vector space V, note that V and {0} are subspaces. The latter iscalled_the zero subspace of V.

Sec. 1.3 Subspaces 15

Fortunately, it isnot necessary to verify all the vector space conditions inorder to provethat a subset W of a vector space V is in fact a subspace.Sinceconditions(VS 1), (VS 2), (VS 5), (VS 6), (VS 7), and (VS8)are known to hold forelements of V, these conditions automatically hold forelementsofa subset of V.Thus a subset W of V is a subspace of V if and only if the following fourconditions hold:

1. x 4- y e W whenever xeW and y s W.2. ax 6 W whenever aeF and xeW.3. The zero vector of V belongs to W.4. The additive inverseof eachelement of W belongs to W.

Actually, condition 4 is redundant,as the following theorem shows.

Theorem 1.3. Let V be a vectorspaceandW a subset of V. Then W is asubspaceof V if and only if the following three conditions holdfor the operationsdefined in V:

(a) OeW.

(b) x + y 6 W whenever x 6 W and yeW.(c) ax 6 W whenever aeF and xeW.Proof If W is a subspace of V, then W is a vector space under the

operations of addition and scalar multiplication defined on V. Hence conditions(b)and (c) hold, and there exists an element 0' e W such that x 4- 0' = x for eachx e W. But also x 4- 0 = x, and thus 0' = 0 by Proposition 1.1. So condition (a)holds.

Conversely, if conditions(a),(b), and (c) hold, the discussion preceding thistheorem shows that W is a subspace of V if the additive inverse of eachelementof W belongs to W. But if x e W, then (\342\200\224l)x belongs to W by condition (c), and\342\200\224

x = (\342\200\224l)x by Proposition 1.2. Hence W is a subspace of V. 1

The theorem above provides a simple method for determining whether ornota given subset of a vector space is a subspace. Normally, it is this result thatis used to prove that a subset is, in fact,a subspace. >

The transpose M1 of an m x n matrix M is the n x m matrix obtained fromM by interchanging the rows with the columns;that is, (M%* = Mn. Forexample,

/ i o\\\342\226\240

Gi-iH-' 4\\

3-1/

A symmetric matrix is a matrix M such that M1 = M. Clearly, a symmetricmatrix must be square. The set W of all symmetric matrices in Mnxn(F) is a

16 Chap.1 Vector Spaces

subspace of Mnxn(F) since the conditions of Theorem 1.3hold:

(a) The zero matrix is equal to its transpose and hence belongsto W.It is easily proved that for any matrices A and B and any scalars a and b,(aA + bBY = aAl + bBl (see Exercise3).Usingthis fact, we can easily establishconditions (b) and (c) of Theorem 1.3as follows.

(b) If yie-W and BeW, then A = A* and B = Bl. Thus (A + BJ =A1 + B1 = A + B, so that A + B e W.

(c) If A 6 W, then A1 = A. So for any asF, (aAJ= aA1 = aA. Thus aA e W.The following examples providefurtherillustrationsof the concept of a

subspace. The first three are particularlyimportant.Example1Let M be an n xjn matrix. The (main) diagonal ofM consistsof the entries MllsM22, \342\200\242- -, Mnn. An n x n matrix D is calleda diagonalmatrix if each entry not onthe diagonal of Dis zero,that is,if DtJ = 0 whenever i # j- The setofalldiagonalmatrices in Mnxn(F) is a subspace of Mnxn(F). 1

Example2Let n be a nonnegative integer, and let Pn(F) consist of allpolynomialsin P(F)having degree tess than or equal to n. (Noticethat the zero polynomial is anelement of Pn(F) since its degree isn\342\200\2241.) Then Pn(F) is a subspace of P(F). 1

Example3The set C(R) consisting of all the continuous real-valued functions definedon Ris a subspace of ^(R, R), where ^{R, R) is as defined in Example 3 of Section

Example 4

The traceofan n x n matrix M, denoted tr(M), is the sum ofall the entries of Mlying on the diagonal; that is,

tr(M)= MU + M22 + - + Mnn.The setof n x n matrices having trace equal fo zero is a subspaceof Mnxn(^)(see Exercise 6). 1

Example 5

The set ofmatricesin Mmxn(F) having nonnegative entries is not a subspace ofMmxn(F)becausecondition(c) of Theorem 1.3 does not hold. 1

Sec.1.3 Subspaces 17

The next theorem provides a method of forming a new subspace fromother subspaces.

Theorem 1.4. Any intersection of subspaces of a vector space V is asubspace of V.

Proof Let ^ be a collection of subspaces of V, and let W denote theintersectionof all the subspaces in #. Since every subspace contains the zerovector,0 e W.Let a e F and x, y be elements of W; then x and y are elements ofeach subspace in #. Hence x + y and ax are elements of each subspace in ^(becausethe sum of vectors in a subspace and the product of a scalarandavector from the subspace both belong to that subspace). Thus x + y e\\N andaxE\\N, so that W is a subspace by Theorem1.3. 1

Having shown that the intersection of subspaces is a subspace, it isnaturalto consider the question of whether or not the union of subspacesisa subspace.It is easily seen that the union of subspaces must satisfyconditions(a) and (c) ofTheorem 1.3 but that condition (b)neednothold.In fact, it can be readily shown(see Exercise 18)that theunionof two subspaces is a subspace if and only if oneof thesubspacesis a subset of the other. It is natural, however, to expect thatthereshouldbe a method of combining two subspaces Wx and W2 to obtain alarger subspace (that is, one that contains both W^ and VV2). As we havesuggested above, the key to finding sucha subspaceis condition (b) of Theorem1.3. We explore this idea in Exercise21.

EXERCISES

1. Label the following statements as being true or false.(a) If V is a vector space and W is a subset of V that is a vector space, then

W is a subspaceof V.(b) The empty set is a subspace of every vector space.(c) If V is a vector space other than the zero vectorspace{0}, then V

contains a subspace W such that W ^ V.(d) The intersection of any two subsetsof V is a subspace of V.(e) Ann x n diagonal matrix can neverhave more than n nonzero entries.(f) The trace of a square matrix is the product of its entries on the

diagonal.2. Determinethe transposeof each of the following matrices. In addition, if

the matrix is square,compute its trace.


(e) (1, -1, 3, 5)

3. Prove that (aA + bBf = aAl + bBl for any A, B e Mmxn(F) and anya,b eF.

4. Provethat (AJ = A for each A e Mm x n(F).5. Prove that-A + A1 is symmetric for any square matrix A.6. Provethat-tr(a/l+ bB) = a tx(A) + ft tr(\302\243) for any A, B e Mn xn(F).7. Prove that diagonalmatrices are symmetric matrices.8. Determine if the following sets are subspacesof R3 under the operations of

addition and scalar multiplication definedon R3.Justify your answers.(a) Wj = {(aua2,a3) e R3: ax = 3a2 and a3=-a2}(b) W2 = {(a^a^aj e R3:a, = a3+ 2}(c) W3 - {(aua2,a3) e R3:2a, - la2+ a3 = 0}(d) W4 ^.((0^02, fl3) 6 R3:a., ~ 4a2 ~ a3 = 0}(e) W5 = {(aua2,a3) e R3:aJh- 2a2 ~ 3a3 = 1}(f) W6 = {(aua2,a3) e R3:5a\\ - 3a\\ + 6a\\ = 0}

9. Let Wx,W3,andW4 be as in Exercise 8. Describe Wj n W3j Wj n W4jand VV3 n VV4 and observe that each is a subspaceof R3.

10. Verify that W^ = {(a,,..., an) 6 Fn:a, + \342\200\242\342\200\242\342\200\242+ an = 0} is a subspace of Fnbut that W2 = {(au..., an) 6 Fn: ax + \342\200\242\342\200\242\342\200\242+ an = 1} is not.

11. Is the set W = {/ 6 P(F):./ = 0 or / has degree n} a subspace of P(F) ifn > 1?Justifyyour answer.

12. An m.x n matrix A is called upper triangularif all entries lying below thediagonal are zero, that is, if Ai} = 0 whenever i > j. Verify that the uppertriangularmatrices form a subspace of Mmxn(F).

13. Verify that for any s0e 5, W = {/ 6 &(S9 F): f(s0) = 0} is a subspaceof3?(S, F).

14. Is the set of all differentiablereal-valuedfunctions defined on R a subspaceof C(R)1 Justify your answer. .

15. Let Cn(jR) denote the set of all real-valued functions definedon thereallinethat have a continuous nth derivative (and hence continuous derivatives oforders1,2,..., n). Verify that Cn(JR) is a subspace of &(R, R).

Sec.1.3 Subspaces 19

16. Prove that a subset W of a vector space V is a subspace of V if and only ifW # 0, ax e W, and x + ye\\N whenever a e F and x,ye\\N.

17. Prove that a subsetW of a vector space V is a subspace of V if and only if0 eW and ax + y e\\N whenever ae F and xjeW.

18. Let Wj andVV2 be subspaces of a vector space V. Prove that \\Nl \\j VV2 is asubspace of V if and only if Wx Q W2 or W2 \302\243WV

19. Let Fj and F2 be fields. A function g s^r(F1,F2) is called an evenfunction if g(\342\200\224x) \342\200\224g(x) for each xe Fl and is called an oddfunction ifg(\342\200\224x)

=

\342\200\224g(x) for each x e Fx. Prove that the set ofalleven functions in2F(FU F2) and the set of all odd functionsin J**(Fls F2) are subspaces of

20.f Prove that if W is a subspace of V and x\302\261,..., xn are elements of W, thenfli*i + \342\200\242\342\200\242\342\200\242+ anxn is an element of W for any scalars au..., an in F.

Definition. If Sj and S2 are nonemptysubsetsofa vector space V, then thesum of Sj anrf S2) denoted Sj + S2,is t/ie set {x + y:xe Sj an


24. Let W1= {^6Mmxn(F):Afj = 0 whenever i > j} and W2 ={Ae Mm x n(F): AtJ = 0 whenever i ^ /}. QNl is the setofuppertriangularmatrices defined in Exercise 12.) Show that Mmxn(F) = \\Nl \302\251W2.

25. Let V denote the vector space consisting of all upper triangularn x nmatrices (as defined in Exercise 12), and let \\Nl denote the subspace of Vconsisting of all diagonal matrices. Show that V = Wx \302\251VV2, whereVV2 = {A e V: AfJ = 0 whenever i > j}.

26. A matrix M is called skew-symmetric if M1 = \342\200\224M. Clearly, a skew-symmetric matrix is square. Prove that the set of all skew-symmetric n x nmatrices is a subspace Wx of Mn x n(R). Let VV2 be the subspace of Mn x n(R)consisting of the symmetricnxn matrices. Prove that Mnxn(jR) =Wx \302\251W2. _

27. Let Wx = {A e Mnxn(F): ASJ = 0 whenever i ! + W) +{v2 + W) = fa + W) + (y2 + W)and

a(Vl + W) = a(vi + W)for all a e F.

(d) Prove that the set 5 is a vector space under the operations definedabove.This vector space is called the quotient space of V modulo VV andis denoted by V/W.

Sec.1.4 Linear Combinations and Systems of Linear Equations 21

1.4 LINEAR COMBINATIONS ANDSYSTEMSOF LINEAREQUATIONS

In Section 1.1 it was shown that the equationof the plane through threenoncollinear points P, Q, and R in spaceis x = P + txu 4- t2v, where u and vdenote the vectors beginning at P and ending at Q and jR, respectively, and tjand t2 denote arbitrary real numbers. An important specialcaseoccurs when Pis the origin. In this case the equation of the planesimplifies to x = t^u + t2v,and the setofallpoints in this plane is a subspace of R 3. (This will be proved asTheorem 1.5.) Expressions of the form t^u + t2v, where tx and t2 are scalars andu and v are vectors, play a central role in the theory of vectorspaces.Theappropriate generalization of such expressions is presented in the followingdefinition.

Definition. Let V be a vector space and S a nonempty subsetof V.A vectorx in V is said to be a linear combinationof elements of S if there exist a finitenumber of elementsyi,..., yn iw S and scalars a!,...,an in F such thatx = a^ + \342\200\242*\342\200\2424- anyn. In this situation it is also customary to say that x is alinear combination of yl5..., yn.

Observe that in any vectorspaceV, Ox = 0 for each x 6 V. Thus the zerovector is a linear combination of any nonempty subset of V.

Example1Table1.1shows the vitamin content of 100 grams of 12 foods with respecttovitamins A, Bj (thiamine), B2 (riboflavin), niacin, and C (ascorbic acid).

We will record the vitamin content of 100 grams of each foodasa columnvector in R5\342\200\224for example, the vitamin vector for apple butter is

/o.oo\\0.01

Consideringthe vitamin vectors for cupcake, coconut custard pie, brown rice,soysauce,and wild rice, we see that

/o.oo\\ /o.oo\\ /o.oo\\/nno\\ /-0.34* 'nno*

0.054.70

\\o.oo,

/o.oo\\'

0.05*

0.06

. 0.30

\\o.oo/

+ + 20.020.25

.0.40\\o.oo/

7o.oo\\'0.45'

0.63

. 6.20

\\o.oo/


TABLE 1.1 Vitamin Content of 100 Grams of Certain Foods

Apple butter

Raw, unpared apples (freshly harvested)Chocolate-coated candy with coconut

center

Clams (meat only)Cupcake from mix (dry form)Cooked farina (unenriched)Jams and preservesCoconutcustardpie (baked from mix)Raw brown rice \"

Soy sauce

Cooked spaghetti (unenriched)Raw wild rice

A

(units)

0

90

0

1000

(Of100

(0)

0

0

(0)

B,(nig)

0.010.030.02

0.100.050.010.010.02

0.34

0.02

0.01

0.45

B2(mg)

0.020.020.07

0.180.060.010.03

0.02

0.05

0.25

0.010.63

Niacin(mg)

0.20.10.2

1.30.30.10.2

0.4

4.7

0.4

0.36.2

C(mg)

240

100

(0)

2

0

(0)00

(0)

Source: Bernice'K.Watt and Annabel L. Merrill, Composition of Foods (AgricultureHandbookNumber 8), Consumer and Food Economics Research Division, U.S.Departmentof Agriculture,Washington, D.C., 1963.

\"Zeros in parentheses indicate that the amount ofa vitamin present is either none or too small tomeasure.

Thus the vitaminvectorfor raw wild rice is a linear combination of the vitaminvectorsfor

cupcake, coconut custard pie, raw brown rice, and soy sauce.So100grams of cupcake, 100 grams of coconut custard pie, 100 grams of raw brownrice, and 200 grams of soy sauce provide exactlythe same amounts of the fivevitamins as 100 grams of raw wild rice. Similarly, since

/o.oo\\ / 90.oo\\ /o.oo\\ /o.oo\\/nni\\ I nrv, \\ I Qm * - -0.01

0.02

0.20

\\2.00/

+

0.030.020.10

\\ 4.00/

+ 0.07

0.20.

\\o.oo/

+

0.010.010.10

\\o.oo/

+

/io.oo\\ /o.oo\\

nm *'0.01

0.01

0.30

\\o.oo/200 grams of applebutter, 100 grams of apples, 100 grams of chocolate candy,100gramsof farina,100grams of jam, and 100 grams of spaghetti provideexactly the sameamountsof the five vitamins as 100 grams of clams. 1

ThroughoutChapters1 and 2 we will encounter many different situationsin which it is

necessary to determine whether or not a vector can be expressedasa linear combination of other vectors, and if so, how. This questionreducestothe problem of solving a system of linear equations. To illustratethisimportanttechnique, we will determine if the vector (2,6,8) can be expressedas a linear

Sec. 1.4 Linear Combinations and Systems of Linear Equations 23

combination of

yx= (1, 2, 1), y2 - (-2, -4, -2), ^3 = (0, 2, 3),

3\302\274= (2, 0,-3), and y3 =(-3, 8, 16).

Thuswe must determine if there are scalars ax, a2, a3, a4, and a5 such that(2,6,8) = axyx + a2y2 + azyz + 043\302\274 + asy5

= flx(l,2,1) + a2(-2, -4, -2) + a3(0,2,3)+ fl4(2,0,-3)+ a5(-3,8,16)

= (ax \342\200\2242a2 4- 2a4 \342\200\2243a5,2a1 \342\200\224Aa2 4- 2a3 + 8a5,

ax\342\200\224

2a2 + 3a3 \342\200\2243a4 + 16a5).

Hence (2,6,8) can be expressedas a linearcombination of yl9 y2, y3, yA, and y5if and only if thereisa 5-tuple of scalars {ax ,a2,az,aA, a5) satisfying the systemof linearequations

-

ax\342\200\224

2a2 + 2a4 \342\200\2243a5 = 2

< 2ax\342\200\224

Aa2 4- 2a3 + 8a5 = 6 (1)ax - 2a2+ 3a3 - 3a4 + 16a5 = 8

obtained by equatingthe corresponding coordinates in the preceding equation.To solve system (1),we will replace the system by another system with the

same solutionsbut which is easier to solve. The procedure to be used expressessomeof the unknowns in terms of others by eliminating certain unknowns fromall the equations except one. To begin, we eliminate ax from everyequationexcept the first by adding \342\200\2242 times the first equation to the second and \342\200\2241times the first equation to the third. The result is the following new system:

\\ax\342\200\224

2a2 4- 2a4 \342\200\2243a5 = 2

2x2 - 6, x3 - 2x2+ 4x + 1, 3x3 - 6x2 + x + 4(c) -2x3 - llx2+ 3x + 2, x3 - 2x2 + 3x - 1,2x3 + x2 + 3x - 2(d) x3 + x2 + 2x+ 13,2x3 - 3x2 + 4x + 1, x3 - x2 + 2x + 3(e) x3 - 8x2+ 4x, x3 - 2x2 + 3x - 1,x3- 2x.+ 3(f) 6x3 - 3x2 + x + 2, x3 - x2 + 2x + 3, 2x3 + x2 - 3x+ 1

5. In Fn leffej denote the vector whose 7 th coordinate is 1 and whose othercoordinates are 0. Provethat{e1,e2,..-,e\342\200\236}generates Fn.

6. Show that Pn{F) is generated by {l,x,x2,. .\\,x\"}.7. Show that the matrices

and

generate M2x2(F).8. Show that if

and Mi =3'0 r

,1 0

then the span of {Ml9M2,M3} is the set of all symmetric 2x2 matrices.9.T For any elementx in a vector space, prove that span({x}) = {ax: a e F}.

Interpret this result geometrically in R3.10. Show that a subset W of a vector space V is a subspace of V if and only if

span(W)= W.II.1\" Show that if 5X and S2 are subsets of a vectorspaceV such that 5X \302\243S2,

then span(Sj) \302\243span(S2). In particular, if S1 \302\243S2 and span(5x) = V,deduce that span(S2) = V.

Sec.1.5 Linear Dependence and Linear Independence 31

12. Show that if S1 and S2 are arbitrary subsets of a vector space V, thenspan(5x u S2)= span(5x) + span(S2).

13. Let 5X and S2 be subsetsof a vector space V. Prove that span(5x n S2)\302\243span(5x) n span(S2), Give an example in which span(SxnS2) and

span(Sx) n span(S2) are equal and an example in which they are unequal.

1.5 LINEAR DEPENDENCE AND LINEAR INDEPENDENCE

At the beginning of Section 1.4 we remarked that the equation of a planethrough three noncollinearpointsin space,one of which is the origin, is of theformx = txu + t2v, where u, v e R3 and tx and t2 are scalars.Thusa vector x inR3 is a linear combination of u, v e R3 if and only if x lies in theplanecontainingu and v (see Figure \\ .5). We see, therefore, that in R 3 the span of two nonparallelvectors has a simplegeometricinterpretation.A similar interpretation can begiven for the span of a singlenonzero vector in R3 (see Exercise 9 of Section 1.4).

Figure 1.5 hv *>

In the equation x = txu + t2v, x dependson u and v in the sense that xis a linear combinationof u and v. A set in which at least one vector is alinear combination of the others is called a linearly dependent set. Consider,for example, the set S = {xu x2, x3, x4} c R3, where xx = (2, \342\200\2241,4),

x2 = (1, -1,3), x3 = (1,1, -1), and xA = (1, -2, -1). To see if S is linearlydependent, we must check whether or not there is a vectorin S that is a linearcombination of the others. Now the vector xA is a linear combination of xl3 x2,and x3 if and onlyif there are scalars a, b, and c such that

x4.= axl +bx2 + oc3,that is, if and only if

x4 == (2a + b + c, \342\200\224a\342\200\224b + c,4a + 3b \342\200\224c).Thus x4 is a linear combination of xx,x2, and x3 if and only if the system

[ 2a + b + c = 1-a- t + c= -24a + 3b - c = -1

has a solution. The reader should verify that no such solution exists.Notice,however, that this does not show that the set S is not linearly dependent, for we


must now checkwhether or not xXs x2, or x3 can be written as a linearcombination of the other vectors in S. It can be shown,in fact, that x3 is a linearcombination of xx, x2, and x4; specifically,x3 = 2xx \342\200\2243x2 + 0x4. So S isindeed linearly dependent.

Wesee from this example that the condition for linear dependence givenaboveis inconvenient to use because it may require checking several vectors toseeifsome vector in S is a linear combination of theothers.By reformulating thedefinition in the following way, we obtain a definitionoflineardependence thatis easier to use.

Definition. A subset Sofa vectorspaceV is said to be linearly dependentif there exist a finite'numberof distinctvectors xx,x2,...,xn in S and scalarsax, a2,..., anJ not all zero, such that

axxx + a2x2 + \342\200\242\342\200\242\342\200\242+ anxn = 0.

In this case we will also say that the elements of S are linearly dependent.

To show that the subset S of R3 defined above is linearlydependentusingthis definition, we must find scalars ax, a2, az, and a4, not all zero, such that

\302\2531*1 + fl2*2 + ^3*3 + fl4*4 = ftthat is,suchthat

(2ax + a2 + a3 + a4, - ax - a2 + a3 - 2a4,4ax+ 3a2- a3 - a4) = (0,0,0).Thus we mustffind-a solution to the system

ilax

+ a2 + a3 + a4 = 0\342\200\224

ax\342\200\224

a2 + a3 \342\200\2242a4 = 0

4ax + 3a2 \342\200\224a3 \342\200\224a4 = 0,in which not all the unknowns are zero.Usingthe techniques discussed inSection 1.4, we find that ax = 2, a2 = \342\200\2243, a3 = \342\200\2241, a4 = 0 is one such solution.Notice that by using the definition of linear dependence stated above, we areable to check that S is linearly dependent by solving only one systemofequations.The reader should verify that the two conditions for lineardependence discussed above are, in fact, equivalent (see Exercise 11).

It is easilyseen that in any vector space a subset S that containsthe zerovector must be linearly dependent. For since 0 = 1\342\200\2420, the zero vector is a linearcombination of elementsofS in which some coefficient is nonzero.Example 1

In R4 the setS = {(1,3, -4, 2), (2, 2, -4, 0), (1, -3, 2, -4), (- 1,0, 1,0)}

is linearly dependent because

4(1, 3, -4, 2) - 3(2,2, -4, 0) + 2(1, -3, 2, -4) + 0(-1, 0, 1, 0) = (0, 0, 0, 0).

Sec. 1.5 LinearDependenceand Linear Independence

Similarly, in M2x3{R) the set

(7 1 -3 2\\ /-3 7 4\\(V-4 0 5j5V 6 -2 -7J5

is linearly dependent since/ 1 -3 2\\ /-3 7 4\\ /-2

5(-4 0 5J+3( 6 -2 -l)-\\-XDefinition. A subset Sofa vector space that is not linearlydependentis

said to be linearly independent. As before, we will also say that the elements of Sare linearly independent in this case.

The followingfacts about linearly independent sets are true in any vectorspace.

1. The empty set is linearly independent, for linearly dependent sets must benonempty.

2. A set consisting of a single nonzero vector is linearlyindependent.For if{x} is linearly dependent, then ax = 0 for some nonzero scalara.Thus

x = a-\\ax) = a~l0-=0.

3. For any vectorsxx,x2,..., xn, we have a1xi + a2x2 + \342\200\242\342\200\242\342\200\242+ anxn = 0 ifax \342\200\2240, a2 = 0,..., an = 0. We call this the trivialrepresentation of 0 as alinear combination of xx, x2,.. -, x\342\200\236.A set is linearly independent if and onlyiftheonly representations ofO as linear combinations of its distinct elementsare the trivial

representations.

The condition in item 3 provides a very usefulmethodfordeterminingifafinite set is linearly independent. This technique is illustrated in the followingexample.

Example 2

Let xk denote the vector in Fn whose first k \342\200\2241 coordinates are zero and whpselast n \342\200\224k + 1 coordinates are 1. Then {xl9x2,..., xn}is linearly independent,for if axxx + a2x2 + \342\200\242\342\200\242* + anx\342\200\236= 0, equating the corresponding coordinates ofthe left and rightsidesof this equality gives the following system of equations:

\"ax =0-

ax + a2 = 0\342\200\242\342\200\242\342\200\242< fli +a2 +a3 =0

ax '+a2r+a3 + \342\200\242\342\200\242\342\200\242+ an = 0.

33

3 ll\\_/0 0 0'\342\226\2403 2/ U 0 0


Clearly, the only solution of this system is ax = \342\200\242\342\200\242\342\200\242= an = 0. 1

The followingusefulresultsare immediate consequences of the definitionsof linear dependence and linear independence.

Theorem 1.6. Let V be a vector space, and let Sx Q S2 \302\243'V. If Sx islinearly dependent, then S2 is linearly dependent.

Proof.Exercise. 1

Corollary. Let V be a vector space, and let Sx Q S2 \302\243V. If S2 is linearlyindependent, then Sx:is linearly independent.

Proof Exercise. 1

EXERCISES

1. Label the following statements as beingtrue or false.(a) If ~S is..a linearly dependent set, then each elementof S is a linear

combination of other elements of S.(b) Any set containing the zero vector is linearly dependent.(c) The emptyset is linearly dependent.(d) Subsets of linearly dependent sets are linearlydependent.(e) Subsets of linearly independent sets are linearly independent.(f) If axxA + a2x2 + \342\200\242\342\200\242\342\200\242+ anx\342\200\236= 0 and xx, x2, . \342\200\242.,x\342\200\236are linearly

independent, then all the scalars at equal zero.2. In Fn let e,- denote the vector whose jth coordinate is 1 and whose other

coordinates are 0. Prove that {e1,e2,..., e\342\200\236}is linearly independent.3. Show that the set {1, x, x2,..., x\"} is linearly independent in Pn(F).4. Prove that the matrices

(J o)'(o 5)(5 o)' and (o 5)are linearly independent in M2x2(F).

5. Find a set of linearly independent diagonal matrices that generate thevector spaceof2 x 2 diagonal matrices.

6.T Show that {x, y] is linearlydependentifandonly if x or y is a multiple ofthe other.

7. Give an example of three linearly dependent vectors in R3 such that noneofthe three is a multiple of another,

8. Let S = {xl9x2,..., xn} be a linearly independent subset of a vector spaceV over the field Z2. How many elements are there in span(S)?Justify youranswer.

Sec. 1.6 Bases and Dimension 35

9. Prove Theorem1,6and its corollary.10. Let V be a vector space over a fieldofcharacteristicnot equal to 2.

(a) Prove that {u, v) is linearlyindependent with u and v distinct if andonly if {u + v, u - v} is linearly independent.

(b) Similarly, prove that {u, v, w} is linearly independent if and only if{u + v, u + w, v + w} is linearly independent.

11. Provethata set S is linearly dependent if and only if S = {0}or there existdistinct vectors y, xu x2, - - -, xn in S such that y is a linear combinationof

12. Let S = {xu x2, - - -, xn) be a finite set of vectors. Prove that S is linearlydependent if and only if xx \342\200\2240 or xfc + 1 espa.n({x1,x2,---,xk}) for some/c< n.

13. Prove that a set S of vectors is linearly independent ifand only if each finitesubset of S is linearly independent.

14. Let M be a square upper triangular matrix (as defined in Exercise 12 ofSection1.3)having nonzero diagonal entries. Prove that the columns of Mare linearlyindependent.

15. Let f and g be functions defined by f{t) = ert and g{t) = est, where r # s.Provethatf and g are linearly independent in ^(R, R). Hint: Suppose thataert+ best= 0. Let t = 0 to obtain an equation involving a and b. Thendifferentiate aert + best = 0, and let t \342\200\2240 to* obtain a second equationinvolving a and ft. Solve these equations for a and ft.

1.6 BASES AND DIMENSION

A subset S of a vector spaceV that is linearly independent and generates Vpossesses a very useful property\342\200\224every element of V can be expressed in oneandonlyoneway as a linear combination of elements of S. (Thispropertywill beproved in Theorem 1.7.) It is this result that makes linearly independentgenerating sets the building blocks of vector spaces.

Definition. A basis pfor a vector space V is a linearly independentsubsetofV that generates V. (If fi is a basis for V, we also say that the elements of Pform abasisfor V.)Example 1

Recalling that span(0) = {0} and that 0 is linearly independent, we see that 0is a basis for the vector space {0}. 1Example 2

In Fn, let ex = (l, 0,0,...,0, 0), e2 = (0, 1, 0, ..., 0, 0), .... en =(0,0,0,..., 0,1); {el5e2,-.., en} is readily seen to be a basis for Fn and is calledthestandardbasisfor Fn. 1


Example 3In Mm xn(F), let Mij denote the matrix whose only nonzero entryisa 1 in the ithrow and /th column. Then {MiJ:1 < i < m, 1 < J < n} is a basis forMmxn(n iExample 4

In Pn(F) the set {1, x, x2,..., xn}isa basis.We call this basis the standard basisforPn(F). 1

Example 5In P(F) the set {1, x, x2,...} is a basis. 1

Observethat Example 5 shows that a basis need not be finite. In fact,wewill see later in this section that no basis for P(F)canbefinite.Hence not everyvector space has\" a finite basis.

The followingtheorem,which will be used frequently in Chapter 2, showsthe most significantproperty of a basis.

Theorem 1.7. Let V be a vector spaceand ft = {xl5..., xn} be a subset ofV. Then $ isa basis for V if and only if each vector y in V can be uniquely expressedas a linear combinationof vectorsin p, i.e., can be expressed in the form

V . y = axxx + \342\200\242\342\200\242\342\200\242+ anxn

for unique scalars al5..., an.

Proof Let /? be a basis for V. If y e V, then y e span(/?) since span(/?) = V.Thus y is a linear combination of the elements of /?. Suppose thaty

=

axxx 4-...4- anx\342\200\236and y = bxxx 4--..4- bnx\342\200\236are two such representationsof y. Subtracting the second equalityfrom the first gives

0 = {ax -bx)xx + '--+{an-bn)xn.Since ft is linearly independent, it follows that ax \342\200\224bx = \342\200\242\342\200\242\342\200\242= an \342\200\224bn = 0.Thus ax = bx,...,an = b\342\200\236,so that y is uniquely expressible as a linearcombination of the elements of /?.

The proof of the converseis an exercise. 1

Theorem 1.7 shows that each vector v in a vector space V with basis ft =

{xl9 -.., xn} can be uniquely expressedin the formv = a1x1 +\342\200\242... +anxn

for appropriately chosen scalars al5..., an. Thus v determines a unique n-tupleof scalars {au...,a\342\200\236), and conversely, each n-tuple of scalars determines aunique vector v by using the entries of the n-tuple as the coefficientsofa linear


combination of the vectors in /?.Thisfact suggests that V is like the vector spaceFn,where n is the number of vectors in a basis for V. We will seein Section2.4that this is indeed the case.

Theorem 1.9 identifies a large classof vector spaces having finite bases.First, however, we must prove a preliminaryresult.

Theorem 1.8. Let S be a linearly independent subset of a vectorspaceV,and let x be an element of V that is not inS. Then S u {x} is linearly dependent ifand only if x espan(S).

Proof If 5u {x} is linearly dependent, then there are vectors xx,..., xnin Su{x} and nonzero scalars al9...,an such that axXi+ \342\200\242\342\200\242\342\200\242+ anx\342\200\236= 0.Because S is linearly independent, one of the xh say x^ equals x. Thusaix + fl2*2 + \342\226\240\342\226\240'+ anXn = 0, and so

x = a^\\-a2x2 anxn).Sincex is a linear combination of x2,...,xn, which are elements of S,x e span(S).

Conversely, suppose that x 6 span(S). Then there exist vectors xXl'..., xn in S and scalars au ..., a\342\200\236such that x = axxx + \342\226\240\342\200\242\342\226\240+ anxn.So 0 = axxx + \342\226\240\342\200\242\342\200\242+ anx\342\200\236+ (\342\200\224l)x, and since x # x* for i = 1,..., n,{x1,...,x\342\200\236,x} is linearly dependent. Thus 5u{x} is linearly dependent byTheorem1.6. 1

Theorem 1.9. If a vector space V is generated by a finite set S0, then asubset of S0 is a basis for V. HenceV has afinitebasis.

Proof If S0 = 0 or so = {\302\260}>then V = {0} and 0 is a subsetof S0 that isa basis for V. Otherwise S0 contains a nonzeroelement xx. Recall that {xx} is alinearly independent set. Continue,if possible,choosingelements x2]...,xr inS0 so that {xx, x2, -.., xr}is linearly independent. Since S0 is a finite set, wemust eventuallyreacha stage at which S = {xu..., xr} is a linearlyindependentsubset of 50, but adjoining to S any elementofS0notinSproduces a linearlydependent set. We will show that S is a basis for V. Because S is linearlyindependent, it sufficesto provethat span(S)= V. Since span(50) = V, it sufficesby Theorem 1.5 to show that S0 Q span(S). Let x 6 50. If x 6 5, then clearlyx 6 span(S). Otherwise, if x fi 5, then the construction' above shows that S u {x}is linearly dependent.So x 6span(S)by Theorem 1.8. Thus S0 c span(S). 1

Themethodby which the basis S was obtained in the proof ofTheorem1.9is a useful way of obtaining bases. An example of this procedureis given below.Example 6

The reader should check that the elements(2,-3,5), (8, -12,20), (1,0, -2),(0,2, \342\200\2241),and (7,2,0) generate R3. We will select a basis for R3 from among


these elements.To start,selectany nonzero element from the generating set, say(2, \342\200\2243,5), as one of the elements of the basis.Since4(2,-3,5) = (8, \342\200\22412,20), theset {(2, \342\200\2243,5), (8, \342\200\22412,20)} is linearly dependent (Exercise 6 of Section 1.5).Hence we do not include(8,\342\200\22412,20) in our basis. Since (1,0, \342\200\2242)is not amultiple of (2, \342\200\2243,5), and vice versa, the set {(2, \342\200\2243,5), (1,0, -2)} is linearlyindependent. Thus we include (1,0, \342\200\2242)in our basis. Proceeding to the nextelement in the generatingset,we exclude from or include into our basis theelement (0,2, -1) accordingto whether the set {(2, - 3,5), (1,0,- 2),(0,2,-1)}islinearly dependent or linearly independent. An easy calculation shows that theset is linearly independent; so we include (0,2, \342\200\2241) in our basis. The finalelement of the generating set (7,2,0)is excluded from or included into our basisaccording to whether {(2,-3,5), (1,0,-2), (0,2,-1), (7,2,0)}is linearlydependent or linearly independent. Since

2(2,.-3, 5) + 3(1,0, -2) + 4(0, 2,-1)- (7, 2, 0) = (0, 0, 0),the set is linearly dependent and we exclude (7,2,0) from the basis. So the set{(2,-3,5), (1,0,-2), (0,2,-1)}is a basis for R3. I

The following theorem and itscorollariesare perhaps the most significantresults in Chapter 1.

Theorem1.10(Replacement Theorem). Let V be a vector space having abasis P containing exactly n elements. Let S = {yu...,ym} be a linearlyindependent subset of V containing exactly m elements, where m < n. Then thereexists a subset Sx of ft containing exactly n \342\200\224m elements such that SuSxgenerates V.

Proof The proof will be by induction on m. The induction begins withm = 0; for in this caseS= 0, so 5X = /? clearly satisfies the conclusion of thetheorem.

Now assume that the theorem is true for some m, where m < n. We willprove that the theorem is true form + 1. Let S = {_yx,..., ym, ym+i} be a linearlyindependent subset of V containing exactly m + 1 elements. Since {yl9..., ym} islinearly independent by the corollaryto Theorem 1.6, we may apply theinductive hypothesis to conclude that thereexists a subset {xx,..., x\342\200\236_m} of j?such that {^1,...,ym}u {x1,...,x\342\200\236_m} generates V. Thus there exist scalarsau...,am,b1,b2,..., bn\342\200\236msuch that

Observe that some bh say fcx,is nonzero,forotherwise(9) would imply that ym+lis a linear combination of yu ..., ym, in contradiction to the assumption that{yu* \342\200\242\342\200\242>J'ms J'm+i} is linearly independent. Solving (9) for xx gives*i = (-bll*i)yi + \342\200\242'\342\200\242+ i-b:xam)ym - {-b^l)ym+1 + {-b;lb2)x2

Sec.1.6 Bases and Dimension 39

Hence xx 6 span({ j^,..., ym, ym+u x2, \342\200\242\342\200\242.,xn-m}). But since yu ..., ym3 x2,...,xn_m are clearly elements of span({3>x,..., ym, ym+u x2, \342\200\242..,xn_m}), it followsthat

Xi, X2,. \342\200\242., } cspand^,..., Xn-m})-

Thus Theorem 1.5 implies that

span({3>x,..., *2 Vm}) = V-So the choice of 5X \342\200\224{x2,..., xn_m} proves that the theorem is true for m 4- 1.

This completes the proof. 1

To illustrate the replacementtheorem, note that 5 = {x2 \342\200\2244, x 4- 6} is alinearly independent subset of P2(F).Since/? = {l,x,x2} is a basis for P2(^)sthere must be a subset 5X of /? containing 3 \342\200\2242=1 element such that S\\j 5Xgenerates P2(^). In this example any subset of ft containing one element willsuffice for 5X. Hence we see that theset 5X in Theorem 1.10 need not be unique.

Corollary 1. Let V be a vector space having a basis p containingexactlynelements. Then any linearly independent subset of V containing exactly n elementsisa basis for V.

Proof. Let S = {yu..., yn} be a linearly independent subset of Vcontaining exactly n elements. Applying the replacement theorem, we see that thereexists a subset 5X of /? containing n \342\200\224n = 0 elements such that S u 5X generatesV. Clearly, 5X = 0; so S generatesV.SinceSisalsolinearly independent, S is abasis for V. 1Example7The vectors (1, \342\200\2243,2), (4,1,0), and (0,2, \342\200\2241) form a basis for R3S for if

ax(l, -3, 2)+ a2(4, 1, 0) + a3(0, 2, -1) = (0, 0, 0),then au a2, and a3 must satisfy the system of equations

ax + 4a2 = 0

< \342\200\2243a2 + a2 4- 2a3 = 0

. 2a x \342\200\224a3 = 0. iBut it is easily seen that the only solution of this system is ax = 0, a2 = 0, anda3 = 0. Hence (1, \342\200\2243,2), (4,1,0), and (0,2, \342\200\2241)are linearly independent andtherefore form a basis for R3by Corollary1.Note that we do not have to checkthat the givenvectorsspanR3. I

Corollary 2. Let V be a vector space havinga basisp containing exactly nelements. Then any subset of V containing more than n elements is linearlydependent. Consequently,any linearlyindependentsubset of V contains at most nelements.

40 Chap,1 Vector Spaces

Proof Let S be a subset of V containing more than n elements. In order toreach a contradiction,assume that S is linearly independent. Let 5X be anysubset of S containing exactly n elements; then 5X is a basis for V by Corollary 1.Because 5X is a proper subset of S, we can select an elementx ofS that is not anelement of 5X. Since 5X is a basis for V}xe span(5x) = V. Thus Theorem 1.8impliesthat 5X u {x} is linearly dependent. But 5X u {x} \302\243S; so 5 is linearlydependent\342\200\224a contradiction. We conclude therefore that S is linearlydependent. 1Example S

Let S = {x2 + 7, 8x2 \342\200\2242x, 4x \342\200\2243, Ix 4- 2}. Although we can prove directlythat S is a linearly dependent subset of P2{F), this conclusion followsimmediately from Corollary 2 since ft = {l,x,x2} is a basis for P2(*r)containingfewer elementsthan S. I

Corollary 3, Let V be a vector space having a basisft containing exactly nelements. Then every basis for V contains exactly n elements.

Proof. Let S be a basisforV.SinceSis linearly independent, S contains atmost n elements by Corollary2. Supposethat S contains exactly m elements;then m ^ n.But alsoSisabasisfor V and /? is a linearly independent subset of V.So Corollary2 may be applied with the roles of /? and S interchanged to yieldn ^m. Thus m = n. I

If a vector space has a basiscontaininga finite number of elements, thenthe corollary 'above asserts that the numberof elements in each basis for thespace is the same. This resultmakes the following definitions possible.

Definitions. A vector space V is called finite-dimensional if it has a basisconsisting of a finite number of elements; the unique number of elements in eachbasisfor V is called the dimension of V and is denoted dim(V). If a vectorspaceisnot finite-dimensional, then it is called infinite-dimensional.

The following results are consequencesof Examples 1 through 5.Example 9

The vector space {0}has dimensionzero. I

Example 10-

\342\226\240

The vector space Fn has dimension n. 1

Example11The vector space Mmxn(F) has dimension mn.

Sec. 1.6 Basesand Dimension 41

Example 12

The vector space Pn(^) has dimension n + 1. I

Example 13

The vector space P(F) is infinite-dimensional. I

The followingtwo examples show that the dimension of a vector spacedependson its field of scalars.Example 14

Over the field of complex numbers,the vector space of complex numbers hasdimension 1. (A basis is {1}.) 1

Example 15

Over the fieldof real numbers, the vector space of complex numbers hasdimension 2. (A basis is {l,z}.) I

Corollary 4. Let V be a vector space having dimension n, and let S be asubset of V that generates V and contains at most n elements.ThenS isa basis forV and hence contains exactly n elements.

Proof. Thereexistsa subset 5X of S such that 5X is a basis for V (Theorem1.9). By Corollary 3,5X contains exactly n elements. But 5X Q S and S contains atmost n elements. Hence S= 5X; so S is a basis for V. 1Example16It follows from Example 4 of Section 1.4 and Corollary 4 that

{x2 + 3x- 2, 2x2 + 5x - 3, -x2 - Ax + 4}

.is a basis for P2tR). I

Example17It follows from Example 5 of Section 1.4 and Corollary 4 that >

l oMo iMi lMi l

is a basisfor M2 x 2(8)- B

Corollary 5. Let P be a basisfor a finite-dimensional vector space V, andlet Sbea linearly independent subsetof V. There exists a subset Sx of fi such thatSuSx is a basis for V. Thus every linearly independentsubset of V can beextended to a basis for V.


Proof. Let dim(V) = n. By Corollary 2 we know that S must contain melements, where m ^n. Hence the replacement theorem guarantees that there isa subset5X of /? containing exactly n \342\200\224m elements such that S \\j 5X generates V.Clearly, S u 5X contains at most n elements;so Corollary4 implies that S u 5X isa basis for V. I

Theorem1.9,the replacement theorem, and the five corollaries of thereplacement theoremcontaina wealth of information about the relationshipsamong linearly independent sets, bases, and generatingsets.For this reason wewill summarize here the main results of thissectionin order to put them intobetter perspective.

A basis for a vector space V is a linearly independent subsetthat generatesV. If V has a finite basis, then every basis for V contains the same number ofvectors. This number is called the dimensionof V, and V is said to be finite-dimensional. Thus ifthedimensionof V is n, every basis for V contains exactly nvectors. Moreover, every linearlyindependentsubsetof V contains no morethan n vectors'and can be extended to a basisfor V by including appropriatelychosen vectors. Also, each generating set for V contains at least n vectors andcan be reduced to a basisfor V by excluding appropriately chosen vectors. TheVenn diagram in Figure1.6depictsthese relationships. We will see in Section 2.4that every vectorspaceover F of dimension n is essentially Fn.

Figure1.6

The LagrangeInterpolationFormula

The preceding results can be applied to obtain a usefulformula.Let c0, cx,cn, be distinct elements in an infinite fieldF. Thepolynomialsf0{x\\ /x(x),fn{x\\ where

Mx) ={x - c0)\342\200\242\342\200\242\342\200\242(x - Cj-tX* - c{+1)\342\200\242\342\200\242\342\200\242(x - cn)fe - Cn) \342\200\242\342\200\242\342\200\242fo - Cj- i){Ci- Ci+ x) \342\200\242\342\200\242\342\200\242{Ci - cn)

Sec. 1.6 Basesand Dimension 43

are called the Lagrange polynomials (associated with c0, cu ..., cn). Note thateach ft(x) is a polynomial of degree n and henceis an element of Pn(F). Byregarding fi(x) as a polynomial function fx\\ F ~* F, we see that

We will use this propertyof the Lagrange polynomials to show thatP = {To*fu - \342\200\242\342\200\242* fn) *s a linearly independent subset of Pn(F). Since thedimension of Pn(F) is n + 1, it will follow from Corollary1ofTheorem 1,10 that ft is abasis for Pn(F). To show that ft is linearly independent, suppose that

71

E aJt= 0 for some scalars a0, au..., an,

i~0

where 0 denotes the'zero function. Then71

E aift(cj) = \302\260 for 7 = 0, 1,...,\302\253.1=0

But also71

E flt/l(cj) = a;

by (10). Hence a, = 0 for j = 0, 1, ..., n; so /? is linearly independent.Because j? is a basis for Pn(F),every polynomialfunctiong in Pn(F) is a

linear combination of elements of /?, say

Then

9 = I bj.

9(cj)=

E bJt(cj) = by,i=0so

9 = E fifcJ/ii~0

is the unique representation of g as a linearcombinationof elements of /?. Thisrepresentation is called the Lagrange interpolation formula* Notice that theargument above shows that if b0, bu ..., bn are any n + 1 elements of F (notnecessarilydistinct),then the polynomial function

g = t bJii-ois the uniqueelement of Pn(F) such that g(cj) = by Thus we have found theunique polynomial of degree not exceeding n that has specifiedvaluesbjat given


points Cj in its domain (j = 0,1,..., n). For example, let us construct the realpolynomial g of degreeat most 2 whose graph contains the points (1,8), (2,5),and (3, \342\200\2244).(Thus in the notation above c0 = 1, cx = 2, c2 = 3, b0 = 8, fcx = 5,and b2 \342\200\224\342\200\2244.)The Lagrange polynomials associated with c0, cx, and c2 are

,,. (x-2Xx-3) 1, 2 . ,./oW = (l-2)(l-3)=2(X \"5X + 6)-

^ , v (x \342\200\224l)(x \342\200\2243) ,,-/lW = (2-l)(2-3)

=-1(x2-4x + 3)-

and

(x-l)(x-2) 1^W \" (3 _ 1)(3_ 2)

-

2(X3X + 2)l

Hence the desired polynomial is2

9(x) = E &\302\253/,(x) = 8/o(x) + 5/i(x) - 4/2(*),=o

= 4{x2- 5x + 6) - 5(x2 - 4x + 3) - 2(x2 - 3x + 2)=

-3x2 + 6x + 5.An important consequence of the Lagrange interpolation formula is the

following result: If f e Pn(F) and f(cj) = 0 for n + 1 distinct elements c0, cx,...,cn

in F, then f is the zero function.

The Dimensionof SubspacesOur next result relates the dimension of a subspace to thedimensionof the

vector space that contains it.

Theorem 1.11. Let W fte a subspace of a finite-dimensional vector space V.Then W is finite-dimensionaland dim(VV) ^ dim(V). Moreover, ifdim(W) = dim(V)| then W = V.

Proof Let dim(V) = n. If W = {0}, then W is finite-dimensional anddim(VV)= 0 ^ n. Otherwise, W contains a nonzero element xx; so {xx} is alinearlyindependentset.Continuing in this way, choose elements xu x2,..., xkin W such that {xt,x2, \342\200\242\342\200\242\342\200\242,xk} is linearly independent. Since no linearlyindependent subset of V can contain more than n elements, this process muststop at a stagewhere k ^ n and {xlt x2,..., xk} is linearly independentbutadjoiningany other element of W produces a linearly dependent set. Theorem1.8now implies that {xu x2,..., xk} generates W, and hence it is a basisfor W.Therefore, dim(VV) = k ^ n.

Sec.1.6 Basesand Dimension 45

If dim(W) = n, then a basis for W is a linearly independent subset of Vcontaining n elements. But Corollary 1of Theorem 1.10 implies that this basisfor W is also a basisfor V; so W = V. I

Example IS

LetW = {{au a2, a3i'aAi as) e F5: ax + az + a5= 0, a2 = a4}.

It is easily shown that W is a subspace of F5 having

{(1, 0, 0, 0, -1), (0,0, 1,0,-1), (0, 1, 0, 1, 0)}as a basis. Thus dim(VV)= 3. 1

Example 19

The set of diagonal n x n matrices is a subspaceW of Mn xn(F) (see Example 1 ofSection 1.3).A basis for W is

{M11,M22,...,MBB},where MlJ is the matrixin which the only nonzero entry is a 1 in the ith row andjth column. Thus dim(VV) = n. I

Example20We saw in Section 1.3 that the set of symmetric n x n matricesis a subspace Wof MnxnW A Dasis f\302\260rw is

{Aij: 1 < i ^ j < n},whereAij is the n x n matrix having 1 in the ith row and J th column, 1 in the j throw and ith column, and 0 elsewhere. It follows that

dim(W)= \302\253+ (\302\253- 1) + \342\226\240\342\226\240\342\200\242+ 1 = |\302\253(\302\253-4- 1). I

Corollary. IfWisa subspace of a finite-dimensionalvectorspaceV, thenW has a finite basis, and any basisfor W is a subset of a basis for V.

Proof Theorem 1.11showsthat W has a finite basis S. If j? is any basis forV, then the replacement theorem shows that there exists a subset 5X of /? suchthat S\\j 5X is a basis for V. HenceS is a subset of the basis Su 5X for V. IExample21The set of all polynomials of the form

ai8x18 + ai6x16+ \342\200\242\342\200\242\342\200\242+ a2x2 + a0>where a18, a16,..., a2s a0 6 F, is a subspace W of Pi8(F). A basis for W is{1,x2,..., x16,.x18},which is a subset of the standard basis for Pia(^)- 1


We conclude this section by using Theorem 1.11 to determine thesubspaces of R2 and R3. Since R2 has dimension 2, subspacesof R2 can be ofdimensions 0,1, or 2 only. The onlysubspacesof dimensions 0 or 2 are {0} andR2, respectively.Any subspace of R2 having dimension 1 consists of all scalarmultiplesof some nonzero vector in R2 (Exercise 9 of Section 1.4).

If a point of R2 is identified in the natural way with a point on theEuclideanplane, then it is possible to describe the subspaces of R2geometrically:A subspace of R2 having dimension 0 consists of the origin of the Euclideanplane, a subspace of R2 with dimension 1 consists of a line through the origin,and a subspace of R2 having dimension 2 is the entire Euclidean plane.

As above, the subspaces of R3 must have dimensions 0, 1, 2, or 3.Interpreting thesepossibilitiesgeometrically, we see that a subspace ofdimension zero must be the origin of Euclidean 3-space, a subspaceofdimension1 is aline through the origin, a subspace of dimension 2 isa plane through the origin,and a subspace of dimension 3 is Euclidean3-spaceitself.

EXERCISES

1. Label the following statements as being true or false.(a) The zero vector space has no basis.(b) Everyrvector space that is generated by a finite set has a basis.(c) Every .Vectorspacehasa finite basis.(d) A vector space cannot have more than one basis.(e) If a vector space has a finite basis, then the numberof vectors in every

basis is the same.

(f) The dimension of P\342\200\236(F)is n.(g) The dimension of Mmxn(F) is m + n.(h) Supposethat V is a finite-dimensional vector space, that 5X is a linearly

independent subset of V, and that S2 is a subsetof V that generates V.Then 5X cannot contain more elements than S2>

(i) If S generates the vector space V, then every vector in V can be writtenas a linear combination of elements of S in only one way.

(j) Every subspace of a finite-dimensional space is finite-dimensional.(k) If V is a vector space having dimension n, then V has exactly one

subspace with dimension 0 and exactly onesubspacewith dimension n.2. Determine which of the following sets are basesfor R3.

(a) {(1,0,-1),(2,5,1),(0,-4,3)}(b) {(2,-4,1),(0,3,-1),(6,0,-1)}(c) {(1,2,-1),(1,0,2),(2,1,1)}(d) {(-1,3,1),(2,-4,-3),(-3,8,2)}(e) {(1,-3, -2),(-3, 1,3),(-2,-10,-2)}


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