Elementary Linear Algebra [5 ed.] 0471848190, 9780471848196

Citation preview

Howard Anton

Elenwntar]/ Linear Algebra Se

?-

ELEMENTARY LINEAR ALGEBRA EDITION FIFTH

HOWARD ANTON Drexel University

JOHN WILEY & SONS New York

•

Chichester Brisbane Toronto Singapore •

•

•

Copyright

©

1973, 1977, 1981, 1984,

All rights reserved.

and 1987, by Anton Textbooks,

Published simultaneously

in

Canada.

Reproduction or translation of any part of this

work beyond

that permitted by Sections

107 and 108 of the 1976 United States Copyright

Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department,

John Wiley

&

Sons.

Library of Congress Cataloging in Publication Data

Anton, Howard.

Elementary linear algebra. Includes index. 1.

Algebra, Linear.

QA184.A57

ISBN

1987

I.

Title.

512'.5

0-471-84819-0

Printed in the United States of America

10

9

8

7

86-11087

Inc.

To Pat and

my

Brian, David,

children:

and Lauren

Digitized by the Internet Archive in

2010

http://www.archive.org/details/elementarylinearOOanto

Preface

This textbook provides an elementary treatment of linear algebra that is suitable for students in their freshman or sophomore year. Calculus is not a prerequisite. I have, however, included a number of exercises for students with a calculus background; these are clearly marked: "For students who have studied calculus." My aim in writing this book is to present the fundamentals of hnear algebra in the clearest possible way. Pedagogy is the main consideration; formalism is

secondary. Where possible, basic ideas are studied by means of computational examples (over 200 of them) and geometrical interpretation. My treatment of proofs varies. Those proofs that are elementary and have significant pedagogical content are presented precisely, in a style tailored for be-

A few

more difficult, but pedagogically valuable, are placed at the ends of the sections and marked "Optional." Still other proofs are omitted completely, with emphasis placed on applying the theorem. Whenever a proof is ginners.

omitted,

I

proofs that are

try to

motivate the

result, often

with a discussion about

its

interpretation

in 2-space or 3-space. It is

my

experience that Z-notation

is

more

of a hindrance than a help for

have generally avoided its use. should proceed from the familiar to teacher pedagogical axiom that a It is a The ordering of the chapters the abstract. from the concrete to unfamiliar and the adherence to this tenet. reflects my

beginners in linear algebra. Therefore,

Chapter

1

I

deals with systems of linear equations,

of their properties.

It

how

also contains the basic material

to solve them,

on matrices and

and some

their arith-

metic properties.

have used the classical permutation than the approach through n-linear alternating forms, gives the student a better intuitive grasp of the subject than an Chapter 2 introduces determinants.

approach. In

my

opinion,

it

is

I

less abstract

vi

I

Preface

inductive development,

vanced topics

and provides the best foundation

for a future study of ad-

in linear algebra.

Chapter 3 introduces vectors in 2-space and 3-space as arrows and develops the analytic geometry of lines

and planes

in 3-space.

Depending on the background

of the students, this chapter can be omitted without a loss of continuity (see the

guide for the instructor that follows this preface).

Chapter 4 and Chapter 5 develop the basic results about real finite-dimenand linear transformations. I begin with a study of /?" and

sional vector spaces

proceed slowly to the general concept of a vector.

Chapter 6 deals with the eigenvalue problem and diagonalization. Chapter 7 gives some applications of linear algebra to problems of approximasystems of differential equations, Fourier series, and identifying conic sections and quadric surfaces. Applications to business, biology, engineering, economics, the social sciences, and the physical sciences are included in an optional paperback tion,

supplement to this text. Applications of Linear Algebra, by Chris Rorres and the author. Included in the supplement are such topics as curves of best fit to empirical data, population dynamics, Markov processes, optimal harvesting, Leontief models in economics, engineering applications, graph theory, and an introduction to linear programming. There is also an alternate version of this text, called Elementary Linear Algebra with Applications, that combines material in Applications of Linear Algebra.

this text

and most of the

Chapter 8 introduces numerical methods of linear algebra; it does not require computing facilities since the exercises can be solved by hand computa-

access to

tion or with the use of a pocket calculator. This chapter gives the student a basic

understanding of

many

how

certain linear algebra problems are solved practically.

Too

students complete their linear algebra studies with the naive belief that

ei-

Some inLIN EAR-KIT'^^

genvalues are found in practice by solving the characteristic equation. structors

may

wish to use

software package that

is

this section in

conjunction with the

available as a supplement to this text.

Chapter 9 gives a brief introduction to complex numbers and then proceeds development of complex vector spaces and inner product spaces. Unitary,

to the

Hermitian, and normal matrices are studied, and the chapter concludes with the

proof that symmetric matrices have real eigenvalues.

I

have included a large number of exercises. Each exercise set begins with drill problems and progresses towards more theoretical problems. Answers the computational problems are given at the end of the text.

routine to all

Preface

Since there

is

more material in

this

or one-quarter course, the instructor help in this selection,

I

/

vii

book than can be covered

will

in a one-semester have to make a selection of topics. To

have provided a guide to the instructor following

this

preface.

NEW FEATURES

IN

The wide acceptance of grateful for the

from •

users. In

many

THE FIFTH EDITION the

first

favorable

four editions has been most gratifying and

comments and

response to these suggestions

I

I

am

constructive suggestions received

have made the following changes:

During the 14 years that have passed since this text was first pubhshed, the widespread use of computers has significantly increased the importance of numerical methods in linear algebra. In recognition of this I have rewritten Chapter 8, which deals with this topic. There is a new section on LU-decompositions and a new section in which the various methods for solving linear systems are compared in terms of the number of operations they require. The inverse power method for finding "smallest" eigenvalues has been introduced, and the student is given more information on the advantages and disadvantages of various numerical methods. However, this chapter is still intended only as a brief introduction to the topic.

•

The Supplementary

Exercises have been greatly expanded to provide a richer

variety of problems. •

Examples have been added and exposition improved

•

To

in

many parts

of the text.

reduce the level of abstraction, the material on length and angle

in

inner

product spaces has been rewritten, and more examples of inner products have been added. •

As a general rule, I prefer to explain concepts in words rather than symbols. However, so many readers asked that I introduce the notation [Tjg for the matrix of a Hnear operator with respect to a basis B and [TJ^ ^ for the matrix of a linear transformation with respect to bases B and B' that I have done so.

•

New

material on positive definite matrices and quadratic forms has been

added.

SUPPLEMENTARY MATERIALS Manual for Elementary Linear Algebra by Elizabeth M. Grobe and Charles A. Grobe, Jr. contains detailed solutions to most theoret-

• Student Solutions

ical exercises

and many computational

exercises.

via

Preface

I

• Applications of Linear

Algebra by Chris Rorres and

Howard Anton

contains

applications to business, biology, engineering, economics, the social sciences,

and the physical sciences. This supplement also contains a "minicourse" linear programming which can be covered in about six lectures. • Solutions

Manual for Applications of Linear Algebra contains

in

detailed solu-

tions to all exercises. •

Elementary Linear Algebra with Applications by Rorres

is

an expanded version of

Howard Anton and

this text that includes

Chris

most of the material

from Applications of Linear Algebra. •

LINEAR-KIT^^ — A new most of the basic mals.

Apple •

It is

lie.

Hnear algebra software package that can perform

linear algebra

available for

IBM

lie,

and the

This software requires 128K of memory.

LINEAR-KIT^^ LINEAR-KIT'^^

—

A book of linear algebra problems LIN EAR-KIT'^^ software package.

Problem Book

signed to be solved using the •

computations using either fractions or deci-

compatible computers, the Apple

Applications Problem

Book

— A book of

plications problems designed to be solved using the

ware package. Intended

de-

linear algebra ap-

LINEAR-KIT'^

soft-

use with Applications of Linear Algebra or Elementary Linear Algebra with Applications.

If

any of these supplements

is

for

not in stock, ask your bookstore manager to order

a copy for you.

HOWARD ANTON

A Guide for the Instructor

STANDARD COURSE Chapter

3

can be omitted without loss of continuity

if

the students have previ-

and geometric vectors in 2-space and 3-space. Depending on the available time and the background of the students, the instructor may wish to add all or part of this chapter to the following suggested core material: ously studied

lines, planes,

Chapter 1 Chapter 2 Chapter 4 Chapter 5 Chapter 6

7 lectures 5 lectures

14 lectures 6 lectures 5 lectures

This schedule

is

rather liberal;

it

allows a

fair

discussion of homework problems but assumes that to the material

marked

"optional."

The

amount little

instructor can build

permits by including lectures on optional material, Chapter 8,

and Chapter

of classroom time for

classroom time

3,

on

this

is

devoted

core as time

Chapter

7,

Chapter

9.

APPLICATIONS-ORIENTED COURSE Instructors

who want

to

emphasize applications or numerical methods

the following program, which reaches eigenvalues

Chapter 1 Chapter 2

7 lectures

Sections 4.1-4.6

8 lectures

Sections 5.1-5.2

3 lectures

Sections 6.1-6.2

3 lectures

may

and eigenvectors more

prefer

quickly:

5 lectures

IX

X

I

A Guide for

the Instructor

Whenever Sections

5.4

material at the end of nalization problem,

Once Chapter

7,

and 6.1,

and

5.5 are deleted, the instructor

delete

the above material

Chapter

8,

should omit the optional

begin Section 6.2 with the matrix form of the diago-

Example 9 is

in that section.

covered, the instructor can choose topics from

or Chapter 9 in the text or else choose topics from the

supplement, Applications of Linear Algebra. Depending on which topics are selected, it may be necessary to cover some of the material in Sections 4.7-4.10 or 5.3-5.5.

COMPUTER-ORIENTED COURSE Either the standard course or the application-oriented course can be taught in con-

junction with the

LINEAR-KIT^^ software together with the LINEAR-KIT^^ LINEAR-KIT^^ Applications Problem Book to supplement

Problem Book or the the exercises.

will

Because problems can be solved so quickly using LINEAR-KIT'^, students much less time to solve homework problems, which should allow them

need

to cover

more material or

the

same material

in

more depth.

A ckno wiedgments

I

express

my

appreciation for the helpful guidance provided by the following

people:

Joseph Buckley, Western Michigan University Harold S. Engelsohn, Kingsborough Community College Lawrence D. Kugler, University of Michigan

Robert W. Negus, Rio Hondo Junior College Hal G. Moore, Brigham Young University William A. Brown, University of Maine at Portland-Gorham Ralph P. Grimaldi, Rose-Hulman Institute of Technology Robert M. McConnel, University of Tennessee James R. Wall, Auburn University Roger H. Marty, Cleveland State University Donald R. Sherbert, University of Illinois Joseph L. Ullman, University of Michigan Arthur G. Wasserman, University of Michigan J. Hightower, University of Colorado Marjorie E. Fitting, San Jose State University

Collin

Bruce Edwards, University of Florida Garret Etgen, University of Houston

Donald P. Minassian, Butler University David E. Flesner, Gettysburg College Arlene Kleinstein Bart

S.

Ng, Purdue University

Mathew Gould, C.

S.

Ballantine,

Vanderbilt University

Oregon State University

Douglas McLeod, Drexel University Craig Miller, University of Pennsylvania William Scott, University of Utah F. P. J. Rimrott, University of Toronto

xii

I

Acknowledgments I

am

also

grateful

to

William

F.

Trench of Trinity University, whose

suggestions immeasurably improved both the style and content of the text and

Steven C. Althoen of the University of Michigan-Flint trary to is

pointed out that, con-

named

statement in previous editions, the Gauss-Jordan reduction procedure after the German engineer Wilhelm Jordan and not the famous mathe-

matician, Camille Jordan.

work

who

my

due to Dale Lick who encouraged my Corey who helped make the first edition Gary W. Ostedt, who guided this book through

Thanks are

also

in its early stages, to Frederick C.

a reahty, and to

my

former editor,

three successful editions. Finally,

I

am

Wiley and especially to my editor, the development of this new edition.

indebted to the entire production staff of

Robert

Pirtle, for his

invaluable help during

H. A.

Contents

Chapter One SYSTEMS OF LINEAR EQUATIONS AND MATRICES 1.1

1.2

1.3 1.4 1.5 1.6 1.7

Introduction to Systems of Linear Equations 1 Gaussian Elimination 8 Homogeneous Systems of Linear Equations 19 Matrices and Matrix Operations 23 Rules of Matrix Arithmetic 31 Elementary Matrices and a Method for Finding A~^ 43 Further Results on Systems of Equations and Invertibility

Chapter

57

Two

DETERMINANTS 2.1

1

65

The Determinant Function

65

12 Evaluating Determinants by Row Reduction Determinant Function lA Cofactor Expansion; Cramer's Rule 84

2.3 Properties of the

72

78

\\

Chapter Three

VECTORS IN 2-SPACE AND 3-SPACE

99

3.1

Introduction to Vectors (Geometric)

3.2

Norm

3.3

Dot Product; Projections

3.4

Cross Product

3.5 Lines

of a Vector; Vector Arithmetic

99 109

112

122

and Planes

in 3-space

130

Xlll

xiv

I

Contents

Chapter Four VECTOR SPACES 143 143

4.1

Euclidean n-Space

4.2

General Vector Spaces

4.3 4.4

Subspaces 755 Linear Independence

164

4.5

Basis and Dimension

171

4.6

Row and Column

4.7

Inner Product Spaces

150

Space; Rank; Finding Bases

779

797

Length and Angle in Inner Product Spaces 198 Orthonormal Bases; Gram-Schmidt Process 209 227 4.10 Coordinates; Change of Basis 4.8 4.9

I"

Chapter Five LINEAR TRANSFORMATIONS 5.1

245

Introduction to Linear Transformations

5.2 Properties of Linear Transformations; 5.3

5.4

254

Linear Transformations from R" to R!"; Geometry of Linear Transformations from R^ to R^ 263 Matrices of Linear Transformations 280

5.5 Similarity

290

Chapter Six EIGENVALUES, EIGENVECTORS 6.1

245

Kernel and Range

Eigenvalues and Eigenvectors

6.2 Diagonalization

301

301

309

v\6.3 Orthogonal Diagonalization; Symmetric Matrices

318

Chapter Seven APPLICATIONS* 329 329 Approximation Problems; Fourier Series 13 Quadratic Forms 342 lA Diagonalizing Quadratic Forms; Application to Conic

7.1

Application to Differential Equations

7.2 Application to

Sections

335

i57

7.5 Application to

Quadric Surfaces

364

/^ *

Additional applications to business, economics, and the physical and social sciences are available in

the supplement to this text. Applications of Linear Algebra or in the

Linear Algebra with Applications.

expanded version of this

text.

Elementary

Contents

Chapter Eight

INTRODUCTION TO NUMERICAL METHODS OF LINEAR ALGEBRA 371 8.1

Comparison of Procedures

8.2

LL/-Decompositions

8.3

The Gauss-Seidel and Jacobi Methods 391 Reduction of Roundoff Error 398 Approximating Eigenvalues by the Power Method 404 Approximating Nondominant Eigenvalues; Deflation and Inverse Power Methods 413

for Solving Linear

Systems

8.4 Partial Pivoting; 8.5

8.6 ,

./

Chapter Nine

COMPLEX VECTOR SPACES 9.1

9.2

9.3 9.4 9.5

Normal, and Hermitian Matrices

^\ ANSWERS TO EXERCISES INDEX

421

Complex Numbers 421 Modulus; Complex Conjugate; Division 429 Polar Form; DeMoivre's Theorem 437 Complex Vector Spaces 447 Complex Inner Product Spaces 454

9.6 Unitary,

II

371

382

Al

463

j

xv

CHAPTER ONE

Systems of Linear Equations and Matrices

1 .1

INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS In this section

we introduce

and discuss a method

basic terminology

for solving

systems of linear equations.

A

line in the xj'-plane

can be represented algebraically by an equation of the

form

An

equation of this kind

we

generally,

is

called a linear equation in the variables

define a linear equation in the n variables x^, X2,

that can be expressed in the

Example

,a„

,

y.

More

x„ to be one

form

a 1X1

where 0^,02,

x and

and b are

+

a 2X2

+

•

•

+

•

a„x„

=

b

real constants.

1

The following

are linear equations .V

_v

+ 3v = 7 = ix + 3z +

-

Xi 1

xi

-

2x2

+

X2

3x3

+

•

•

+ X4 = + x„ =

7 1

any products or roots of variables. power and do not appear as arguments for trigonometric, logarithmic, or exponential functions. The following are not linear

Observe that a

linear equation does not involve

All variables occur only to the first

equations:

+ 3.V- = — sin X =

X y

7

3x

+

^ x^

2_v

+

-

z

2x2

+ +

xz -^'3

= =

4 1

2

Systems of Linear Equations and Matrices

I

A Xi

=

,

s„

+ QjXj + such that the equation is

x„

=

s„.

solution of a linear equation AjXi

of « numbers Si,S2, Si,

Xj

=

S2,

.

.

.

.

,

.

.

The

of

set

•

•

+

a„x„

=

ft

solutions of the equation

all

a sequence

is

when we

satisfied

substitute

is

called

its

solution set.

Example 2 Find the solution (i)

set

-2y=\

Ax

To

of each of the following:

(ii)

-

x^

find solutions of

(i),

+

Axj

we can

7x3

=

5

assign an arbitrary value to x

or choose an arbitrary value for y and solve for and assign x an arbitrary value t, we obtain

X

=

x. If

we

and solve for >', first approach

follow the

y^2t-^

f,

These formulas describe the solution set in terms of the arbitrary parameter t. Particular numerical solutions can be obtained by substituting specific values for t. For example, t — 3 yields the solution x = 3, 3/ = 11/2 and t = —Ijl yields the solution X If

we

= —\l2,y— — 3/2. follow the second approach and assign y the arbitrary value •X

Although these formulas are

=

if

+

y

i,

=

f,

we obtain

t

from those obtained above, they yield the For example, the previous formulas gave the solution x = 3, 3; = 11/2 when t = 3, while these

same solution

set as

t

formulas yield this solution

To

different

varies over all possible real numbers.

when

=

t

11/2.

we can

assign arbitrary values to any two and solve for the third variable. In particular, if we assign arbitrary values to X2 and X3, respectively, and solve for Xj, we obtain find the solution set of

(ii)

variables s

and

t

Xj

=

5

+

—

4s

X2

7f,

=

A finite set of linear equations in the variables x^-, of

linear equations or a linear system.

called a solution of the system

x^

if

=

A

X3

s,

X2,

= .

.

t

.

,

x„

is

sequence of numbers

5^,

X2

=

^2

^n

=

^n

called a system

^i, ^2 is

5„ is

a solution of

every equation in the system. For example, the system

4Xi 3xi

has the solution Xj

However, x^ the

first

=

1,

= X2

1,X2

=

8,

-X2 + 3X3= -1 + X2 + 9x3 = -4

=

2,

X3

X3

=

1

= —1

is

since these values satisfy

both equations.

not a solution since these values satisfy only

of the two equations in the system.

Introduction to Systems of Linear Equations

/./

Not

all

systems of linear equations have solutions. For example,

/

3

we multiply

if

the second equation of the system

x+

V

+

2y

2.\-

by

1/2,

it

becomes evident

that there are .V

X

= =

4 6

no solutions since the resulting system

+y= +y=

4 3

has contradictory equations.

A is

system of equations that has no solutions

at least

can occur

one solution, in

is

it

called consistent.

is

To

said to be inconsistent.

If

there

illustrate the possibilities that

solving systems of linear equations, consider a general system of two

hnear equations

in the

unknowns

flj-v

ih.y

+ ^1 r = + h^y =

.y

and

v:

t'l

{a^.b^ both not zero)

c'2

(cij.hn

both not zero)

The graphs of these equations are lines; call them /j and l2- Since a lies on a line if and only if the numbers x and y satisfy the equation

point {x,y) of the

line,

the solutions of the system of equations will correspond to points of intersection of (a)

/i

and

Ij.

the lines

There are three /j

and

Ij

may

possibilitiees (Figure

1.1):

be parallel, in which case there

is

no intersection and

consequently no solution to the system; (b) the lines

and

/^

/,

may

intersect at only

one point,

in

which case the system

has exactly one solution; (c)

the lines

/j

and

I2

may

coincide, in which case there are infinitely

of intersection and consequently infinitely

many

/j

Ay

(c)

Figure 1.1 (a)

No

solution, (b)

One

solution, (c) Infinitely

many

points

solutions to the system.

many

solutions.

y

and

1

4

I

Systems of Linear Equations and Matrices

Although we have considered only two equations with two unknowns show later that this same result holds for arbitrary systems; that is,

here,

we

will

every system of linear equations has either no solutions, exactly one solution, or infinitely

An

many

solutions.

arbitrary system of

m

«ll-^l

«21-^1

where

x^, Xj,

.

.

.

,

+ +

+ +

^12-^2 «22-^2

•

•

•

•

•

•

+ +

unknowns and

x„ are the

unknowns

linear equations in n

dln^r,

a2„X„

= =

be written

will

^1

bj

the subscripted a's

and

6's

denote

constants.

For example, a general system of three

linear equations in four

unknowns

will

be written

«21^1

+ +

«22^2

«31^'l

+

«32'^2

«ll-^l

^12^2

The double subscripting on the that

is

on the

+ + +

«23-^3

«33^3

«14-^4

«24'^4 «34-^4

coefficients of the

= = =

^1

^2 ^3

unknowns

is

a useful device

used to specify the location of the coefficient in the system. The coefficient a,^ indicates the

equation and multiplies

we mentally keep

first

subscript

equation in which the coefficient occurs, and the

second subscript indicates which unknown If

+ + +

«13-^3

unknown

it

multiplies. Thus, a ^2

is

in the first

X2.

track of the location of the

+ 's,

the x's,

and the

= 's,

a

system of m linear equations in n unknowns can be abbreviated by writing only the rectangular array of numbers: flu

fll2

fliIn

b:

fl,,

a-,-,

a-2n

b2

b,n_

'mil

is called the augmented matrix for the system. (The term matrix is used in mathematics to denote a rectangular array of numbers. Matrices arise in many contexts; we shall study them in more detail in later sections.) For example, the augmented matrix for the system of equations

This

Xi

2xi 3xi

+ + +

X2

4x2 6x2

+ -

2x3 3x3 5x3

= = =

9 1

IS

'1

1

2

4

3

6

2

-3 -5

9" 1

1. 1

Introduction to Systems of Linear Equations

/

5

REMARK. When constructing an augmented matrix, the unknowns must be written same order in each equation.

in the

The

method for solving a system of Hnear equations is to replace the new system that has the same solution set but which is easier to solve. This new system is generally obtained in a series of steps by applying the following three types of operations to eliminate unknowns systematically. basic

given system by a

1.

Multiply an equation through by a nonzero constant.

2.

Interchange two equations.

3.

Add

a multiple of one equation to another.

Since the rows (horizontal Hnes) of an augmented matrix correspond to the

equations in the associated system, these three operations correspond to the following operations on the rows of the augmented matrix.

These are called elementary row operations. The following example illustrates these operations can be used to solve systems of linear equations. Since a systematic procedure for finding solutions will be derived in the next section, it is not necessary to worry about how the steps in this example were selected. The main effort at this time should be devoted to understanding the computations

how

and the

discussion.

Example 3 In the

left

column below we solve

on the same system by

a system of linear equations by operating

equations in the system, and in the right column

we

solve the

operating on the rows of the augmented matrix.

X

2x 3x

Add —2

+ + +

>•

4v 6v

+ -

times the

2z 3z 5r

= = =

first

9

1

1

1

2

4

3

6

equation to

the second to obtain

9 + 2r= 2.V- lz= -17 + 6v - 5r =

.\'+

3.Y

y

Add —2

2

-3 -5

times the

first

9 1

row

second to obtain 1

1

2 3

6

2

9

-7 -5

-17

to the

6

Systems of Linear Equations and Matrices

I

Add (—3

times the

first

Add —3

equatioiy to

the third to obtain

X

+

times the

row

first

to the

third to obtain "1

+ 2z= 9 2y- lz= -11 ^y- llz= -27 y

9"

2

1

2

-7

3

-11

-17 -27

Multiply the second equation by 1/2

Multiply the second row by 1/2 to

to obtain

obtain

+

X

y

+

y-

7

z

3>'- llz

Add

'1

Iz

= =

-V

i

2

27

3

-11

Add — 3

second equation/to

f- 3 times the

the third to obtain

X

+

+

2

-27

times the second

row

2z

'1

9 17 2 3 2

1

2

1

^

^

9"

^

-i2

-^2_

Multiply the third equation by

Multiply the third row by

to obtain

obtain

X

+

= 9 17 = y-\z ~ 2 );

+

Add^—

1

first

"1

2z

z= the

Add —

times the second equation\:o '

-z

=

first

1

i.^^

^

1

-2

-^

1

3

times the second

11

"1

2 1

3

Add(— ^ times the third equation] to the first and^ times the third equatlon'^

Add — V^

the second to obtain

second to obtain

= z

=

solution

is

now

evident, h.

1,

y

=

2.

z

=

3

35

1

3

times the third

2

row

1

2 1

3

=

to the

2

1

X

row

17

"1

1

=2

y

17

~1

7

2

z=

to

to obtain

17

y-

2

1

1

-2

9"

1

3

to obtain

+

The

to the

third to obtain

= y-iz = -\z = y

9"

2

1

3

to the

I

Introduction to Systems of Linear Equations

/./

EXERCISE SET 1.

N

.Vi

(c)

-Vi

(e)

Xi

|g L.

^L

y

2.

(a)

3.

2.V1.V2

+

=

2

(b)

+ 2xV' = 4 x7' - 3x3 = 5 set of

- 7 = --3xi + 4x2 -

6x

2xi

X,

(f)

7x3 4-^X4

=

for

(b)

+ —

4x2 X2

+

5">o'^^., -

= -1 = 3

+A-5

+

^-

X4

= =

a constant)

2x,

(d)

2v

I'

^

V "^ ^

" "Z—-

'

+ 4x2 - 7x3 - w + 3x + -

(b)

y»

and Xj'

+ .Y2 + X3 = sin k {k is = ^2x3 - Xj + 7 ^L = X3 ^^

7

o

4-

each of the following systems of linear equations.

+

xi

=1

.X-3

2x3

5

-Xi +

-X-3

-y,, .Yj,

each oftifeTollowing:

1'

X,

3xi

.Vi

(d) Xi

Find the augmented matrix (a)

4.

.V3

are linear equations in

3X2

Find the solution (c)

o

+ +

7

1.1

Which of the following

L- (a)

/

X3

2x2 --^3

= =

1

=

(d) X,

r

1

-2

2

O

I

\

3

O

I

O

I

I

2-

3

Find a system of linear equations corresponding to each of the following augmented

3

matrices 1

I

X3 -2.

-1