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2013 Solutions Manual for Introductory Linear Algebra Author:Dr.Mahmoud Al-Beik ىحتفتيعت انقذس ان جاALQUDS OPEN UNIVERSITY

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• 2013

Solutions Manual for Introductory Linear Algebra

Author:Dr.Mahmoud Al-Beik

ALQUDS OPEN UNIVERSITY

• 1

Al-Quds Open University

SolutionsManual forIntroductory Linear

Algebra

Author:Dr.Mahmoud Al-Beik

Evaluator :DrMukhtar Al-Shellah

2013

• 2

• 3

Introduction

The SolutionsManual for Introductory Linear Algebra has

been preparedto be complementary to the textbook to help

students to understand the scientific material.

In this Manual there are many exercises and solutions on

each section of each unit.As well as the glossary has been

prepared in both Arabic and English languages in addition to

from the textbook.

In addition to this manual and the textbook there is a

Dr.Mahmoud Al-Beik

April 2013

• 4

• 5

Textbook: Introductory Linear Algebra with Applications.

Contents

(Required Material from The Textbook)

1.Chapter1:Linear Equations and Matrices:

1.1.Exercises: 9

1.2.Exercises: 10

1.3.Exercises: 12

1.4.Exercises: 14

1.5.Exercises: 15

1.7.Glossary: 29

2. Chapter2:Determinants:

2.1.Exercises: 33

2.2.Exercises: 35

2.4 Glossary: 45

3. Chapter3:Vectors and Vector Spaces

3.1.Exercises: 49

3.2.Exercises: 50

3.4.Exercises: 51

3.5.Exercises: 52

• 6

3.6.Exercises: 53

3.7.Exercises: 54

3.9 Glossary:69

4. Chapter4:Linear Transformations and Matrices:

4.1.Exercises: 72

4.2.Exercises: 73

4.3.Exercises: 74

4.5 Glossary:89

5. Chapter5:Eigenvalues and Eigenvectors:

5.1.Exercises: 93

5.2.Exercises: 94

5.4 Glossary:102

6. Chapter6:Linear Programming:

6.1.Exercises: 105

6.3 Glossary:111

• 7

• 8

• 9

1.1.Exercises:

1.Solve the given linear systems by the method of elimination:

(a). x-2y=8

3x+4y=4

(b). 3x+2y+z=2

4x+2y+2z =8

x -y+z =4

(c). 2x+3y=13

x-2y =3

5x+2y =27

2. An oil refinery processes low-sulfur and high-sulfur fuel.Each ton of

low-sulfur fuel requires 5 minutes in the blending plant and 4 minutes in

the refining plant.Each ton of high sulfer fuel requires 4 minutes in the

plending plant and 2 minutes in the refining plant .If the plending plant

is available for 3 houresand the refining plant is available for 2 houres

,how many tons of each typ of fuel should be manufactured?.

3.A dietician is preparing a meal consisting of foods A,B , and C.Each

ounce of foodA contains 2 units of protien, 3 units offat , and 4 units of

carbohydrate.Each unit of food B contains 3units of protein ,2 units

of fat , and 1 unit of carbohydrate. Each unit of food C contains 3

units of protein ,3 units of fat , and 2 unit of carbohydrate.If the meal

must provide exactly 25 units of protein ,24 units of fat , and 21 units of

carbohydrate,how many ounces of each type of food should be used?.

• 10

1.2.Exercises:

1. Let

312

514

313

23

12

01

,412

321CandBA

If possible ,compute: BTC+A

2.If dandcbafinddcdc

baba,,,

34

24

22

22

3. Consider the following linear system:

53

3432

2323

72

zx

zyx

zyx

wx

(a) Find the coefficient matrix.

(b) Write the linear system in matrix form.

(c) Find the augmented matrix.

4. Write the linear system with augmented matrix?

6:3103

4:2001

3:8723

5:4012

5.A manufacturer makes two kinds of products,P and Q,at each of two

plants,X and Y.In making these products , the pollutants

sulfurdioxide,nitric oxide,and particulate matter are produced.The

amounts of pollutants produced are given (in kilograms)by the matrix

• 11

SulfurNitricParticulate

Dioxideoxidematter

Stateand federal ordinances require that these pollutantsbe

removed.The daily cost of removing each kilogram of pollutant is given

(in dollars)by the matrix

Plant XPlant Y

What do the entries in the matrix ProductAB tell the manufacturer?

6. A photography business has a store in each of the following cities:

New York,Denver,and Los Angeles.AParicular make of camera is

available in

automatic,semiautomic,andnonautomaticmodels.Moreover,each

camera has a matched flash unit and a camera is usually sold together

with the corresponding flash unit.The selling prices of the cameras and

flash units are given (in dollars)by the matrix

AutomaticsemiautomaticNonautomatic

The number of sets (camera and flashunit )available at each store is

given by the matrix:

oductQ

oductPA

Pr

Pr

400250200

150100300

mattereParticulat

OxideNitric

dioxideSulfur

B

1015

97

128

unitFlash

CameraA

254050

120150200

• 12

New York Denver Los Angeles

cNoautomati

ticSemiautoma

Automatic

B

250320120

120250300

100180220

(a) What is the total value of the Camera in New York?

(b) What is the total value of the flash units in Los Angeles?

1.3.Exercises:

1.If ,42

63

32

32

BandA show that AB=0.

2.If ,40

34,

02

31

32

32

CandBandA

show that AB = AC.

3. Consider two quick food companies,M and N.Each year ,companyM

keeps 3

1of its customers,while

3

2 switch to N.Each year ,N keeps

2

1 of

its customers,while2

1switch to M.Suppose that the initial distribution of

the market is given by

3

23

1

0X

(a) Find the distribution of the market after 1 year.

(b) Find the stable distribution of the market.

• 13

4.Let A and B be symmetric matrices.

(a) Show that A+B is symmetric.

(b) Show that AB is symmetric if and only ifAB=BA

5. If A is annxn matrix ,prove that

(a)AATand AT.Aare symmetric.

(b) A+AT is symmetric.

(c)A - AT is skew symmetric.

1.4.Exercises:

1.Which of the following matrices are in reduced row echelon form?

21000

40100

30001

A

31000

40100

50010

B

32010

40100

50010

C

10000

00000

41000

10000

20010

D

00000

31000

00100

20001

E

00000

01000

32100

00000

F

• 14

00000

11000

20010

10001

G

0000

1000

2010

1001

H

2. Let

515

224

413

301

A

Find the matrices obtained by performing the followingelements row

operations on A.

(a)Interchanging the second and fourth rows.

(b)Multiplying the third row by 3.

(c) Adding -3 times the first row to the fourth row .

3. If

2011

5231

5132

2021

A

Find a matrix Cin reduced row echelon form that is row equivalent to A.

4. Find all solutions to the given linear system:

5. Find all values of a for which the resulting of the following linear

systemhas:

22

32

1

zyx

zyx

zyx

• 15

(1)No solution,

(2)A unique solution,

(3)Infinity many solutions

6. Solve the linear system with thefollowing given augmented matrix:

7. A furniture manufacturer makes chairs, coffee tables, and dining-

room tables. Each chair requires 10 minutes of sanding, 6 minutes of

staining, and 12 minutes of varnishing. Each coffee table requires 12

minutes of sanding, 8 minutesof staining,and 12 minutes of

varnishing.Each dining room table requires 15 minutes of sanding, 12

minutesof staining,and 18 minutes of varnishing. The sanding bench is

available 16 hours per week, the staining bench 11 hours per week, and

the varnishing bench 18 hours per week. How many (per week) of each

type of furniture should be made?

1.5. Exercises:

1. Show that

is non-singular.

2.Show that

azayx

zyx

zyx

)5(

32

2

2

1:110

3:011

0:111

32

12A

24

12A

• 16

is singular.

3.Find the inverse of the following matrix, if possible

4. Which of the following linear systems have a nontrivial solution?

(a)

(b)

5.Consider an industrial process whose matrix is

Find the input matrix for the following output matrix

210

211

321

C

032

022

032

zyx

zy

zyx

023

032

02

zyx

zyx

zyx

112

123

312

A

10

20

30

B

• 17

0)1(2

02).1(

yx

yx

6.For what values of does the homogeneous system

have a nontrivial solution?

7. Suppose that A and B are square matrices and AB=0.If B is non-

singular, find A.

8. Let A be nxnmatrix.Prove that if A is non-singular, then the

homogeneous system AX=O has trivial solution.

• 18

2,4:

22

84

2

8

4205

........................

443

)sec(

)2(2./82

yxissolutionThe

xy

xx

yx

byequationtheMultiplyMeansyx

)1......(....................6

___________________

8224

1./223

zx

zyx

zyx

)2.....(..........1035

___________________

2./4

223

zx

zyx

zyx

1.1. Exercises:

1. (a)

b)

• 19

10,2,4

241044

10)4(66482

___________________

1035

3./6

zyx

zxy

xzxx

zx

zx

1,5

27=2(1)+5(5)

523177

_______________________

2/32

1332

yx

yxyy

yx

yx

c)

2. Suppose that: :1x low- sulfur, :2x high- sulfur,and note that 2 hours = 2x60=120 minutes, and 3 hours = 3x60=180 minutes.

12024

18045

21

21

xx

xx

• 20

tonxx

xx

xx

20603

_____________________

2/12024

18045

11

21

21

tonxx 20))20.(4120(2

1)4120(

2

112

3.Suppose that: :1x food A, :2x food B , :3x food C

2124

24323

25332

321

321

321

xxx

xxx

xxx

2945

_____________________

2124

2./25332

32

321

221

xx

xxx

zxx

2.32

33252.4

5

21

5

)327(2

__________________

2735

)1.(/2945

32

1

3

23

32

32

xxx

xxx

xx

xx

2,2.4,2.3 321 xxx

1.2.Exercises:

1.

1138

22417

312

514

313

.210

321.,

210

321CBB TT

15410

25618. ACBT

• 21

2.

024,2105

___________________

22

2./42

babb

ba

ba

224155

_________

32

2./42

cdcc

dc

dc

2,1,2,0 dcba

3.(a): The coefficient matrix is:

0301

0432

0323

1002

(b): Thelinear system in matrix form:

0301

0432

0323

1002

.

w

z

y

x

=

5

3

2

7

C :The augmented matrix.

5

3

2

7

:

:

:

:

0301

0432

0323

1002

• 22

4.

633

42

38723

542

431

41

4321

421

xxx

xx

xxxx

xxx

5.For each product P or Q, the daily cost of pollution control at plant X

or at Plant Y.

6. (a)

(b)

1.3. Exercises:

1.

00

00

42

63.

32

32.BA

2.

103400\$120

300

220

.120150200

A

16050\$250

120

100

.254050

A

• 23

ACABCA

BA

68

68

40

34.

32

32.

68

68

02

31

32

32.

3. The information can bedisplayed in matrix form as:

2

1

3

22

1

3

1

A

(a)The distribution of the market after 1 year is:

9

59

4

3

23

1

.

2

1

3

22

1

3

1

.1 oXAX

(b)The stable distribution of the market is :

b

aa+b=1

bbaabab

a

b

aA

2

1

3

2,,

2

1

3

1.

2

1

3

22

1

3

1

7

3,

7

41

2

1)1(

3

11 abbbbba

4.

(a): (A+B) is symmetric matrix if:

(A+B)=(A+B)T

• 24

(A+B)T = A

T + B

T

But A=AT and B = B

T because A and B are symmetric

matrices , so (A+B)T = A

T + B

T =A+B

(b):- Suppose that AB=BA.

(AB) is symmetric matrix if (AB)=(AB)T

(AB)T=B

T .A

T=B.A=AB

matrixsymmetricisAB)(

- Suppose that(AB) is symmetric matrix .

Take (AB)=(AB)T=B

T.A

T=B.A as both A and B are symmetric

matrices.

BAAB

5.

a. AAT issymmetric if ( AA

T)=(A.A

T)

T

(A.AT)

T= (A

T)

T.A

T=A.A

T

Similarly for AT.A.

b.A+AT is symmetric if A+A

T=(A+A

T)

T

(A+AT)

T = (A

T)

T+A

T= A+A

T

c. A - ATis skew symmetric if A - A

T = - (A-A

T)

T

- (A-AT)

T= -(A

T-(A

T)

T)=-A

T+A=A-A

T

1.4.Exercises:

1. A,E,G

2.

• 25

(a)

413

224

515

301

(b)

515

6612

413

301

(c)

412

224

413

301

3.

1000

0100

0010

0001

..........

3300

2700

1110

4201

3

5

2

0030

3250

1110

2021

2

2011

5231

5132

2021

42

32

12

41

31

21

RR

RR

RR

RR

RR

RR

4.

x1=1 , x2 =0.667 , x3 =-0.667 ,

3

2:100

3

2:010

1:001

3

2:100

0:110

1:001

3

12:300

0:110

1:001

0:110

2:300

1:001

0:110

2:300

1:111

22:112

3:211

1:111

23

3

223

23

32

21

RR

R

RRR

RR

RR

RR

• 26

5.

(1) No solution:

(b)) A unique solution:

(c) Infinity many solutions:

(6)

x1=-1 , x2=4 , x3 = -3

(7)

Let: x=number of chairs

y=number of coffee tables

z= number of dining room tables.

2:400

3:121

2:111

:511

3:121

2:111

2

21

2 aaRRaa

202,042 aaa

2,2/,042 Raa

202,042 aaa

3:100

4:010

1:001

3:100

4:010

1:001

3:100

1:110

1:001

1:110

3:100

1:001

1:110

3:100

0:111

1:110

3:011

0:111

3

2332

13

21

R

RRRR

RR

RR

1080181212

6601286

960151210

zyx

zyx

zyx

• 27

x=30 , y = 30 z = 20

1.5. Exercises:

1.A is non-singular matrix because there exist matrix B such that

A.B=B.A=I

2. Suppose that

Is an inverse of A. Then

Equating corresponding elements of these two matrices, we obtained

the linear system

20:100

30:010

30:001

1080:181212

6601286

960:151210

10

01

4

1

4

18

1

8

3

.32

12

32

12.

4

1

4

18

1

8

3

dc

baB

10

01

2424

22.

24

12. 2I

dbca

dbca

dc

baBA

02

______________

024

2./12

ca

ca

• 28

1,30)1)(3(032

0:320

0:422

0:)1(22

0:422

)1(

2

0:12

0:21

2

22

1

R

3,1

So this linear system has no solution .Hence there is no such matrix B

and A is singular matrix.

3.

4.

5. X= A-1.B

6.

111

122

1101C

5.005.0

5.525.2

5.315.11A

10

60

30

10

20

30

.

5.005.0

5.525.2

5.315.1

.1 BAX

solutionnontrivialhassystemThe

MatrixSingular

RR

a

000

220

321

321

220

321

)(

21

solutionnontrivialhassystemThe

MatrixSingular

RR

RR

RR

RR

b

000

550

321

5

7

770

550

321

3

2

213

112

321

213

321

112

)(

32

31

21

21

• 29

OAOBOBBAOBA 11 .....7

OXOXAAOXA ....8 1

1.7Glossary:

Matrix:

Zero matrix:

Square Matrix:

Reduced Row Echelon form:

Diagonal Matrix:

Identity Matrix:

Triangular Matrix:

Coefficient Matrix:

Augmented Matrix :

Linear System:

Inverse of a Matrix: ( )

Transpose of a Matrix:

Rank of a Matrix:

Elementary Row Operations:

• 30

• 31

• 32

• 33

5,4,3,2,1s

2.1. Exercises:

1. Find the number of inversions in each of following permutations of

.

(a) 52134

(b) 45213

(c) 42135

2.Determinate whether each of the following permutations of

Is even or odd

(a) 4213

(b) 1243

(c) 1234

3. Compute thefollowing determinants:

(a)

(b)

4. If

4,3,2,1s

4

321

321

321

ccc

bbb

aaa

A

23

12

300

520

024

• 34

Find the determinants of the following matrices:

And

5. Evaluate the following determinants using the properties of this

Section:

(a)

(b)

(c)

123

123

123

ccc

bbb

aaa

B

321

321

321

222 ccc

bbb

aaa

C

321

332211

321

444

ccc

cbcbcb

aaa

D

402

025

534

A

64811

0132

2423

4102

B

300

320

214

C

• 35

6. If A and B are 3x3 matrices with

Calculate

(a)

(b)

(c)

(d)

(e)

7. Show that if A=A-1, then

8.Show that if AT=A, then

9. Show that

This determinant is called a Vandermonde determinant

2.2.Exercises:

1. Let

Compute all the cofactors.

2. Compute the determinants of the following matrices:

(a)

,42 BandA

A3

1)3( BA

13 ABTBA .1

2A

1A

1A

))()((

1

1

1

2

2

2

bcacab

cc

bb

aa

325

413

201

A

023

251

321

A

• 36

(b)

3. Let

(b) Compute

4. Compute the inverse of the following matrix:

5.Use theorem 2.12 to determine which of the following matrices are

non-singular:

(a)

(b)

1230

4302

3021

1244

B

544

143

826

A

A

210

430

204

A

264

312

534

A

2564

3153

1462

2131

B

273

251

422

C

• 37

(c)

(d)

6. Use Corollary 2.3 to find out whether the following homogeneous

systems have nontrivial solutions:

(a) x+y-z=0

2x+y+2z=0

3x-y+z=0

(b) x+y+2z+w=0

2x-y+z-w=0

3x+y+2z+3w=0

2x-y-z+w=0

7. Solve the following linear system by Cramer's rule:

X+y+z-2w=-4

2y+z+3w=4

2x+y-z+2w=5

x-y+w=4

8. Prove that

431

021

210

D

1

• 38

2.1.Exercises:

1. (a) five inversions: 52,51,53,54,21.

(b) Seven inversions: 42,41,43,52,51,53,21

(c) Fourinversions : 42,41,43,21.

2. (a) even(n=4: 42,41,43,21)

(b) Odd (n=1: 43)

(c) Even( n=0)

3. (a) (2 ).(2)-(-1)(3)=7

(b) (4)(-2)(3)=-24

4.

5. (a)

4.1 AB

8.2 AC

72)46

144)(

4

23(4

46

14400

4

25

4

230

534

)23

4)(

2

3(

2

3

2

30

4

25

4

230

534

2

14

5

402

025

534

3231

21

A

RRRR

RR

• 39

(b)

(c)

6.

54.33)( 3 AAa

54

1

)4(27

1.2

.3

1.

3

1.)3()(

3

1

BA

BABAb

0

64811

0132

8204

00002

1

64811

0132

8204

4102

2

64811

0132

8068

4102

64811

0133

2423

4102

12

23

24

B

RR

RRB

RRB

24)3)(2(4

300

320

214

C

• 40

7.

2.2.Exercises:

1.

2

27

4

1).2.(27

1..33)( 31

BAABc

22

4.

1..)( 11 B

4.)( 2 AAAe

))()((

00

10

1

))((

10

10

1

))((

0

0

1

1

1

1

.9

1

2

22

22

2

31

21

2

2

2

bcacab

bc

ba

aa

acab

ac

ba

aa

acab

acac

abab

aa

RR

RR

cc

bb

aa

111 211 AAA

AAAA

111

.8211 AA

AAAAAA TT

,1132

41.)1( 1111

A 29

35

43.)1( 2112

A

,125

13.)1( 3113

A 432

20.)1( 1221

A

,735

21.)1( 2222

A

225

01.)1( 3223

A

• 41

2.

3.

,241

20.)1( 1331

A 10

43

21.)1( 2332

A

113

01.)1( 3333

A

4323

513

03

212

02

25.1)(

Aa

75

302

021

244

.1

402

321

144

2

432

301

124

.3)(

Bb

,2454

14.)1()( 1111

Aa 1954

13.)1( 2112

A

444

43.)1( 3113

A 42

54

82.)1( 1221

A

254

86.)1( 2222

A 3244

26.)1( 3223

A

3013

86.)1( 2332

A

• 42

(c)

4.

303030

32242

41924

3043

26.)1( 3333 CA

30324

30219

304224

150)4(8)19)(2()24)(6()( Ab

15000

01500

00150

020

401,10

21

431211

AA

221

201,0

10

302113

AA

410

041,8

20

242322

AA

1640

241,6

43

203231 AA

• 43

5.

6.

40)4).(10(

12166

482

0010

1230

0433

ACA

1240

1680

6210

10

3

10

10

5

2

5

10

20

3

20

1

4

1

A

MatrixSingularAa 0)(

MatrixSingularBb 0)(

MatrixSingularCc 0)(

MatrixgularNonDd sin2)(

SolutionNontrivialHasa

4

113

212

111

)(

• 44

7.

8.

.0

1112

3213

1112

1211

)( SolutionTrivialHasb

4

5

4

4

,

1011

2112

3120

2111

BA

1041

2152

3140

2141

,

1014

2115

3124

2114

21 AA

4011

5112

4120

4111

,

1411

2512

3420

2411

43 AA

0,30,30,30 321 AAAA

604 A

2,0,1,14321

A

Aw

A

Az

A

Ay

A

Ax

1)(.)(.

.)(..)(.

n

n

n

n

AA

• 45

2.4Glossary:

The Minor of aij :aij

Cofactor:

Determinant:

Cramer's Rule :

• 46

• 47

• 48

• 49

3.1.Exercises:

1. Determine the head of the vector whose tail is at

(3,2).

2.Determine the tail of the vector whose head is at

(1,2).

3.Let X=(1,2) , Y=(-3,4),Z=(x,4),and U=(-2,y).Find x and y so that:

(a) Z=2X

(b)

(c )

4. Find the length of the following vectors:

(a) (1,2)

(b)( -3,-4)

5. Find the distance between pairs of points:

(a) (4,2) , (1,2)

(b) (-2,-3), (0,1)

6. Find a unit vector in the direction of X

(a) X=(2,4)

5

2

5

2

XUZ

YU 2

3

• 50

(b) X=(0,-4)

7. Find X.Y.

(a) X=(0,-1),Y=(1,0)

(b) X=(2,2) , Y = (4,-4)

8. Write each of the following vectors in terms of i and j.

(a) (1,3)

(b) (-2,-3)

9. A ship is being pushed by a tugboat with a force of 300 pounds

along the negative y-axis while another tugboat is pushing along the

negative x-axis with a force of 400 ponds. Find the magnitude and

sketch the direction of the resultant force.

3.2.Exercises:

1. Find X+Y,X-Y,2X, and 3X-2Y if X=(1,2,-3),Y=(0,1,-2).

2. Let X=(1,-2,3),Y=(-3,-1,3),Z=(x,-1,y),and U=(3,u,2).Find x,y,and u so

that:

(a) Z+Y=X

(b) Z+U=Y

3. Find the length of the following vectors:

(a) (2,3,4)

(b) (0,-1,2,3)

4.Find the distance between the following pairs of points.

(a) (1,1,0),(2,-3,1)

(b) (4,2,-1,6),(4,3,1,5)

5. Find X.Y.

(a) X=(2,3,1),Y=(3,-2,0)

• 51

(b) X=(1,2,-1,3),Y=(0,0,-1,-2)

6.Which of the following vectors:

X1=(4,2,6,-8),X2=(-2,3,-1,-1),X3=(-2,-1,-3,4),X4=(1,0,0,2),X5=(1,2,3,-4),

X6=(0,-3,1,0)are:

(a) Orthogonal?

(b) Parallel?

(c) In the same direction?

7. Find a unit vector in the direction of X.

(a) X = (1,2,-1)

(b) X = (1,2,3,4)

8.Write each of the following vectorsin R3in terms of i,j,and k.

(a) (1,2,-3).

(b) (2,3,-1).

9.Write each of the following vectorsin R3as a3x1 matrix.

(a) 2i+3j-4k.

(b) i+2j.

10. A large steel manufacturer, who has 2000 employees, lists each

employee's salary as a component of a vector S in R2000.If an 8 percent

across-the-board salary increase has been approved, find an

expression involving S giving all the new salaries.

3.4.Exercises:

1. Determine whether thegiven set together with the given operations

is a vector space. If it is not a vector space, list the properties of

Definition 1 that fail to hold.

(a)The set of all ordered triples of real numbers (x,y,z)with the

operations (x,y,z)and ),,(),,( zyyxzyx

),,(),,.( czcycxzyxc

• 52

(b) The set of all ordered triples of real numbers of the form (0,0,z)with

the operations

(c)The set of all positive real numbers with the operations

And

2. Which of the following subsets of R4 are subspaces of R4? The set of

all vectors of the form

(a) (a,b,c,d),where a-b=2

(b) (a,b,c,d),where c=a+2b and d=a-3b.

(c) (a,b,c,d),where a=0 and b = -d

3. Which of the following subsets of the vector space of all 2x3

matrices under the usual operations of matrix addition and scalar

multiplication are subspaces?

(a)

(b)

3.5.Exercises:

1.Which of the following vectors are linear combinations of X1=(4,2,-3),

X2=(2,1,-2),and X3=(-2,-1,0)?

(a) (4,2,-6)

(b) (-2,-1,1)

2.Which of the following vectors are linear combinations of

P1(t)=t2+2t+1, p2(t)=t

2+3, p3(t)=t-1?

(a) t2+t+2.

),0,0(),0,0.(),0,0(),0,0(),0,0( czzandczzzz

xyyx

cxxc .

cabwhered

cba

,

00

.22, defandcawherefed

cba

• 53

(b) -t2+t-4.

3. Which of the following sets of vectors span R4?

(a) (1,0,0,1),(0,1,0,0),(1,1,1,1),(1,1,1,0).

(b) (1,2,1,0),(1,1,-1,0),(0,0,0,1).

(c) (6,4,-2,4),(2,0,0,1),(3,2,-1,2),(5,6,-3,2),(0,4,-2,-1).

(d)(1,1,0,0),(1,2,-1,1),(0,0,1,1,),(2,1,2,1)

4. Do the polynomials t3+2t+1,t2-t+2,t3+2,-t3+t2-5t+2 span P3?

5.Which of the following sets of vectors in R3 are linearly dependent?

For those that are ,express one vector as a linear combination of the

rest.

(a) X1=(4,2,1),X2=(2,6,-5),X3=(1,-2,3)

(b) ) X1 =(1,2,-1),X2=(3,2, 5)

(c ) X1 =(1,1,0),X2=(0,2,3),X3=(1, 2,3),X4=(3,6,6)

(d) X1 =(1,2,3),X2=(1,1,1),X3=(1,0,1)

3.6.Exercises:

1. Which of the following vectors are bases for R2?

(a) (1,3),(1,-1).

(b) (1,2),(2,-3),(3,2)

(c) (1,3),(-2,6)

2. Which of the following vectors are bases for P3?

(a) t3+2t2+3t, 2t3+1, 6t3+8t2+6t+4, t3+2t2+t+1

(b) t3+t2+t+1, t3+2t2+t+3, 2t3+t2+3t+2, t3+t2+2t+2

3. Let W be the subspace of P3 spanned by

3432,2,1,12 233223 ttttttttt

• 54

Find a basis for What is the dimension of W ?

4.Find a basis for R4 that includes the vectors (1,0,1,0),and (0,1,-1,0)

5. Find the dimension of the solution space of

3.7.Exercises:

1. Find a basis for the subspace V of R4 spanned by the vectors

2. Find the row and column ranks of

3. Is

a linearly independent set of vectors in R3?

4. Determine which of the linear systems have a solution by

comparing the ranks of the coefficient and augmented matrices

(a)

0

0

0

.

213

312

201

3

2

1

x

x

x

3

5

3

5

,

3

3

3

3

,

2

3

2

3

,

1

2

1

2

,

2

1

2

1

and

52187

12513

12321

A

3

1

2

,

5

5

2

,

2

1

4

S

0

0

0

.

2015

4232

2521

4

3

2

1

x

x

x

x

• 55

(b)

3.1.Exercises

1. x=x1+m=3+(-2)=1

y=y1+n=2+5=7

The head of the vector is (1,7)

2. x=x2-m=1-(2)=-1

y=y2-n=2-5=-3

The tail of the vector is (-1,-3)

3. (a) (x,4)=2(1,2)=(2,4) , x=2

(b)

4. (a)

(b)

5

2

5

2

3

8,

2

344)

2

3,3()4,3(),2(

2

3 yyyy

224

312

)2,1()4,2()2,1(),2()4,()(

yy

xx

yxyxc

521 2

525)4()3( 22

2

2

1

.

2132

6514

4321

4

3

2

1

x

x

x

x

• 56

5.(a)

(b)

6.

7. (a) X.Y = (0).(1)+(-1).(0)=0

(b) X.Y= (2).(4)+(2).(-4)=0

8. (a)

(b)

9.

3.2.Exercises:

1. X+Y=(1,2,-3)+(0,1,-2)=(1,3,-5)

X-Y=(1,2,-3)-(0,1,-2)=(1,1,-1)

3)22()14( 22

5220164

)4,2(20

1)( Xa

)4,0(4

1)( Xb

ji 3

ji 32

• 57

2X=(2,4,-6)

3X-2Y=3(1,2,-3)-2(0,1,-2)=(3,4,-5)

2. (a) (x,-1,y)+(-3,-1,3) = (1,-2,3)

(x-3,-2,y+3)=(1,-2,3),

x-3=1 , x = 4

y+3 = 3 , y = 0

(b) (x,-1,y)+(3,u,2)= (-3,-1, 3)

(x+3,u-1,y+2)= (-3,-1,3)

x+3=-3, x=-6

u-1=-1, u=0

y+2=3,y=1

3. (a)

(b)

4. (a)

(b)

5. (a) (2,3,1).(3,-2,0) = 6 + (-6)+0=0

(b) (1,2,-1,3).(0,0,-1,-2) = 0 + 0 + 1+ (-6)= - 5

6. (a) X1 and X2 , X1 and X6 , X2 and X3 , X3 and X6 , X4 and X6

(X1.X2=0, X1.X6=0,X2.X3=0, X3.X6=0,X4.X6=0)

(b) X1 and X3

(C) None.

291694

149410

23181161)01()13()12( 222

6141)65())1(1()23()44( 2222

)..( 3131 XXXXbecause

)..( jiji XXXXbecause

)1,2,1(6

1 X

• 58

7. (a)

(b)

8. (a)

(b)

9. (a) (b)

10. 1.08 . S

3.4.Exercises:

1. (a)

Let X=(x1,y1,z1) , Y = (x2,y2,z2) , Z(x3,y3,z3),c and d in R.

Not a vector Space : (a) , (c) , (d) , and (f) do not hold .

(b) Vector Space.

(c) Vector Space.

2.

(a) Let X=(a1,b1,c1): a1-b1=2, Y = (a2,b2,c2): a2-b2=2

)4,3,2,1(30

1 X

kji 32

kji 32

4

3

2

2

1

),,(),,( 11212212 zyyxXYzyyxYX

ZYXzyxzyxzyxZYX )()),,(),,((),,()( 333222111

)0,0,0(),0,(),,(),,(),,()( 111111111111 zxzyyxzyxzyxXX

),,.(..))(,)(,)(().( 1111111 dzdycyxdXdXczdcydcxdcXdc

• 59

X+Y=(a1+a2,b1+b2,c1+c2):a1+a2-b1-b2=4

c.X=(ca1,cb1,cc1):c(a1-b1)=2c

So W is not a subspace of R4 .

(b) is Subspace of R4.

(c) isSubspace of R4.

3.(a) Let

W is subspace.

(b) Subspace

3.5.Exercises:

1. (a) -2X1+6X2+0.X3=(4,2,-6)

(b) -X1+X2+0.X3=(-2,-1,1)

WYX

2

2

WXc .

)1...(:00

111

1

111cab

d

cbaX

)2......(:00

222

2

222cab

d

cbaY

0021

212121

dd

ccbbaaYX

WYXccaabb 212121)2()1(

00.

1

111

cd

cccbcaXc

WXccccacbccab ../)1( 111111

• 60

2. (a)

(b)

3. (a)To show that (1,0,0,1),(0,1,0,0),(1,1,1,1),(1,1,1,0) spans R4,we let

X=(a,b,c,d) be any vector in R4 .We now seek constants k1

, k2,k3,and k4 such that:

K1(1,0,0,1)+k2(0,1,0,0)+k3(1,1,1,1)+k4(1,1,1,0)=(a,b,c,d)

K1=d-a+c

K2=c-a

K3=b-c

K4=a-c

So (1,0,0,1),(0,1,0,0),(1,1,1,1),(1,1,1,0) spans R4

(b) To show that(1,2,1,0),(1,1,-1,0),(0,0,0,1)spans R3,we let

X=(a,b,c,d) be any vector in R4 .We now seek constants k1

,k2 and k3 such that:

K1(1,2,1,0)+k2(1,1,-1,0)+k3(0,0,0,1)=(a,b,c,d)

2)(.0)(2

1)(

2

1 2321 tttPtPtP

4)(.0)(2

3)(

2

1 2321 tttPtPtP

dkk

ckk

bkkk

akkk

31

43

432

431

dk

ckk

bkk

akk

3

21

21

21

2

• 61

(c)To show that (6,4,-2,4),(2,0,0,1),(3,2,-1,2),(5,6,-3,2),(0,4,-2,-1)

spans R4,we let X=(a,b,c,d) be any vector in R4 .We now seek

constants k1 , k2,k3,k4 and k5 such that:

K1(6,4,-2,4)+k2(2,0,0,1)+k3(3,2,-1,2)+k4(5,6,-3,2)+k5(0,4,-2,-1)=(a,b,c,d)

So the vector (1,1,1,1) is not spanned by the vectors (6,4,-

2,4),(2,0,0,1),(3,2,-1,2),(5,6,-3,2),(0,4,-2,-1)

.4notdovectorsthese (0,0,0,1),(1,1,-1,0)(1,2,1,0),

)1,1,1,1(32

02323

2

3

;0100

;0000

:0010

:0001

:0100

:0011

:0012

:0011

Rspansvectorsthe

byspannednotisvectorthesoabc

cba

d

cba

ab

ab

d

c

b

a

dkkkkk

ckkkk

bkkkk

akkkk

54321

5421

5431

4321

224

232

4624

5326

cb

cb

cd

cb

cb

c

d

c

b

a

2

02

2:64010

3:24020

2:000002

:12

3

2

101

:12214

:23102

:46204

:05324

• 62

These vectors do not spans R4.

(d)To show that (1,1,0,0),(1,2,-1,1),(0,0,1,1,),(2,1,2,1) spans R4,we let

X=(a,b,c,d) be any vector in R4 .We now seek constants k1 , k2,k3 and

k4 and such that:

K1(1,1,0,0)+k2 (1,2,-1,1)+k3(0,0,1,1,)+k4(2,1,2,1)=(a,b,c,d)

K1+k2+2k4=a

K1+2k2+k4=b

-k2+k3+2k4=c

K2+k3+k4=d

So the vectors (1,1,0,0),(1,2,-1,1),(0,0,1,1,),(2,1,2,1) spans R4 .

4.To show that t3+2t+1,t2-t+2,t3+2,-t3+t2-5t+2 spans P3,we let

X=at3+bt2+ct+d be any vector in P3 .We now seek constants k1 ,

k2,k3 and k4 such that:

K1(t3+2t+1)+k2(t

2-t+2,t3+2)+k3(t3+2)+ k4(-t

3+t2-5t+2)=at3+bt2+ct+d

K1(1,0,2,1)+k2(0,1,-1,2)+ k3(1,0,0,2))+ k4(-1,1,-5,2)=(a,b,c,d)

dcba

dcba

dcba

dcba

d

c

b

a

22:1000

233:0100

:0010

3354:0001

:1110

:2110

:1021

:2011

dcbak

dcbak

dcbak

dcbak

22

233

3354

4

3

2

1

2

34

22

233

3354

4

3

2

1

dcbak

dcbak

dcbak

dcbak

• 63

So a vector like t3+t2+t+1 is not spanned by the vectorst3+2t+1,t2-

t+2,t3+2,-t3+t2-5t+2.

The vectors t3+2t+1,t2-t+2,t3+2,-t3+t2-5t+2 do not spans P3.

5. (a)To show thatX1=(4,2,1),X2=(2,6,-5),X3=(1,-2,3) are linearly

depenent, we seek constants k1 , k2 and k3 such that:

K1(4,2,1)+k2(2,6,-5),k3(1,-2,3)=(0,0,0)

4k1+2k2+k3=0

2k1+6k2-2k3=0

K1-5k2+3k3=0

Take

(b)To show thatX1 =(1,2,-1),X2=(3,2, 5)are linearly independent, we

seek constants k1 and k2 such that:

K1(1,2,-1)+k2(3,2,5) = (0,0,0)

The vectors X1=(1,2,-1),X2=(3,2, 5) are linearly independent.

tkktkkRttk

213232

1,

2

1,:2

0:000

0:2

110

0:2

101

0:351

0:262

0:124

dependentlinearlyarevectorsgivenThekkk 2,1,1 321

0,0

0:00

0:10

0:01

0:51

0:22

0:31

21 kk

• 64

( c ) To show that X1 =(1,1,0),X2=(0,2,3),X3=(1, 2,3),X4=(3,6,6) are

linearly dependent, we seek constants k1 , k2,k3 and k4such that:

K1(1,1,0)+k2 (0,2,3)+k3 (1, 2,3)+k4 (3,6,6) = (0,0,0)

K1+k3+3k4 = 0

K1+2k2+ 2k3+6k4 = 0

3k2+3k3+6k4 =0

Let k4=1

K2=1 , k3=-1 , k1 = -2

The vectorsX1 =(1,1,0),X2=(0,2,3),X3=(1, 2,3),X4=(3,6,6) are linearly

dependent.

( d ) To show thatX1 =(1,2,3),X2=(1,1,1),X3=(1,0,1) are linearly

independent, we seek constants k1 , k2 and k3such that:

K1(1,2,3)+k2 ( 1,1,1) +k3(1,0,1) = (0,0,0)

K1+2k2+k3 =0

2k1+k2 = 0

3k1+k2+k3 = 0

0:1100

0:1010

0:2001

0

0

0

:6330

:6221

:3101

0:100

0:010

0:001

0:113

0:012

0:111

• 65

K1 = k2 = k3 = 0

The vectors X1 =(1,2,3),X2=(1,1,1),X3=(1,0,1) are linearly independent.

3.6.Exercises:

1.

(a) To show that (1,3),(1,-1) is linearly independent ,we form the

Equation:

c1X1+c2.X2=O

c1 (1,3)+c2.(1,-1) =O

(c1+c2, 3c1-c2)=(0,0),

So (1,3),(1,-1) is linearly independent.

To show that (1,3),(1,-1) spans R2,we let X=(a,b) be any vector in

R2 .We now seek constants k1 and k2 such that

K1(1,3)+k2.(1,-1)=X=(a,b), (k1+k2,3k1-k2)=(a,b)

0004

____________

03

0

211

21

21

ccc

cc

cc

4

3

444

__________

3

1211

21

21

babaakak

bakbak

bkk

akk

• 66

So spans R2and is a basis for R2.

(b) To show that (1, 2), (2,-3),(3,2) is linearly independent

,weform the equation:

c1X1+c2.X2+c3.X3= O, c1(1,2)+c2.(2,-3) +c3.(3,2) = (0,0)

(c1+2c2+3c3,2c1-3c2+2c3) =(0,0),

c1+2c2+3c3=0

2c1-3c2+2c3

This system has infinity many solutions as

Showing that is linearly dependent.

Hence S does not span R2 and is not a basis for R2.

(c) is a basis for R2.

2. (b)

3.

5.

)1,1(),3,1( S

1,7

4,

7

13321

ccc

(3,2)(2,-3),(1,2),S

0000

0000

1010

1211

3432

0201

1010

1211

1,12 223 tttt )1,0,0,0(),1,1,0,0(),0,1,1,0(),0,1,0,1(

0

0:100

0:010

0:001

0:213

0:312

0:201

321

xxx

• 67

The solution set is

The dimension is zero.

3.7.Exercises:

1.

2.

The row and column rank are 3

3.

Linearly dependent.

4. (a)

)0,0,0(S

0000

0000

0000

1010

0101

3535

3333

2323

1212

2121

1

0

1

0

,

0

1

0

1

077.0462.0100

615.0308.0010

00001

52187

12513

12321

A

000

333.110

833.001

312

552

214

S

723.0100

675.0010

265.0001

.

2015

4232

2521

• 68

The rank of the coefficient and augmented matrices are 3 ,

So the linear system has a solution.

(b)

The rank of the coefficient matrix is 2.

The rank of the augmented matrix coefficient matrix is 3.

So the linear system has no solution.

0000

429.1110

143.1101

.

2015

4232

2521

1:0000

286.0:429.1110

571.0:143.1101

.

2

2

1

:

:

:

2132

6514

4321

• 69

3.9Glossary:

Vector:

N-dimensional:n

Linear Space:

Subspace:

Spanning Set:

Linearly Independent:

Linearly dependent:

Polynomial:

Linear Combination:

Basis:

Dimension:

• 70

• 71

• 72

4.1. Exercises:

1. Which of the following are linear transformation?

(a) L

(b) L

2. Which of the following are linear transformation?

(a) L

(b) L

3. Let: be linear transformation for which we know that:

(a) What is

(b) What is

zx

y

yx

z

y

x

2

2

yy

xx

y

x

zx

yx

z

y

x

2

0

22

22

yx

yx

y

x

2

1

1

0

3

2

1

1LandL

?2

3

L

?

b

aL

22: RRL

• 73

4. Let L: p2 p3 be linear transformation for which we know that

L(1)=1 , L(t) =t2 , and L(t2) =t3+t .

(a) . Find L(2t2-5t+3).

(b) .Find L(at2+bt+c).

4.2. Exercises:

1. Let L: R2 R3 be defined by?

(a). Find ker L.

(b) . Is L one to one?

(c) . Is L onto?

2. Let L : R5 R4 be defined by ?

(a) Find a basis for kerL .

(b) Find a basis for range L .

(c) Verify Theorem 4.7.

3. Let L : R4 R6 be Linear transformation ?

(a) If dim (ker L) =2 ,what is dim (range L)?

(b) ) If dim (range L) =3 ,what is dim (ker L)?

4.Determine the dimension of the solution space to the homogeneous

system AX=0 for the given matrix.

y

yx

x

y

xL

5

4

3

2

1

5

4

3

2

1

01100

15102

12001

13101

x

x

x

x

x

x

x

x

x

x

L

143

321

211

• 74

A =

4.3. Exercises:

1. Let S =

be abasis for R2 . Find the coordinate vectors of the following vectors

with respect to S .

(a) .

(b) .

2. Let

be a basis for P2. Find the coordinate vectors of the following vectors

with respect to S .

(a) t2 + 2

(b) 2t 1

3. Let L : R2R3 be defined by

Let S and T be the natural bases for R2 and R3 , respectively . Also , Let

3

2,

1

1

7

3

7

3

tttS ,1,12

yx

yx

yx

y

xL 2

2

1

1

1

,

1

1

0

,

0

1

1

1

0,

1

1TandS

• 75

be bases for R2 and R3 , respectively, Find the matrix of L with respect

to:

(a) S and T.

(b)

(c)Compute

4. Let L : R3 R2 be defined by

Let S and T be the natural bases for R3 and R2, respectively. Also, let

be bases for R3and R2, respectively.

Find the representation of L with respect to

(a) S and T

(b)

(c) Compute using both representations.

5.Letbe a linear transformation. Suppose that thematrix of L with

respect to the basis

is

).()(sin2

1bandainobtainedmatricestheguL

zy

yx

z

y

x

L

2

1,

1

1

1

1

1

,

0

1

0

,

0

1

1

TandS

TandS

3

2

1

L

TandS

21, XXS

22: RRL

41

32A

• 76

where

(a)Compute

(b) Compute

(c) Compute

6. LetL : R3 R3 be defined by

(a). Find the matrix of L with respect to the natural basis S for R3 .

Using the definition of L and also

using the matrix obtained in (a).

.

1

0

1

1

0

0

,

1

0

2

0

1

0

,

0

1

1

0

0

\1

,

LLL

3

2

1

).( LFindb

1

1

2

121 XandX

SS

XLandXL )()( 21

)()( 21 XLandXL

3

2L

• 77

4.1.Exercises:

1. (a)

Let

Then

Also, if c is a real number, then

Hence L is a linear transformation.

(b) Let

2

2

2

1

1

1

,

z

y

x

Y

z

y

x

X

)()()(

)(

22

2

22

11

1

11

2121

21

2121

21

21

21

2

2

2

1

1

1

YLXL

zx

y

yx

zx

y

yx

YXL

zzxx

yy

yyxx

zz

yy

xx

L

z

y

x

z

y

x

LYXL

)(..)(

11

1

11

11

1

11

1

1

1

XLc

zx

y

yx

c

czcx

cy

cycx

cz

cy

cx

LcXL

2

2

1

1,

y

xY

y

xX

)()()(

)(

)()(

22

212

1

222

112

22

2

222

12

1

112

2

2121

21

2

21

21

21

2

2

1

1

YXLyyyy

xxxx

yy

xx

yy

xxYLXL

yyyy

xxxx

yy

xxL

y

x

y

xLYXL

• 78

Hence L is not a linear transformation.

2. Let

Also, if c is a real number, then

Hence L is a linear transformation.

(b)

Hence L is not a linear transformation.

3. (a)

2

2

2

1

1

1

,

z

y

x

Y

z

y

x

X

)()(

2

0

2

0)(

)()(2

0)(

22

22

11

11

2121

2121

21

21

21

2

2

2

1

1

1

YLXL

zx

yx

zx

yx

YXL

zzxx

yyxx

zz

yy

xx

L

z

y

x

z

y

x

LYXL

)(.

2

0.

2

0)(

11

11

11

11

1

1

1

XLc

zx

yx

c

czcx

cycx

cz

cy

cx

LcXL

)()()()( YLXLYLXL

5,3),()1,0()1,1()2,3( 2121121 ccccccc

19

1

2

1.5

3

2.3

2

3

1

05

1

1.3

1

0.

1

1.

2

321

L

LLcLcL

• 79

(b)

We can solve this question using

abcaccccccba 2121121 ,),()1,0()1,1(),(

ba

baaba

b

aL

abLaLcLcb

aL

252

1).(

3

2.

1

0)(

1

1.

1

0.

1

1. 21

AXXL

3,23

2

3

2

1

1.

1

1

4321

43

21

43

21

cccccc

cc

cc

ccL

53,12

,2,12

1

2

1

1

0.

1

0

4321

42

4

2

43

21

cccc

ccc

c

cc

ccL

19

1

2

3.

25

11

2

3

25

11

L

A

• 80

4. (a) L(2t2-5t+3)=2.L(t2)-5L(t)+3.L(1)

L(2t2-5t+3)= 2.(t3+t)-5(t2)+3.1 = 2t3-5t2+2t+3

(b) L(at2+bt+c)=a.L(t2)+bL(t)+c.L(1)

L(at2+bt+c)= a.(t3+t)+b(t2)+c.1 = at3+bt2+at+c

4.2.Exercises:

1. (a) x = 0

x+y = 0

y = 0

x=y=0,

(b) L is one - to one because

(c) Given any

are any real numbers, can we find

in R2 so that we are seeking a solution to

the linear system:

0

0ker L

ba

ba

b

a

b

aL

2525

11

0

0ker L

321

3

3

2

1

,, yandyywhereRin

y

y

y

Y

2

1

x

xX

?YxL

YXAXL .)(

• 81

We are seeking a solution to the linear system :

The reduced row echelon form of the augmented matrix is :

Thus a solution exists only for

and so L is not onto.

2.(a):

5.00000

01100

5.00000

5.02001

01100

15102

12001

13101

1,0

1,1

0,1

.

6565

4343

2121

65

43

21

65

43

21

ccyycxc

ccyxycxc

ccxycxc

y

yx

x

ycxc

ycxc

ycxc

y

yx

x

y

x

cc

cc

cc

y

xL

10

11

01

A

3

2

1

2

1

2

1.

10

11

01

y

y

y

x

x

x

xL

123

12

1

:00

:10

:01

yyy

yy

y

,0123 yyy

• 82

(b)

(c) L:

dim(ker L)+dim(range L)=dim V

RttxxRttxx :,:,0 4345

0

0

0

1

0

.

0

1

1

0

2

.

0

0

0

0

0

0

2

0

2

5

4

3

2

1

st

s

t

t

t

t

t

s

t

x

x

x

x

x

0

0

0

1

0

,

0

1

1

0

2

:ker Lforbasisposssible

0000

0000

0100

1010

1001

0111

1523

1101

0000

0211

0

1

0

0

,

1

0

1

0

,

1

0

0

1

:rangeLforbasisposssible

• 83

2 + 3 = 5

3. (a) dim(range L)=4-2=2

(b) dim(ker L)=4-3=1

4.

The solution space:

The dimension of the solution space to the homogeneous system AX=0

for the given matrix A is 1.

4.3. Exercises:

1. (a)

The coordinate vector of the vector with respect to S :

(b)

000

510

701

143

321

211

:55

:77,:

32

313

Rs

s

s

s

:5

7

21

21

213

2

3

2.

1

1.

7

3

cc

cccc

167372105

__________

73

32

2122

21

21

cccc

cc

cc

7

3

2

1

21

21

213

2

3

2.

1

1.

7

3

cc

cccc

167372105

__________

73

32

2122

21

21

cccc

cc

cc

• 84

The coordinate vector of the vector with respect to S :

2. (a) t2 + 2 = c1(t2+1)+c2(t-1)+c3(t)=c1.t

2+(c2+c3)t+(c1-c2)

c1 = 1 , c1-c2 =2 , c2 = -1 , c2+c3 = 0 , c3 = - c2=1

The coordinate vector of the vector t2 + 2 with respect

To S :

(b)

3. (a)

The matrix of L with respect to S and T:

(b)

2

1

7

3

1

1

1

1

1

0

11

12

21

,

1

2

1

0

1

L ,

1

1

2

1

0

L

3

2,

3

2

3

7

1

1

1

.

1

1

0

.

0

1

1

.

0

1

3

1

1

32

1321

cc

ccccL

• 85

The matrix of L with respect to S` and T ` :

(c)- For (a) :

- For(b):

4. (a)

The representation of L with respect to S and T:

3

4

3

2

1.

11

12

21

2

1L

3

2,

3

5

3

4

1

1

1

.

1

1

0

.

0

1

1

.

1

1

2

1

0

32

1321

cc

ccccL

3

2

3

23

5

3

23

4

3

7

3

23

83

1

2

1.

3

2

3

23

5

3

23

4

3

7

2

1L

,0

1

0

0

1

L ,1

1

0

1

0

L

,1

0

1

0

0

L

• 86

(b)

ST XXL .110

011)(

21

21

2122

1

1

1.

1

2

0

1

1

cc

ccccL

1,112,2 211221 cccccc

1

1

0

1

1

T

L

21

21

2122

1

1

1.

1

1

0

1

0

cc

ccccL

3

2,

3

112,1 211221

cccccc

3

23

1

0

1

1

T

L

0,022

1

1

1.

0

0

0

1

0

21

21

21

21

cccc

ccccL

0

0

0

1

0

T

L

• 87

The representation of L with respect to:

(c) - For (a) :

- For (b) :

5.(a)

(b)

(c)

SXTXL

.

03

21

03

11

)(

4

3)(

1

2)(

2

1

s

s

XL

XL

5

1

1

1

2

1.2.1.2)( 211 XXXL

10

1

1

14

2

1.3.4.3)( 212 XXXL

TandS

1

3

3

2

1

.110

011

3

2

1

L

3

73

5

3

2

1

.

03

21

03

11

3

2

1

L

21

21

21

2211

21

1..

2

1.

3

2

.3

2

cc

cccc

XcXc

• 88

(6)(a)

(b) -Using the definition of L:

- Using the matrix obtained in (a):

110

001

121

A

3

7,

3

132,2 212121

cccccc

25

2

10

1.

3

7

5

1.

3

1

3

2

..3

22211

L

XLcXLcL

5

1

8

1

0

1

*3

1

0

2

*2

0

1

1

*1

3

2

1

1

0

0

*3

0

1

0

*2

1

0

1

*1

3

2

1

L

LLLL

5

1

8

3

2

1

*

110

001

121

3

2

1

*)(

L

XAXL

• 89

4.5 Glossary:

Linear Transformation:

Kernel:

Range:

One to One:

Onto:

Reflection:

Rotation:

Composition of Linear Transformation:

Transition Matrix:

Matrix Representation of a Linear Transformation:

• 90

• 91

• 92

• 93

5.1.Exercises:

1. Find the characteristic polynomial of each matrix .

(a) .

(b).

2. Find the characteristic polynomial,eigenvalues, and

eigenvectors of each matrix.

(a) .

(b).

3. Find which of the matrices are diagonalizable.

(a) .

(b).

231

210

121

300

120

314

223

031

001

42

11

12

01

200

210

321

• 94

4. Find for each matrix A , if possible , a non-singular matrix

Psuch that P-1AP is diagonal .

(a).

(b).

5. Prove that if A and B are similar matrices, they have the

same characteristic polynomials and hence the same

eigenvalues.

5.2.Exercises:

1. Verify that

P =

Is an orthogonal matrix.

2.Diagonalizable each given matrix A and find an orthogonal

matrix P such that P-1AP is diagonal.

(a) .

(b).

3.Diagonalizable each given matrix.

021

212

324

212

010

321

3

2

3

2

3

13

2

3

1

3

23

1

3

2

3

2

22

22

220

220

000

21

12

• 95

(a) .

(b).

( c) .

4 . Show that if A is an orthogonal matrix, then

5.Show that if A is an orthogonal matrix , then

is orthogonal.

100

011

011

211

121

112

.1A

.1A

• 96

5.1. Exercises:

1. (a)

(a)

2. (a)

74

231

210

12123

AI

24269)3)(2)(4(

300

120

11423

AI

2,3,1

)()2)(3)(1(

223

031

001

321

fAI

8

3

6

8,3,68

8

3

8

3,

4

3

4

3,:

0:000

0:8

310

0:4

301

0:323

0:021

0:000

0)(

1321

32313

11

XxxxsLet

XAI

2

5

0

2,5,02

2

5

2

5,0,:

0:000

0:2

510

0:001

0:523

0:001

0:002

0)(

2321

3213

22

XxxxsLet

XAI

• 97

(b)

3. (a)

1

0

0

1,0,01

0,0,:

0:000

0:010

0:001

0:023

0:051

0:003

0)(

3321

213

33

XxxxsLet

XAI

,3,2

)(652)4)(1(42

11

21

2

fAI

1

1,1,11

,:

0:00

0:11

0:22

0:110)(

121

212

11

XxxsLet

XAI

2

1,2,12

2

1,:

0:00

0:2

11

0:12

0:120)(

121

212

22

XxxsLet

XAI

ablediagonaliznotismatrix

thetheoremguByAEigenvecsAEigenvals 2.5sin1

0)(,

1

1)(

• 98

(b)

4. (a)

(b)

5.

So A and B have the same characteristic polynomials and

hence the same eigenvalues.

5.2. Exercises:

1.

ableDiagonalizismatrixthetheorem

gubyAEigenvecsAEigenvals

2.5

sin

222.000

148.0707.00

964.0707.01

)(,

2

1

1

)(

ablediagonaliznotismatrixthe

theoremgubyAEigenvecsAEigenvals 2.5sin

707.0487.0

0324.0

707.0811.0

)(,

1

1

3

)(

412

600

113

549.0707.0555.0

824.000

137.0707.0832.0

:2.5sin

549.0707.0555.0

824.000

137.0707.0832.0

)(,

1

4

1

)(

PorP

theoremgUableDiagonaliz

AEigenvecsAEigenvals

AIBI

AIPAIP

PAIP

PAIPPAPIPPPAPIBIPAPB

..1

..

).(....

1

11111

3

2

3

2

3

13

2

3

1

3

23

1

3

2

3

2

1PPT

• 99

P Is orthogonalmatrix.

2. (a)

(b)

3. (a)

2

1

2

12

1

2

1

)(,4

0)( AeigenvecsAeigenvals

2

1

2

12

1

2

1

,40

00PD

2

1

2

10

2

1

2

10

001

)(,

4

0

0

)( AeigenvecsAeigenvals

2

1

2

10

2

1

2

10

001

,

400

000

000

PD

10

03

2.5sin.1

3)(

3,1

0)1)(3(034

01)2(

21

120

21

2

2

D

theoremgUAeigenvals

AI

• 100

(b)

(c)

000

020

001

2.5sin.

0

2

1

)(

0,2,1

0)2)(1(

0)2)(1(01)1()1(

011

11).1(

100

011

011

0

321

22

D

theoremgUAeigenvals

AI

0496

0428234

02)34)(2(

0211)2(*11)2()2(

011

21*1

21

11*1

21

12).2(

211

121

112

0

23

223

2

2

AI

• 101

400

010

001

2.5sin.

4

1

1

)(

4,1,1

0)4()1(

321

2

D

theoremgUAeigenvals

4. if A is an orthogonal matrix ,then A-1 = At

5.Need to show that (A-1)-1 = (A-1)T

Suppose thatA is an orthogonal matrix

By taking the transpose both sides of equation(1)we

get ( A-1)T = (AT)T=A=(A-1)-1 , (A-1)-1 = (A-1)T

111 2

11

AAAA

AAAASo T

)........(1 aAA T

matrixorthogonalanisA

• 102

5.4 Glossary:

Eigen Value:

Eigen Vector:

Characteristic Polynomial:

Characteristic Equation:

Eigen Space:

Similarity:

Similar matrices:

Diagonalizable Matrix:

• 103

• 104

• 105

6.1.Exercises:

1. Steel producer makes two types of steel: regular and special .Aton of

regular steel requires 2 hours in the open hearth furnace and 5 hours

in the soaking pit ;a ton of special steel requires 2 hours in the open-

hearth furnace and 3 hours in the soaking pit. The open- hearth

furnace is available 8 hours per day and the soaking pit is available 15

hours per day. The profit on a tonof regular steel is\$120 and it is \$100

on a ton of special steel.Determine how many tons of each type of

steel should be made to maximize the profit.

2.Atelevision producer designsa program based on a comedian and

time for commercials .The advertiser insists on at least 2 minutes of

advertising time , the station insists on no more than 4 minutes of

advertising time, and the comedian insists on at least 24 minutes of the

comedy program .Also , the total time allotted for the advertising and

comedy portions of the program cannot exceed 30 minutes .If it has

been determined that each minute of advertising(very creative)

attracts 40,000 viewers and each minute of the comedy program

attracts 45,000 viewers, how should the time be divided between

advertising and a programming to maximize the number of viewer-

minutes ?

3. The protein Diet Club serves a luncheon consisting of two dishes, A

and B . Suppose that each unit of A has 1 gram of fat , 1 gram of

carbohydrate , and 4 grams of protein , whereas each unit of B has 2

grams of fat, 1 gram of carbohydrate, and 6 grams of protein .If the

dietician planning the luncheon wants to provide no more than 10

• 106

grams of fat or more than 7 grams of carbohydrate, how many units of

A and how many units of B should be served to maximize the amount

of protein consumed ?

4. Sketch the set of points satisfying the given system of inequalities.

a.

b.

5. Minimize z = 3x - y

Subject to

6 .Formulate the following linear programming problem as a

new problem with slack variables.

Maximize z =2x1+3x2 + 7x3

subject to

0

0

102

62

y

x

yx

yx

0

0

84

4

y

x

yx

yx

0

0

2438

2045

623

y

x

yx

yx

yx

0

0

0

9

2162

343

3

2

1

321

321

321

x

x

x

xxx

xxx

xxx

• 107

6.1. Exercises

1.Supposethat x is the mount of regular steel, and y is the

mount of special steel.

Maximize Z= 120 x + 100 y

Subject to

2.Suppose that x is advertising time, and y is the time of the

comedy program

Maximize Z= 40000 x +45000 y

Subject to

0

0

1535

822

y

x

yx

yx

0

0

4

2

24

30

y

x

x

x

y

yx

• 108

3.Suppose that x is the mount units of type A ,and y is the

the mount units of type B.

Maximize Z = 4x+6y

Subject to

4. (a)

(b)

0

0

7

102

y

x

yx

yx

• 109

5.

The point Z

)

11

45,

11

8( 11

21

• 110

6.

Maximize z = 2x1 +3x2 +7 x3

Subject to

11

21min

11

45,

11

8

z

yx

0,0,0,0,0,0

9

2162

343

654321

6321

5321

4321

xxxxxx

xxxx

xxxx

xxxx

)17

40,

17

36( )

17

68(

)5

24,

5

6(

5

6

• 111

6.3 Glossary:

Linear Inequality:

Objective Function:

Constraints: ()

• 112

The End

• 113

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