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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

    the table of contents, which also shows therequired material

    from the textbook.

    In addition to this manual and the textbook there is a

    practical Manualfor Mathcad and QSBprograms.

    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

    1.6Answers to Exercises:18

    1.7.Glossary: 29

    2. Chapter2:Determinants:

    2.1.Exercises: 33

    2.2.Exercises: 35

    2.3Answers to Exercises:38

    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

    3.8Answers to Exercises:55

    3.9 Glossary:69

    4. Chapter4:Linear Transformations and Matrices:

    4.1.Exercises: 72

    4.2.Exercises: 73

    4.3.Exercises: 74

    4.4Answers to Exercises:77

    4.5 Glossary:89

    5. Chapter5:Eigenvalues and Eigenvectors:

    5.1.Exercises: 93

    5.2.Exercises: 94

    5.3Answers to Exercises:96

    5.4 Glossary:102

    6. Chapter6:Linear Programming:

    6.1.Exercises: 105

    6.2Answers to Exercises:107

    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

    equationondthetoaddThen

    byequationtheMultiplyMeansyx

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

    ___________________

    8224

    1./223

    zx

    zyx

    zyx

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

    ___________________

    2./4

    223

    zx

    zyx

    zyx

    1.6 :Answers to Exercises:

    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

    (a) Find adj A.

    (b) Compute

    (c) Compute A.(adj A)

    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

    nAadjA

  • 38

    2.3Answers to Exercises:

    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

    4 AD

    72)46

    144)(

    4

    23(4

    46

    14400

    4

    25

    4

    230

    534

    )23

    4)(

    2

    3(

    2

    3

    2

    30

    4

    25

    4

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    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

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    B

    RR

    RRB

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    24)3)(2(4

    300

    320

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  • 40

    7.

    2.2.Exercises:

    1.

    2

    27

    4

    1).2.(27

    1..33)( 31

    BAABc

    22

    4.

    1..)( 11 B

    ABABAd TT

    4.)( 2 AAAe

    ))()((

    00

    10

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    ))((

    10

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    0

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    1

    2

    22

    22

    2

    31

    21

    2

    2

    2

    bcacab

    bc

    ba

    aa

    acab

    ac

    ba

    aa

    acab

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    abab

    aa

    RR

    RR

    cc

    bb

    aa

    111 211 AAA

    AAAA

    111

    .8211 AA

    AAAAAA TT

    ,1132

    41.)1( 1111

    A 29

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    A

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    20.)1( 1221

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  • 41

    2.

    3.

    ,241

    20.)1( 1331

    A 10

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    21.)1( 2332

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    Aa

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    .3)(

    Bb

    ,2454

    14.)1()( 1111

    Aa 1954

    13.)1( 2112

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    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

    )( TCAAdj

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

    15000

    01500

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  • 43

    5.

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    12166

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    MatrixSingularAa 0)(

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  • 44

    7.

    8.

    .0

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    0,30,30,30 321 AAAA

    604 A

    2,0,1,14321

    A

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    A

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    1)(.)(.

    .)(..)(.

    n

    n

    n

    n

    AA

    AAAdjIAAAdjA

    IAAAdjAIAAAdjA

  • 45

    2.4Glossary:

    The Minor of aij :aij

    Cofactor:

    Determinant:

    Adjoint of a Matrix:

    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

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    2

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    ,

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    2

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    and

    52187

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  • 55

    (b)

    3.8Answers to Exercises:

    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

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    2

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  • 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

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  • 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

    cbad

    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.

    Possible answer: , dimW=2.4.

    Possible answer:

    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.

    Possible answer :

    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

    4.4 Answers to Exercises:

    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

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    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

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    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

    RttxxRssx :22,: 412

    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

    Rssxx

    RssxxRssx

    :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

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    ccccL

    3

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    112,1 211221

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    3

    23

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    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

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    14

    2

    1.3.4.3)( 212 XXXL

    TandS

    1

    3

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    2

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    .110

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    3

    2

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    3

    73

    5

    3

    2

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    .

    03

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    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

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    22211

    L

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    5

    1

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    0

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    *1

    3

    2

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    0

    0

    *3

    0

    1

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    1

    0

    1

    *1

    3

    2

    1

    L

    LLLL

    5

    1

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    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

    5.3 Answers to Exercises:

    5.1. Exercises:

    1. (a)

    (a)

    2. (a)

    74

    231

    210

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    2

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    0)(

    2321

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    22

    XxxxsLet

    sxxxRssx

    XAI

  • 97

    (b)

    3. (a)

    1

    0

    0

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    3321

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    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

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    ).(....

    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

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    12

    1

    2

    1

    ,40

    00PD

    2

    1

    2

    10

    2

    1

    2

    10

    001

    )(,

    4

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    )( 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

    6.2 Answers to Exercises:

    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