Solutions Manual for Introductory Linear Algebra · 2019. 5. 26. · 3 Introduction The...
Transcript of Solutions Manual for Introductory Linear Algebra · 2019. 5. 26. · 3 Introduction The...
2013
Solutions Manual for Introductory
Linear Algebra
Author:Dr.Mahmoud Al-Beik
ALQUDS OPEN UNIVERSITYجايعت انقذس انًفتىحت
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 A
T.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:يصفىفت
Zero matrix: يصفىفت صفريت
Square Matrix: يصفىفت يربعت
Reduced Row Echelon form:انشكم انصفي انًًيز
Diagonal Matrix:انًصفىفت انقطريت
Identity Matrix: يصفىفت انىحذة
Triangular Matrix:يصفىفت يثهثيت
Coefficient Matrix:يصفىفت انًعايلاث
Augmented Matrix : انًصفىفت انًًتذة
Linear System:انُظاو انخطي
Inverse of a Matrix: (يعكىس انًصفىفت) انُظير انضربي نهًصفىفت
Transpose of a Matrix:يُقىل انًصفىفت
Rank of a Matrix:رتبت انًصفىفت
Elementary Row Operations: عًهياث انصف انبسيظ
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
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
ABABAd TT
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( 11
11
A 2935
43.)1( 21
12
A
,125
13.)1( 31
13 A 432
20.)1( 12
21
A
,735
21.)1( 22
22
A
225
01.)1( 32
23 A
41
2.
3.
,241
20.)1( 13
31
A 1043
21.)1( 23
32
A
113
01.)1( 33
33 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()( 11
11
Aa 1954
13.)1( 21
12
A
444
43.)1( 31
13
A 42
54
82.)1( 12
21
A
254
86.)1( 22
22 A 3244
26.)1( 32
23
A
3013
86.)1( 23
32
A
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(c)
4.
303030
32242
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26.)1( 33
33 CA
30324
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)( TCAAdj
150)4(8)19)(2()24)(6()( Ab
15000
01500
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020
401,10
21
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6.
40)4).(10(
12166
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20
3
20
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4
1
)(.11 AAdjA
A
MatrixSingularAa 0)(
MatrixSingularBb 0)(
MatrixSingularCc 0)(
MatrixgularNonDd sin2)(
SolutionNontrivialHasa
4
113
212
111
)(
44
7.
8.
.0
1112
3213
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)( SolutionTrivialHasb
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0,30,30,30 321 AAAA
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2,0,1,14321
A
Aw
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A
Ax
1)(.)(.
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n
n
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AAAdjIAAAdjA
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45
2.4Glossary:
The Minor of aij :aij انًحذد انًتًى نهعُصر
Cofactor:انًتعايم
Determinant: انًحذد
Adjoint of a Matrix: قريٍ انًصفىفت
Cramer's Rule :قاعذة كرايًر
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)=t2+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)
3.8Answers 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
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
1ˆ)( Xa
)4,0(4
1ˆ)( Xb
ji ˆ3ˆ
ji ˆ3ˆ2
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
)..( ˆˆˆˆ jijiXXXXbecause
)1,2,1(6
1ˆ 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
1ˆ X
kji ˆ3ˆ2ˆ
kji ˆˆ3ˆ2
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 2
321 tttPtPtP
4)(.0)(2
3)(
2
1 2
321 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(t2-t+2,t3+2)+k3(t3+2)+ k4(-t3+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:يتجه
N-dimensional:n رو بعذ
Linear Space:فضاء خطي
Subspace: فضاء جزئي
Spanning Set: يجًىعت يىنذة
Linearly Independent:َيستقهت خطيا
Linearly dependent:َيرتبطت خطيا
Polynomial:كثير حذود
Linear Combination:تركيبت خطيت
Basis:أساس
Dimension:بعذ
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 : R2→R3 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
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
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.t2+(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)
SXT
XL
.
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: تحىيم خطي
Kernel: انُىاة
Range:انًذي
One to One:واحذ نىاحذ
Onto:شايم
Reflection: اَعكاس
Rotation:ٌدورا
Composition of Linear Transformation:تركيب انتحىيلاث انخطيت
Transition Matrix: انًصفىفت انُاقهت
Matrix Representation of a Linear Transformation:تًثيم انتحىيم انخطي بًصفىفت
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
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
sxxsxxRssx
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
sxxxRssx
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
xxRssx
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
xxRssx
XAI
2
1,2,12
2
1,:
0:00
0:2
11
0:12
0:120)(
121
212
22
XxxsLet
xxRssx
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: قيًت يًيزة
Eigen Vector:يتجه يًيز
Characteristic Polynomial:كثير انحذود انًًيز
Characteristic Equation:انًعادنت انًًيزة
Eigen Space:انفضاء انًًيز
Similarity: انتشابه
Similar matrices:يصفىفاث يتشابهت
Diagonalizable Matrix: انًصفىفت انقابهت نهتحىيم انً يصفىفت قطريت
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:انًتبايُت انخطيت
Objective Function:اقتراٌ انهذف
Constraints: (شروط) قيىد
112
The End
113