In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 7. 9. 1 3 -3 77 31 0 1 -4 5 [₁ 3-9 -1 6 -9] 3-2 8. 10. inloe artin [ 1-2 -9 5 1 2 6 1 2 3 6 0-4gi ne -8 nortW 0 0
In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 7. 9. 1 3 -3 77 31 0 1 -4 5 [₁ 3-9 -1 6 -9] 3-2 8. 10. inloe artin [ 1-2 -9 5 1 2 6 1 2 3 6 0-4gi ne -8 nortW 0 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
9
![7.
11.
48 CHAPTER 1 Linear Equations in Linear Algebra
9.
12.
1.5 EXERCISES
In Exercises 1-4, determine if the system has a nontrivial solution.
Try to use as few row operations as possible.
2.
1. 2x₁ - 5x2 + 8x3 = 0
-2x₁ - 7x₂ + x3 = 0
4x₁ + 2x2 + 7x3 = 0
3. -3x₁ + 5x2 - 7x3 = 0
-6x₁ + 7x₂ + x3 = 0
In Exercises 5 and 6, follow the method of Examples 1 and 2
to write the solution set of the given homogeneous system in
parametric vector form.
5.
x₁ + 3x₂ + x3 = 0
-4x₁ - 9x2 + 2x3 = 0
- 3x2 - 6x3 = 0
In Exercises 7-12, describe all solutions of Ax = 0 in parametric
vector form, where A is row equivalent to the given matrix.
7
5
}]
9]
3 -9 6 A usu
LEXO 0
xoni 3-57
[
0
13
1
0 to 0
3 -3
1 -4
0
1
0
0
0
-4 -2
000
0
3 -2
01
5
0
0
6.
0
0
0
2
-6
9
nol
1 -7 4 -8
ORGSY 31
0 0 0 1
000 0 0
x₁ - 3x₂ + 7x3 = 0
-2x₁ + x2 - 4x3 = 0
x₁ + 2x2 + 9x3 = 0
4. -5x1 + 7x2 + 9x3 = 0
X₁ - 2x₂ + 6x3 = 0
10.
00-1
1 -4
0 0
8.
oculoz odt.no
ols 0
x₁ + 3x₂ - 5x3 = 0
x₁ + 4x₂ - 8x3 = 0
-3x₁ - 7x₂ + 9x3 = 0
1
· [²2
1 -2 -9
5
_$]
1 2-6
3
0-4
60-8
=]
13. Suppose the solution set of a certain system of linear equa-
tions can be described as x₁ = 5+4x3, x2 = -2-7x3, with
x3 free. Use vectors to describe this set as a line in R3,
10 57
14. Suppose the solution set of a certain system of linear
equations can be described as x₁ = 3x4, x2 = 8 + x4,
X3 = 2-5x4, with x4 free. Use vectors to describe this set
as a "line" in R4.
E-m
15. Follow the method of Example 3 to describe the solutions of
the following system in parametric vector form. Also, give
a geometric description of the solution set and compare it to
that in Exercise 5.
+ix
x₁ + 3x₂ + x3 = 1
-4x1 - 9x2 + 2x3 = -1
28 +0
1=0$
siqmad
- 3x2 - 6x3 = -3
16. As in Exercise 15, describe the solutions of the following
system in parametric vector form, and provide a geometric
comparison with the solution set in Exercise 6.
10
x₁ + 3x₂ - 5x3 = 4
x₁ + 4x₂ - 8x3 =
7
-3x₁ - 7x₂ + 9x3 = -6
17. Describe and compare the solution sets of x₁ + 9x₂ - 4x3 = 0
and x₁ + 9x2 - 4x3 = -2.
18. Describe and compare the solution sets of x₁ - 3x₂ + 5x3 = 0
and x₁ - 3x2 + 5x3 = 4.
In Exercises 19 and 20, find the parametric equation of the line
through a parallel to b.
- [ ²2 ] - [ ²³ ]
0
3
19. a =
gil bno o motos
nad Wq noitu
minogle gmwollol
21. p
8
In Exercises 21 and 22, find a parametric equation of the line M
through p and q. [Hint: M is parallel to the vector q - p. See the
figure below.]
= [ ³ ] - [ ²1 ]
*+-+-]-[HQ
22. p =
x2
P
20. a =
q
M
-4
Mae.
q-p
-P
-6
=
3
q=
X1
The line through p and q.
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
ach de
1000
23./a. A homogeneous equation is always consistent.
b. The equation Ax = 0 gives an explicit description of its
solution set.
c. The homogeneous equation Ax = 0 has the trivial so-
lution if and only if the equation has at least one free
variable.
d. The equation x = p + tv describes a line through v par-
allel to p.
The solution set of Ax = b is the set of all vectors of
the form w = p + Vh, where vh is any solution of the
equation Ax = 0.
24. a. If x is a nontrivial solution of Ax = 0, then every entry
x is nonzero.
b. The equation x = x₂u + x3v, with x2 and x3 free (and
neither u nor v a multiple of the other), describes a plane
through the origin.
is a solution.
c. The equation Ax=b is homogeneous if the zero vector
d. The effect of adding p to a vector is to move the vector in
a direction parallel to p.
25
26
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F35e33330-5427-4f24-bed9-17085c11ebad%2F97d125e3-2a9e-40ac-884d-1b92b8ca0ed0%2Fcg34j9m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7.
11.
48 CHAPTER 1 Linear Equations in Linear Algebra
9.
12.
1.5 EXERCISES
In Exercises 1-4, determine if the system has a nontrivial solution.
Try to use as few row operations as possible.
2.
1. 2x₁ - 5x2 + 8x3 = 0
-2x₁ - 7x₂ + x3 = 0
4x₁ + 2x2 + 7x3 = 0
3. -3x₁ + 5x2 - 7x3 = 0
-6x₁ + 7x₂ + x3 = 0
In Exercises 5 and 6, follow the method of Examples 1 and 2
to write the solution set of the given homogeneous system in
parametric vector form.
5.
x₁ + 3x₂ + x3 = 0
-4x₁ - 9x2 + 2x3 = 0
- 3x2 - 6x3 = 0
In Exercises 7-12, describe all solutions of Ax = 0 in parametric
vector form, where A is row equivalent to the given matrix.
7
5
}]
9]
3 -9 6 A usu
LEXO 0
xoni 3-57
[
0
13
1
0 to 0
3 -3
1 -4
0
1
0
0
0
-4 -2
000
0
3 -2
01
5
0
0
6.
0
0
0
2
-6
9
nol
1 -7 4 -8
ORGSY 31
0 0 0 1
000 0 0
x₁ - 3x₂ + 7x3 = 0
-2x₁ + x2 - 4x3 = 0
x₁ + 2x2 + 9x3 = 0
4. -5x1 + 7x2 + 9x3 = 0
X₁ - 2x₂ + 6x3 = 0
10.
00-1
1 -4
0 0
8.
oculoz odt.no
ols 0
x₁ + 3x₂ - 5x3 = 0
x₁ + 4x₂ - 8x3 = 0
-3x₁ - 7x₂ + 9x3 = 0
1
· [²2
1 -2 -9
5
_$]
1 2-6
3
0-4
60-8
=]
13. Suppose the solution set of a certain system of linear equa-
tions can be described as x₁ = 5+4x3, x2 = -2-7x3, with
x3 free. Use vectors to describe this set as a line in R3,
10 57
14. Suppose the solution set of a certain system of linear
equations can be described as x₁ = 3x4, x2 = 8 + x4,
X3 = 2-5x4, with x4 free. Use vectors to describe this set
as a "line" in R4.
E-m
15. Follow the method of Example 3 to describe the solutions of
the following system in parametric vector form. Also, give
a geometric description of the solution set and compare it to
that in Exercise 5.
+ix
x₁ + 3x₂ + x3 = 1
-4x1 - 9x2 + 2x3 = -1
28 +0
1=0$
siqmad
- 3x2 - 6x3 = -3
16. As in Exercise 15, describe the solutions of the following
system in parametric vector form, and provide a geometric
comparison with the solution set in Exercise 6.
10
x₁ + 3x₂ - 5x3 = 4
x₁ + 4x₂ - 8x3 =
7
-3x₁ - 7x₂ + 9x3 = -6
17. Describe and compare the solution sets of x₁ + 9x₂ - 4x3 = 0
and x₁ + 9x2 - 4x3 = -2.
18. Describe and compare the solution sets of x₁ - 3x₂ + 5x3 = 0
and x₁ - 3x2 + 5x3 = 4.
In Exercises 19 and 20, find the parametric equation of the line
through a parallel to b.
- [ ²2 ] - [ ²³ ]
0
3
19. a =
gil bno o motos
nad Wq noitu
minogle gmwollol
21. p
8
In Exercises 21 and 22, find a parametric equation of the line M
through p and q. [Hint: M is parallel to the vector q - p. See the
figure below.]
= [ ³ ] - [ ²1 ]
*+-+-]-[HQ
22. p =
x2
P
20. a =
q
M
-4
Mae.
q-p
-P
-6
=
3
q=
X1
The line through p and q.
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
ach de
1000
23./a. A homogeneous equation is always consistent.
b. The equation Ax = 0 gives an explicit description of its
solution set.
c. The homogeneous equation Ax = 0 has the trivial so-
lution if and only if the equation has at least one free
variable.
d. The equation x = p + tv describes a line through v par-
allel to p.
The solution set of Ax = b is the set of all vectors of
the form w = p + Vh, where vh is any solution of the
equation Ax = 0.
24. a. If x is a nontrivial solution of Ax = 0, then every entry
x is nonzero.
b. The equation x = x₂u + x3v, with x2 and x3 free (and
neither u nor v a multiple of the other), describes a plane
through the origin.
is a solution.
c. The equation Ax=b is homogeneous if the zero vector
d. The effect of adding p to a vector is to move the vector in
a direction parallel to p.
25
26
2
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