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.
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.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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13
![48 CHAPTER 1 Linear Equations in Linear Algebra
1.5 EXERCISES
In Exercises 1-4, determine if the system has a nontrivial solution.
Try to use as few row operations as possible.
12.
5.
7.
9.
11.
1.
2x₁5x₂ + 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.
x₁ + 3x₂ + x3 = 0
-4x₁9x2 + 2x3 = 0
- 3x₂ - 6x3 = 0
1
In Exercises 7-12, describe all solutions of Ax = 0 in parametric
vector form, where A is row equivalent to the given matrix.
3 -3
[13
1 -4
-1
3 -9
1 -4
0 0
0
0
00
1
0
0
0
6
-9]
5000
3-2
-4 -2
7700
210N
7
3]
5
0000
0
2-6
1 -7
046
6.
3 -5
0-1
1 -4
0 0
9 0
4 -8
1
0
0
0 0
10.
2.
8.
x₁ - 3x₂ + 7x3 = 0
-2x₁ + x₂ - 4x3 = 0
X2
x₁ + 2x2 + 9x3 = 0
4. -5x₁ + 7x2 + 9x3 = 0
X₁ - 2x2 + 6x3 = 0
x₁ + 3x₂ - 5x3 = 0
x₁ + 4x₂8x3 = 0
-3x₁ - 7x₂ + 9x3 = 0
[6
1
[2
-2 -9
5
1 2-6]
30-4
6 0-8
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.
14. Suppose the solution set of a certain system of linear
equations can be described as x₁ = 3x4, X2 = 8 + x4,
as a "line" in R4.
x3 = 2-5x4, with x4 free. Use vectors to describe this set
15. Follow the method of Example 3 to describe the solutions of
the following system in parametric vector form. Also, give
that in Exercise 5.
a geometric description of the solution set and compare it to
x₁ + 3x₂ + x3 = 1
-4x₁ - 9x₂ + 2x3 = -1
- 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.
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.
19. a =
is
[3] » - [ - ]
0
3
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.]
21. p =
*+-[---]
mikolai ir
b.
Р
q
M
20. a =
=
- [ ³ ] - [ - ]
8
x2
22. p =
P=[¯3 ]-9 = [ - ]
9-P
-P
-X₁
The line through p and q.
each answer.
In Exercises 23 and 24, mark each statement True or False. Justify
23/0
23./a. A homogeneous equation is always consistent.
solution set.
The equation Ax = 0 gives an explicit description of its
c. The homogeneous equation Ax = 0 has the trivial so-
variable.
lution if and only if the equation has at least one free
allel to p.
d. The equation x = p + tv describes a line through v par-
e. The solution set of Ax=b is the set of all vectors of
equation Ax = 0.
the form w = p + Vh, where vh is any solution of the
24.
x is nonzero.
24. a. If x is a nontrivial solution of Ax = 0, then every entry in
b. The equation x = x₂u + X3V, with x2 and x3 free (and
through the origin.
neither u nor va multiple of the other), describes a plane
is a solution.
c. The equation Ax = b is homogeneous if the zero vector
a direction parallel to p.
d. The effect of adding p to a vector is to move the vector in
of A
unique
27. Suppos
28 16-0
the organ
In Exercises 29
al solution and
solution for eve
29. Aisa 3 x 3
30. Aisa 3x3
31. Aisa 3x21
32. Aisa 2x4m
33. Given A =
Ar=0 by ins
written as a vec](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad669a50-5f0c-4df5-9260-23ad3558d216%2F32c5583f-2adb-4b26-9e94-0e251aab41f2%2Flpbvrja_processed.jpeg&w=3840&q=75)
Transcribed Image Text:48 CHAPTER 1 Linear Equations in Linear Algebra
1.5 EXERCISES
In Exercises 1-4, determine if the system has a nontrivial solution.
Try to use as few row operations as possible.
12.
5.
7.
9.
11.
1.
2x₁5x₂ + 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.
x₁ + 3x₂ + x3 = 0
-4x₁9x2 + 2x3 = 0
- 3x₂ - 6x3 = 0
1
In Exercises 7-12, describe all solutions of Ax = 0 in parametric
vector form, where A is row equivalent to the given matrix.
3 -3
[13
1 -4
-1
3 -9
1 -4
0 0
0
0
00
1
0
0
0
6
-9]
5000
3-2
-4 -2
7700
210N
7
3]
5
0000
0
2-6
1 -7
046
6.
3 -5
0-1
1 -4
0 0
9 0
4 -8
1
0
0
0 0
10.
2.
8.
x₁ - 3x₂ + 7x3 = 0
-2x₁ + x₂ - 4x3 = 0
X2
x₁ + 2x2 + 9x3 = 0
4. -5x₁ + 7x2 + 9x3 = 0
X₁ - 2x2 + 6x3 = 0
x₁ + 3x₂ - 5x3 = 0
x₁ + 4x₂8x3 = 0
-3x₁ - 7x₂ + 9x3 = 0
[6
1
[2
-2 -9
5
1 2-6]
30-4
6 0-8
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.
14. Suppose the solution set of a certain system of linear
equations can be described as x₁ = 3x4, X2 = 8 + x4,
as a "line" in R4.
x3 = 2-5x4, with x4 free. Use vectors to describe this set
15. Follow the method of Example 3 to describe the solutions of
the following system in parametric vector form. Also, give
that in Exercise 5.
a geometric description of the solution set and compare it to
x₁ + 3x₂ + x3 = 1
-4x₁ - 9x₂ + 2x3 = -1
- 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.
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.
19. a =
is
[3] » - [ - ]
0
3
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.]
21. p =
*+-[---]
mikolai ir
b.
Р
q
M
20. a =
=
- [ ³ ] - [ - ]
8
x2
22. p =
P=[¯3 ]-9 = [ - ]
9-P
-P
-X₁
The line through p and q.
each answer.
In Exercises 23 and 24, mark each statement True or False. Justify
23/0
23./a. A homogeneous equation is always consistent.
solution set.
The equation Ax = 0 gives an explicit description of its
c. The homogeneous equation Ax = 0 has the trivial so-
variable.
lution if and only if the equation has at least one free
allel to p.
d. The equation x = p + tv describes a line through v par-
e. The solution set of Ax=b is the set of all vectors of
equation Ax = 0.
the form w = p + Vh, where vh is any solution of the
24.
x is nonzero.
24. a. If x is a nontrivial solution of Ax = 0, then every entry in
b. The equation x = x₂u + X3V, with x2 and x3 free (and
through the origin.
neither u nor va multiple of the other), describes a plane
is a solution.
c. The equation Ax = b is homogeneous if the zero vector
a direction parallel to p.
d. The effect of adding p to a vector is to move the vector in
of A
unique
27. Suppos
28 16-0
the organ
In Exercises 29
al solution and
solution for eve
29. Aisa 3 x 3
30. Aisa 3x3
31. Aisa 3x21
32. Aisa 2x4m
33. Given A =
Ar=0 by ins
written as a vec
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