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. [- 3 -3 7 1 -4 5 3-9 3 -1 -2 8. 10. 1-2-9 1 2 3 6 2 5 0-4 0-8

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Chapter2: Second-order Linear Odes
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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
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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