6:12 Done elementary_linear_algebra_8th_edi... O 4.6 Exercises 205 4.6 Exercises See CalcChat.com for worked-out solutions to odd-numbered exercises. Row Vectors and Column Vectors In Exercises 1-4, Finding a Basis for a Column Space and Rank In Exercises 21-26, find (a) a basis for the column space and (b) the rank of the matrix. write (a) the row vectors and (b) the column vectors of the matrix. 1. [ 2. [6 5 -1] 21. 22. [1 31 -4 Г4 20 31 1 24. 6 2 -11 - 16 3. 4. 4 -1 23. -5 -9- -4 -6 1 4 -3 Finding a Basis for a Row Space and Rank In Exercises 5-12, find (a) a basis for the row space and 14 9. -3 25. -2 (b) the rank of the matrix. 4 -2 5. 6. [0 - 2] | 2 4 -2 2 4 -2 -3 7. 8. -2 -5 26. 4 1 -6 15 -4 -1 1. 18 -3 1 4 9. 40 116 10. 5 10 Finding the Nullspace of a Matrix In Exercises 27–40, -3 12 - 27 -7 find the nullspace of the matrix. -4 4 -1 27. A = 28. A = 11. -4 9. -2 -4 4 29. A = [1 3] 30. A = [1 4 2] 4 3 [1 31. A = Г1 32. A = 2 1| -1 1 12. Г1 -3 3 -6 34. A =-2 21 1 33. A = |2 -1 4 - 14 2 -2 3 -2 - 2 [5 - 16 Finding a Basis for a Subspace In Exercises 13–16, find a basis for the subspace of R³ spanned by S. 35. A =3 –1 36. A = 48 -3 [2 - 80 13. S = {(1, 2, 4), (– 1, 3, 4), (2, 3, 1)} 14. S = {(2, 3, – 1), (1, 3, –9), (0, 1, 5)} 15. S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)} 16. S = {(1, 2, 2), (– 1, 0, 0), (1, 1, 1)} 5. 1 3 -2 37. A = -1 -2 -6 4. -8 4 Finding a Basis for a Subspace In Exercises 17–20, find a basis for the subspace of R' spanned by S. 38. A = |-2 -8 -4 -2 17. S = {(2, 9, –2, 53), (– 3, 2, 3, - 2), (8, – 3, – 8, 17), (0, – 3, 0, 15)} 1 18. S = {(6, –3, 6, 34), (3, – 2, 3, 19), (8, 3, –9, 6), 39. A = 3 -2 1 (- 2, 0, 6, – 5)} 19. S = {(-3, 2, 5, 28), (-6, 1, – 8, – 1), (14, – 10, 12, – 10), (0, 5, 12, 50)} 20. S = {(2, 5, –- 3, – 2), (–2, – 3, 2, – 5), (1, 3, – 2, 2), 4 - 1 40. A = 4 (-1, -5, 3, 5)} 4 Copyright 2017 Cengage Leaming All Rights Reserved. May not be copied scanned, or duplicated, in whole ce in part. Due to dectronic rights, some third party content may be suppeessed from the eBook andior eChapter(s) Editarial sevieu has deemed that any suppressed contet does not materially affect the verall learning experience. Cengage Leaming reserves the right to remove additional content at any time if suhsequent rights restrictions require it. :06 Chapter 4 Vector Spaces ank, Nullity, Bases, and Linear Independence In Exercises 41 and 42, use the fact that matrices A and BR re row-equivalent. Nonhomogeneous System In Exercises 49-56, determine whether the nonhomogeneous system Ax = h is consistent. If it is, write the solution in the form x = x, + x, where x, is a particular solution of Ax = b and x, is a solution of Ax = (0. a) Find the rank and nullity of A. b) Find a basis for the nullspace of A. 49. 4y = 17 50. x + 2y - 4z = -1 c) Find a basis for the row space of A. Зх — 12у - 51 —Зх — бу + 12: — d) Find a basis for the column space of A. - 2x + 8y = - 34 e) Determine whether the rows of A are linearly x + 3y + 10z = 18 52. - 2x + 7y + 32z = 29 51. 2x - 4y + 5z = - Zx + 14y + 4z = - 28. independent.
6:12 Done elementary_linear_algebra_8th_edi... O 4.6 Exercises 205 4.6 Exercises See CalcChat.com for worked-out solutions to odd-numbered exercises. Row Vectors and Column Vectors In Exercises 1-4, Finding a Basis for a Column Space and Rank In Exercises 21-26, find (a) a basis for the column space and (b) the rank of the matrix. write (a) the row vectors and (b) the column vectors of the matrix. 1. [ 2. [6 5 -1] 21. 22. [1 31 -4 Г4 20 31 1 24. 6 2 -11 - 16 3. 4. 4 -1 23. -5 -9- -4 -6 1 4 -3 Finding a Basis for a Row Space and Rank In Exercises 5-12, find (a) a basis for the row space and 14 9. -3 25. -2 (b) the rank of the matrix. 4 -2 5. 6. [0 - 2] | 2 4 -2 2 4 -2 -3 7. 8. -2 -5 26. 4 1 -6 15 -4 -1 1. 18 -3 1 4 9. 40 116 10. 5 10 Finding the Nullspace of a Matrix In Exercises 27–40, -3 12 - 27 -7 find the nullspace of the matrix. -4 4 -1 27. A = 28. A = 11. -4 9. -2 -4 4 29. A = [1 3] 30. A = [1 4 2] 4 3 [1 31. A = Г1 32. A = 2 1| -1 1 12. Г1 -3 3 -6 34. A =-2 21 1 33. A = |2 -1 4 - 14 2 -2 3 -2 - 2 [5 - 16 Finding a Basis for a Subspace In Exercises 13–16, find a basis for the subspace of R³ spanned by S. 35. A =3 –1 36. A = 48 -3 [2 - 80 13. S = {(1, 2, 4), (– 1, 3, 4), (2, 3, 1)} 14. S = {(2, 3, – 1), (1, 3, –9), (0, 1, 5)} 15. S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)} 16. S = {(1, 2, 2), (– 1, 0, 0), (1, 1, 1)} 5. 1 3 -2 37. A = -1 -2 -6 4. -8 4 Finding a Basis for a Subspace In Exercises 17–20, find a basis for the subspace of R' spanned by S. 38. A = |-2 -8 -4 -2 17. S = {(2, 9, –2, 53), (– 3, 2, 3, - 2), (8, – 3, – 8, 17), (0, – 3, 0, 15)} 1 18. S = {(6, –3, 6, 34), (3, – 2, 3, 19), (8, 3, –9, 6), 39. A = 3 -2 1 (- 2, 0, 6, – 5)} 19. S = {(-3, 2, 5, 28), (-6, 1, – 8, – 1), (14, – 10, 12, – 10), (0, 5, 12, 50)} 20. S = {(2, 5, –- 3, – 2), (–2, – 3, 2, – 5), (1, 3, – 2, 2), 4 - 1 40. A = 4 (-1, -5, 3, 5)} 4 Copyright 2017 Cengage Leaming All Rights Reserved. May not be copied scanned, or duplicated, in whole ce in part. Due to dectronic rights, some third party content may be suppeessed from the eBook andior eChapter(s) Editarial sevieu has deemed that any suppressed contet does not materially affect the verall learning experience. Cengage Leaming reserves the right to remove additional content at any time if suhsequent rights restrictions require it. :06 Chapter 4 Vector Spaces ank, Nullity, Bases, and Linear Independence In Exercises 41 and 42, use the fact that matrices A and BR re row-equivalent. Nonhomogeneous System In Exercises 49-56, determine whether the nonhomogeneous system Ax = h is consistent. If it is, write the solution in the form x = x, + x, where x, is a particular solution of Ax = b and x, is a solution of Ax = (0. a) Find the rank and nullity of A. b) Find a basis for the nullspace of A. 49. 4y = 17 50. x + 2y - 4z = -1 c) Find a basis for the row space of A. Зх — 12у - 51 —Зх — бу + 12: — d) Find a basis for the column space of A. - 2x + 8y = - 34 e) Determine whether the rows of A are linearly x + 3y + 10z = 18 52. - 2x + 7y + 32z = 29 51. 2x - 4y + 5z = - Zx + 14y + 4z = - 28. independent.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![6:12
Done elementary_linear_algebra_8th_edi... O
4.6 Exercises
205
4.6
Exercises
See CalcChat.com for worked-out solutions to odd-numbered exercises.
Row Vectors and Column Vectors In Exercises 1-4,
Finding a Basis for a Column Space and Rank In
Exercises 21-26, find (a) a basis for the column space
and (b) the rank of the matrix.
write (a) the row vectors and (b) the column vectors of
the matrix.
1. [
2. [6
5 -1]
21.
22. [1
31
-4
Г4
20
31
1
24. 6
2 -11 - 16
3.
4.
4
-1
23.
-5
-9-
-4
-6
1
4
-3
Finding a Basis for a Row Space and Rank In
Exercises 5-12, find (a) a basis for the row space and
14
9.
-3
25.
-2
(b) the rank of the matrix.
4
-2
5.
6. [0
- 2]
| 2
4
-2
2
4
-2
-3
7.
8.
-2
-5
26. 4
1
-6
15
-4
-1
1.
18
-3
1
4
9.
40 116
10. 5
10
Finding the Nullspace of a Matrix In Exercises 27–40,
-3
12
- 27
-7
find the nullspace of the matrix.
-4
4
-1
27. A =
28. A =
11.
-4
9.
-2
-4
4
29. A = [1
3]
30. A = [1
4
2]
4
3
[1
31. A =
Г1
32. A =
2
1|
-1
1
12.
Г1
-3
3 -6
34. A =-2
21
1
33. A = |2
-1
4 - 14
2 -2
3
-2
- 2
[5
- 16
Finding a Basis for a Subspace In Exercises 13–16,
find a basis for the subspace of R³ spanned by S.
35. A =3
–1
36. A =
48
-3
[2
- 80
13. S = {(1, 2, 4), (– 1, 3, 4), (2, 3, 1)}
14. S = {(2, 3, – 1), (1, 3, –9), (0, 1, 5)}
15. S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)}
16. S = {(1, 2, 2), (– 1, 0, 0), (1, 1, 1)}
5.
1
3
-2
37. A =
-1
-2
-6
4.
-8
4
Finding a Basis for a Subspace In Exercises 17–20,
find a basis for the subspace of R' spanned by S.
38. A =
|-2
-8
-4
-2
17. S = {(2, 9, –2, 53), (– 3, 2, 3, - 2), (8, – 3, – 8, 17),
(0, – 3, 0, 15)}
1
18. S = {(6, –3, 6, 34), (3, – 2, 3, 19), (8, 3, –9, 6),
39. A =
3 -2
1
(- 2, 0, 6, – 5)}
19. S = {(-3, 2, 5, 28), (-6, 1, – 8, – 1),
(14, – 10, 12, – 10), (0, 5, 12, 50)}
20. S = {(2, 5, –- 3, – 2), (–2, – 3, 2, – 5), (1, 3, – 2, 2),
4
- 1
40. A =
4
(-1, -5, 3, 5)}
4
Copyright 2017 Cengage Leaming All Rights Reserved. May not be copied scanned, or duplicated, in whole ce in part. Due to dectronic rights, some third party content may be suppeessed from the eBook andior eChapter(s)
Editarial sevieu has deemed that any suppressed contet does not materially affect the verall learning experience. Cengage Leaming reserves the right to remove additional content at any time if suhsequent rights restrictions require it.
:06
Chapter 4 Vector Spaces
ank, Nullity, Bases, and Linear Independence In
Exercises 41 and 42, use the fact that matrices A and BR
re row-equivalent.
Nonhomogeneous System In Exercises 49-56,
determine whether the nonhomogeneous system Ax = h
is consistent. If it is, write the solution in the form
x = x, + x, where x, is a particular solution of Ax = b
and x, is a solution of Ax = (0.
a) Find the rank and nullity of A.
b) Find a basis for the nullspace of A.
49.
4y =
17
50.
x + 2y - 4z = -1
c) Find a basis for the row space of A.
Зх — 12у -
51
—Зх — бу + 12: —
d) Find a basis for the column space of A.
- 2x + 8y = - 34
e) Determine whether the rows of A are linearly
x + 3y + 10z = 18 52.
- 2x + 7y + 32z = 29
51.
2x - 4y + 5z =
- Zx + 14y + 4z = - 28.
independent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2Fc00f8ea7-f3c7-4f97-8c91-a46c574f3567%2Fhd125ud.png&w=3840&q=75)
Transcribed Image Text:6:12
Done elementary_linear_algebra_8th_edi... O
4.6 Exercises
205
4.6
Exercises
See CalcChat.com for worked-out solutions to odd-numbered exercises.
Row Vectors and Column Vectors In Exercises 1-4,
Finding a Basis for a Column Space and Rank In
Exercises 21-26, find (a) a basis for the column space
and (b) the rank of the matrix.
write (a) the row vectors and (b) the column vectors of
the matrix.
1. [
2. [6
5 -1]
21.
22. [1
31
-4
Г4
20
31
1
24. 6
2 -11 - 16
3.
4.
4
-1
23.
-5
-9-
-4
-6
1
4
-3
Finding a Basis for a Row Space and Rank In
Exercises 5-12, find (a) a basis for the row space and
14
9.
-3
25.
-2
(b) the rank of the matrix.
4
-2
5.
6. [0
- 2]
| 2
4
-2
2
4
-2
-3
7.
8.
-2
-5
26. 4
1
-6
15
-4
-1
1.
18
-3
1
4
9.
40 116
10. 5
10
Finding the Nullspace of a Matrix In Exercises 27–40,
-3
12
- 27
-7
find the nullspace of the matrix.
-4
4
-1
27. A =
28. A =
11.
-4
9.
-2
-4
4
29. A = [1
3]
30. A = [1
4
2]
4
3
[1
31. A =
Г1
32. A =
2
1|
-1
1
12.
Г1
-3
3 -6
34. A =-2
21
1
33. A = |2
-1
4 - 14
2 -2
3
-2
- 2
[5
- 16
Finding a Basis for a Subspace In Exercises 13–16,
find a basis for the subspace of R³ spanned by S.
35. A =3
–1
36. A =
48
-3
[2
- 80
13. S = {(1, 2, 4), (– 1, 3, 4), (2, 3, 1)}
14. S = {(2, 3, – 1), (1, 3, –9), (0, 1, 5)}
15. S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)}
16. S = {(1, 2, 2), (– 1, 0, 0), (1, 1, 1)}
5.
1
3
-2
37. A =
-1
-2
-6
4.
-8
4
Finding a Basis for a Subspace In Exercises 17–20,
find a basis for the subspace of R' spanned by S.
38. A =
|-2
-8
-4
-2
17. S = {(2, 9, –2, 53), (– 3, 2, 3, - 2), (8, – 3, – 8, 17),
(0, – 3, 0, 15)}
1
18. S = {(6, –3, 6, 34), (3, – 2, 3, 19), (8, 3, –9, 6),
39. A =
3 -2
1
(- 2, 0, 6, – 5)}
19. S = {(-3, 2, 5, 28), (-6, 1, – 8, – 1),
(14, – 10, 12, – 10), (0, 5, 12, 50)}
20. S = {(2, 5, –- 3, – 2), (–2, – 3, 2, – 5), (1, 3, – 2, 2),
4
- 1
40. A =
4
(-1, -5, 3, 5)}
4
Copyright 2017 Cengage Leaming All Rights Reserved. May not be copied scanned, or duplicated, in whole ce in part. Due to dectronic rights, some third party content may be suppeessed from the eBook andior eChapter(s)
Editarial sevieu has deemed that any suppressed contet does not materially affect the verall learning experience. Cengage Leaming reserves the right to remove additional content at any time if suhsequent rights restrictions require it.
:06
Chapter 4 Vector Spaces
ank, Nullity, Bases, and Linear Independence In
Exercises 41 and 42, use the fact that matrices A and BR
re row-equivalent.
Nonhomogeneous System In Exercises 49-56,
determine whether the nonhomogeneous system Ax = h
is consistent. If it is, write the solution in the form
x = x, + x, where x, is a particular solution of Ax = b
and x, is a solution of Ax = (0.
a) Find the rank and nullity of A.
b) Find a basis for the nullspace of A.
49.
4y =
17
50.
x + 2y - 4z = -1
c) Find a basis for the row space of A.
Зх — 12у -
51
—Зх — бу + 12: —
d) Find a basis for the column space of A.
- 2x + 8y = - 34
e) Determine whether the rows of A are linearly
x + 3y + 10z = 18 52.
- 2x + 7y + 32z = 29
51.
2x - 4y + 5z =
- Zx + 14y + 4z = - 28.
independent.
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