Transcribed Image Text:10:41
Done elementary_linear_algebra_8th_...
166
Chapter 4 Vector Spaces
回洲回
4.2 Exercises
See CalcChat.com for worked-out solutions to odd-numbered exercises.
Describing the Additive Identity In Exercises 1-6,
describe the zero vector (the additive identity) of the
vector space.
1. R
3. Ma
5. P.
27. The set of all 3 x 3 matrices of the form
d
2. C[-1,0]
4. M.
6. M21
28. The set of all 3 x 3 matrices of the form
a
Describing the Additive Inverse In Exercises 7-12,
describe the additive inverse of a vector in the vector
space.
29. The set of all 4 x 4 matrices of the form
7. R"
8. C(-0o, o0)
a
9. M21
10. Мд
12. Mas
a
a
a
11. Р.
Testing for a Vector Space In Exercises 13-36,
determine whether the set, together with the standard
operations, is a vector space. Ifr it is not, identify at least
one of the ten vector space axioms that fails.
13. М
14. M
30. The set of all 4 x 4 matrices of the form
a
a
a b 0
a b e
15. The set of all third-degree polynomials
31. The set of all 2 x 2 singular matrices
32. The set of all 2 x 2 nonsingular matrices
33. The set of all 2 x 2 diagonal matrices
34. The set of all 3 x 3 upper triangular matrices
35. C[0, 11. the set of all continuous functions defined on
the interval [0,
36. C[-1, 1). the set of all continuous functions defined on
the interval [-1, I]
16. The set of all fifth-degree polynomials
17. The set of all first-degree polynomial functions ax,
a + 0. whose graphs pass through the origin
18. The set of all first-degree polynomial functions ar + h
a. b+ 0. whose graphs do not pass through the origin
19. The set of all polynomials of degree four or less
20. The set of all quadratic functions whose graphs pass
through the origin
37. Let V be the set of all positive real numbers. Determine
whether V is a vector space with the operations shown
21. The set
{(x, y): x 2 0, y is a real number)
below.
22. The set
x+y= KY
Addition
{(x, v): x 2 0. y z 0}
Scalar multiplication
If it is, verify each vector space axiom; if it is not, state
all vector space axioms that fail.
38. Determine whether the set R with the operations
23. The set
{(x, x): x is a real number}
24. The set
(x. y) + (x, y,) = (x,xz, ¥;V)
{(x, £x): x is a real number}
and
25. The set of all 2 x 2 matrices of the form
dx, y;) = (cx,, cy,)
is a vector space. If it is, verify each vector space axiom;
if it is not, state all vector space axioms that fail.
26. The set of all 2 x 2 matrices of the form
39. Proof Prove in full detail that the set {(x, 2x): r is a
real number), with the standard operations in R. is a
vector space.
Cea Cap Le AR d May ed d d depn the p my he d e po
l a dd id d y e gpe Cngap tengaeg and im
dtddepn th pa ey he d
4.2 Exercises
16
40. Proof Prove in full detail that M,, with the standard
operations, is a vector space.
46. CAPSTONE
41. Rather than use the standard definitions of addition
and scalar multiplication in R, let these two operations
be defined as shown below.
(a) (x. y) + (x y) - (x, + X, Y, + y,)
(a) Describe the conditions under which a set may be
classified as a vector space.
(b) Give an example of a set that is a vector space and
an example of a set that is not a vector space.
с(х, у) (сх, у)
(b) (x, y,) + (x,. y,) = (x,. 0)
с(х, у) - (сх, су)
47. Proof Complete the proof of the cancellation proper
of vector addition by justifying each step.
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![10:41
Done elementary_linear_algebra_8th_...
166
Chapter 4 Vector Spaces
回洲回
4.2 Exercises
See CalcChat.com for worked-out solutions to odd-numbered exercises.
Describing the Additive Identity In Exercises 1-6,
describe the zero vector (the additive identity) of the
vector space.
1. R
3. Ma
5. P.
27. The set of all 3 x 3 matrices of the form
d
2. C[-1,0]
4. M.
6. M21
28. The set of all 3 x 3 matrices of the form
a
Describing the Additive Inverse In Exercises 7-12,
describe the additive inverse of a vector in the vector
space.
29. The set of all 4 x 4 matrices of the form
7. R"
8. C(-0o, o0)
a
9. M21
10. Мд
12. Mas
a
a
a
11. Р.
Testing for a Vector Space In Exercises 13-36,
determine whether the set, together with the standard
operations, is a vector space. Ifr it is not, identify at least
one of the ten vector space axioms that fails.
13. М
14. M
30. The set of all 4 x 4 matrices of the form
a
a
a b 0
a b e
15. The set of all third-degree polynomials
31. The set of all 2 x 2 singular matrices
32. The set of all 2 x 2 nonsingular matrices
33. The set of all 2 x 2 diagonal matrices
34. The set of all 3 x 3 upper triangular matrices
35. C[0, 11. the set of all continuous functions defined on
the interval [0,
36. C[-1, 1). the set of all continuous functions defined on
the interval [-1, I]
16. The set of all fifth-degree polynomials
17. The set of all first-degree polynomial functions ax,
a + 0. whose graphs pass through the origin
18. The set of all first-degree polynomial functions ar + h
a. b+ 0. whose graphs do not pass through the origin
19. The set of all polynomials of degree four or less
20. The set of all quadratic functions whose graphs pass
through the origin
37. Let V be the set of all positive real numbers. Determine
whether V is a vector space with the operations shown
21. The set
{(x, y): x 2 0, y is a real number)
below.
22. The set
x+y= KY
Addition
{(x, v): x 2 0. y z 0}
Scalar multiplication
If it is, verify each vector space axiom; if it is not, state
all vector space axioms that fail.
38. Determine whether the set R with the operations
23. The set
{(x, x): x is a real number}
24. The set
(x. y) + (x, y,) = (x,xz, ¥;V)
{(x, £x): x is a real number}
and
25. The set of all 2 x 2 matrices of the form
dx, y;) = (cx,, cy,)
is a vector space. If it is, verify each vector space axiom;
if it is not, state all vector space axioms that fail.
26. The set of all 2 x 2 matrices of the form
39. Proof Prove in full detail that the set {(x, 2x): r is a
real number), with the standard operations in R. is a
vector space.
Cea Cap Le AR d May ed d d depn the p my he d e po
l a dd id d y e gpe Cngap tengaeg and im
dtddepn th pa ey he d
4.2 Exercises
16
40. Proof Prove in full detail that M,, with the standard
operations, is a vector space.
46. CAPSTONE
41. Rather than use the standard definitions of addition
and scalar multiplication in R, let these two operations
be defined as shown below.
(a) (x. y) + (x y) - (x, + X, Y, + y,)
(a) Describe the conditions under which a set may be
classified as a vector space.
(b) Give an example of a set that is a vector space and
an example of a set that is not a vector space.
с(х, у) (сх, у)
(b) (x, y,) + (x,. y,) = (x,. 0)
с(х, у) - (сх, су)
47. Proof Complete the proof of the cancellation proper
of vector addition by justifying each step.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F47a8b8fc-027a-4a12-ab4d-0bbb9d9d31c0%2Fgpok0b_processed.jpeg&w=3840&q=75)
