Deck 4: Vector Spaces

Question 1

Find the sum of the vectors and .

A)
B)
C)

D)
E)

Answer 1

Question 2

Find the vector v for which given that and .

A)
B)
C)
D)
E)

Answer 2

Question 3

Let , and . Find .

A)
B)
C)
D)
E)

Answer 3

Question 4

Which of the following vectors is the scalar multiple of

A)
B)
C)
D)
E)

Question 5

Provided and , solve for w.

A)

B)


C)

D)

E)

Question 6

Provided that and , write as a linear combination of u and w, if possible.

A)
B)
C)
D)
E) It is not possible to write v as a linear combination of u and w.

Question 7

Provided and , find w such that .

A)
B)
C)
D)
E)

Question 8

Provided that and , write as a linear combination of , and , if possible.

A)
B)
C)
D)
E) It is not possible to write v as a linear combination of , and .

Question 9

Provide that , and , write as a linear combination of , and , if possible.

A)
B)
C)
D)
E) It is not possible to write v as a linear combination of , and .

Question 10

Describe the zero vector (the additive identity) of the vector space .

A) (0, 0, 0, 0, 0)
B) (0, 0, 0, 0)
C) (0, 0, 0, 0, 0, 0)
D) (0, 0)
E) (0, 0, 0)

Question 11

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 12

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 13

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 14

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 15

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 16

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 17

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 18

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 19

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 20

Describe the zero vector (the additive identity) of the vector space .

A)
B)
C)
D)
E)

Question 21

Determine whether the set, with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is a vector space. All ten vector space axioms hold.
D) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 22

Determine whether the set of all first-degree polynomial functions as given below, whose graphs pass through the origin with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
D) This set is a vector space. All ten vector space axioms hold.
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 23

Determine whether the set of all second-degree polynomial functions as given below, whose graphs pass through the origin with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
D) This set is a vector space. All ten vector space axioms hold.
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 24

Determine whether the set of all third-degree polynomial functions as given below, whose graphs pass through the origin with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
D) This set is a vector space. All ten vector space axioms hold.
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 25

Determine whether the set of all fourth-degree polynomial functions as given below, whose graphs pass through the origin with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
D) This set is a vector space. All ten vector space axioms hold.
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 26

Determine whether the following set with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is a vector space. All ten vector space axioms hold.
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
D) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 27

Determine whether the following set with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is a vector space. All ten vector space axioms hold.
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
D) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 28

Determine whether the following set with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is a vector space. All ten vector space axioms hold.
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
D) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 29

Determine whether the following set with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is a vector space. All ten vector space axioms hold.
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
D) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
E) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication

Question 30

Determine whether the set of all diagonal matrices with the standard operations, is a vector space. If it is not, then determine the set of axioms that it fails.

A) This set is not a vector space. It fails the following axioms.
Commutative property
Additive identity
Distributive property
B) This set is not a vector space. It fails the following axioms.
Scalar identity
Associative property
Distributive property
Additive identity
C) This set is not a vector space. It fails the following axioms.
Closer under addition
Closer under scalar multiplication
D) This set is not a vector space. It fails the following axioms.
Additive identity
Additive inverse
Associative property
Scalar identity
E) This set is a vector space. All ten vector space axioms hold.

Question 31

Verify is a subspace of .​

Question 32

W is the set of all functions that are differentiable on . V is the set of all functions that are continuous on . Verify W is a subspace of V.​

Question 33

the following subset of is a subspace of .The set of all nonnegative functions:

Question 34

the following subset of is a subspace of .The set of all nonpositive functions:

Question 35

the following subset of is a subspace of .The set of all positive functions:

Question 36

the following subset of is a subspace of .The set of all negative functions:

Question 37

the following subset of is a subspace of .The set of all zero functions:

Question 38

the following subset of is a subspace of .The set of all constant functions:

Question 39

The set of all functions such that is a subspace of .

Question 40

The set of all lower triangular matrices is a subspace of .

Question 41

The set of all singular matrices is a subspace of .

Question 42

? The set is a subspace of with the standard operations.

Question 43

The set is a subspace of with the standard operations.

Question 44

The set spans .

Question 45

The set spans .

Question 46

The set spans .

Question 47

The set spans .

Question 48

The set spans .

Question 49

The set is linearly independent.

Question 50

The set is linearly independent.

Question 51

The set is linearly dependent. Express the first vector as a linear combination of the other two vectors.

A)
B)
C)
D)
E)

Question 52

The set spans .

Question 53

Write the standard basis for the vector space .

A)
B)
C)
D)
E)

Question 54

Write the standard basis for the vector space .

A)
B)
C)
D)
E)

Question 55

Write the standard basis for the vector space .

A)
B)
C)
D)
E)

Question 56

Write the standard basis for the vector space .

A)
B)
C)
D)
E)

Question 57

Write the standard basis for the vector space .

A)
B)
C)
D)
E)

Question 58

Explain why is not a basis for .

A) S is linearly independent and spans
B) S is linearly dependent and does not span
C) S is linearly independent and does not span
D) S is linearly dependent and spans
E) none of these choices

Question 59

Explain why is not a basis for .

A) S is linearly independent and spans
B) S is linearly dependent and spans
C) S is linearly independent and does not span
D) S is linearly dependent and does not span
E) none of these choices

Question 60

Explain why is not a basis for .

A) S is linearly independent and spans
B) S is linearly dependent and spans
C) S is linearly dependent and does not span
D) S is linearly independent and does not span
E) none of these choices

Question 61

Explain why is not a basis for .

A) S is linearly dependent and does not span
B) S is linearly dependent and spans
C) S is linearly independent and spans
D) S is linearly independent and does not span
E) none of these choices

Question 62

The set is a basis for .

Question 63

The set is a basis for .

Question 64

Determine whether is a basis for . If it is, write as a linear combination of the vectors in S.?

A) S is a basis for and
B) S is a basis for and
C) S is a basis for and
D) S is a basis for and
E) S is not a basis for

Question 65

Determine whether is a basis for . If it is, write as a linear combination of the vectors in S.?

A) S is a basis for and
B) S is a basis for and ?
C) S is a basis for and
D) S is a basis for and
E) S is not a basis for

Question 66

Determine whether is a basis for . If it is, write as a linear combination of the vectors in S.?

A) S is a basis for and
B) S is a basis for and
C) S is a basis for and
D) S is a basis for and
E) S is not a basis for

Question 67

Find a basis for the column space of the matrix .

A)
B)
C)
D)
E)

Question 68

Find a basis for the row space of a matrix .

A)
B)
C)
D)
E)

Question 69

Find the rank of the matrix

A) 2
B) 0
C) 4
D) 3
E) 1

Question 70

Find a basis for the subspace of spanned by .

A)
B)
C)
D)
E)

Question 71

Find a basis for the subspace of spanned by .

A)
B)
C)
D)
E)

Question 72

Find a basis for and the dimension of the solution space of .

A) , dim = 1
B) , dim = 3
C) , dim = 1
D) ? , dim = 2
E) , dim = 3

Question 73

Find a basis for the solution space of the following homogeneous system of linear equations.

A)
B)
C)
D)
E)

Question 74

Determine whether B is in the column space of A. If it is, write B as a linear combination of the column vectors of A..

A) is in the column space of A and
B) is in the column space of A and
C) is in the column space of A and
D) does not belong to the column space of
E) none of these choices

Question 75

Given that , the coordinate matrix of x relative to a (nonstandard) basis find the coordinate vector of x relative to the standard basis in .

A)
B)
C)
D)
E)

Question 76

Given that , the coordinate matrix of x relative to a (nonstandard) basis . Find the coordinate vector of x relative to the standard basis in .

A)
B)
C)
D)
E)

Question 77

Given that , the coordinate matrix of x relative to a (nonstandard) basis . Find the coordinate vector of x relative to the standard basis in .

A)
B)
C)
D)
E)

Question 78

Find the transition matrix from to .

A)
B)
C)
D)
E)

Question 79

Find the coordinate matrix of in relative to the basis .

A)
B)
C)
D)
E)

Question 80

Find the transition matrix from to by hand.

A)
B)
C)
D)
E)