Spanning Sets: Definition, Properties, and Examples

Question 1

Consider the set of vectors S = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} in R^3. Is S a basis for R^3?

Accepted Answer

Yes, because the vectors in S are linearly independent and span R^3.

Answer

No, because the vectors in S are linearly independent but do not span R^3.

Answer

Yes, because the vectors in S span R^3 but are not linearly independent.

Answer

No, because the vectors in S are neither linearly independent nor span R^3.

Question 2

If a set of vectors S spans a vector space V, then which of the following statements is true?

Accepted Answer

Every vector in V can be written as a linear combination of the vectors in S

Answer

S is a linearly independent set

Answer

The number of vectors in S is less than or equal to the dimension of V

Answer

S is a basis for V

Question 3

Which statement is equivalent to saying that the columns of a matrix A span R^n?

Accepted Answer

The matrix A has rank n

Answer

The null space of A is nontrivial

Answer

The determinant of A is zero

Answer

The column space of A is a subspace of R^n

Question 4

Given vectors v1 = (1, 2, 3) and v2 = (4, 5, 6), determine if the set {v1, v2} spans R^3.

Accepted Answer

The set {v1, v2} spans R^3 because it contains two linearly independent vectors.

Answer

The set {v1, v2} does not span R^3 because it contains only two vectors.

Answer

The set {v1, v2} spans R^3 because it generates all possible linear combinations in R^3.

Answer

The set {v1, v2} does not span R^3 because it is not a basis for R^3.

Question 5

What is the minimum number of vectors required to span a vector space of dimension n?

Accepted Answer

n

Answer

n + 1

Answer

n - 1

Question 6

If a vector space has dimension n, then any spanning set for the vector space must contain:

Accepted Answer

Exactly n vectors

Answer

At least n vectors

Answer

At most n vectors

Answer

Any number of vectors

Question 7

If a vector space has a finite spanning set, then its dimension is:

Accepted Answer

Finite

Answer

Infinite

Answer

Cannot be determined

Question 8

Consider the vector space ( mathbb{R} ^ n ). Which of the following sets is a spanning set for ( mathbb{R} ^ n )?

Accepted Answer

{ (1, 0, ..., 0), (0, 1, ..., 0), ..., (0, 0, ..., 1) }

Answer

{ (1, 1, ..., 1), (0, 1, ..., 1), ..., (0, 0, ..., 1) }

Answer

{ (1, 2, 3, ..., n), (1, 3, 5, ..., n), ..., (1, n, n, ..., n) }

Answer

{ (1, 0, 0, ..., 0), (1, 1, 0, ..., 0), ... (1, 1, 1, ..., 1) }

Question 9

Identify the consequences that may arise if a set of vectors S in a vector space V is not a spanning set for V.

Accepted Answer

There exists a vector in V that cannot be expressed as a linear combination of vectors in S.

Answer

V has a finite dimension.

Answer

S can be extended to form a basis for V.

Answer

S is linearly independent.

Question 10

Determine whether the following set of vectors is a spanning set for R^3: {v1, v2, v3}, where v1 = (1, 0, 0), v2 = (0, 1, 0), and v3 = (0, 0, 1).

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

Requires more information

Question 11

Which of the following statements about linear combinations and spanning sets is true?

Accepted Answer

Every vector in a vector space can be expressed as a linear combination of the vectors in a spanning set.

Answer

Every spanning set is linearly independent.

Question 12

If a set of vectors spans a vector space, then the set is necessarily:

Accepted Answer

Generative

Answer

Redundant

Answer

Linearly dependent

Answer

Invertible

Question 13

Which of the following properties does a spanning set possess?

Accepted Answer

Any linear combination of the vectors in the set can represent any vector in the vector space.

Answer

The vectors in the set are linearly independent.

Answer

The set contains the zero vector.

Question 14

Which of the following sets of vectors spans R³?

Accepted Answer

{ (1, 0, 0), (0, 1, 0), (0, 0, 1) }

Answer

{ (1, 1, 1), (0, 0, 1), (0, 0, 0) }

Answer

{ (1, 0, 0), (1, 1, 0), (0, 1, 1) }

Answer

{ (1, 0, 1), (0, 1, 0), (0, 0, 1) }

Question 15

Determine if the set {(1, 2), (2, 3)} is a spanning set for R²

Accepted Answer

Yes

Answer

No

Answer

Not enough information given

Answer

Depends on the context

Question 16

Which of the following sets of vectors does NOT span the entire vector space ℝ² (two-dimensional Euclidean space)?

Accepted Answer

{(1, 0), (0, 2)}

Answer

{(1, 0), (1, 1)}

Answer

{(1, 1), (1, -1)}

Answer

{(2, 1), (0, 3)}

Question 17

What is the definition of a spanning set in a vector space V?

Accepted Answer

A set of vectors whose linear combinations generate all vectors in V.

Answer

A set of vectors that is linearly independent.

Answer

A set of vectors that has the same dimension as V.

Answer

A set of vectors that is orthonormal.

Question 18

Does the set of vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)} span R³?

Accepted Answer

Yes, it spans R³.

Answer

No, it does not span R³.

Answer

Cannot be determined without additional information.

Answer

It spans a subspace of R³ but not all of R³.

Question 19

Which of the following sets of vectors does NOT span R⁴ (four-dimensional Euclidean space)?

Accepted Answer

{(1, 0, 0, 0), (0, 1, 0, 0), (1, 1, 0, 0)}

Answer

{(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)}

Answer

{(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}

Answer

{(1, 1, 1, 1), (0, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1)}

Question 20

Is the set of vectors S = {(1, 2, 3), (4, 5, 6), (7, 8, 9)} linearly independent?

Accepted Answer

No, S is linearly dependent.

Answer

Yes, S is linearly independent.

Answer

Cannot be determined without additional information.

Answer

S is linearly independent if and only if the zero vector can be expressed as a linear combination of the vectors in S.

Question 21

If a vector space V has dimension n, which statement is always true?

Accepted Answer

Any set of n linearly independent vectors in V forms a basis for V.

Answer

Any set of linearly independent vectors in V forms a basis for V.

Answer

Any set of n vectors in V forms a basis for V.

Question 22

Let W be the subspace of R³ spanned by the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1). What is a basis for W?

Accepted Answer

{(1, 0, 0), (0, 1, 0), (0, 0, 1)}

Answer

{(1, 1, 0), (0, 1, 1), (1, 0, 1)}

Answer

{(1, 0, 0), (0, 1, 0)}

Question 23

Which set of vectors spans R³?

Accepted Answer

{ (1, 0, 0), (0, 1, 0), (0, 0, 1) }

Answer

{ (1, 0, 0), (0, 1, 1), (0, 1, 2) }

Answer

{ (1, 2, 3), (4, 5, 6), (7, 8, 9) }

Answer

{ (1, 0, 0), (0, 1, 0), (1, 1, 0) }

Question 24

In R³ with the standard inner product, which set of vectors is an orthogonal spanning set?

Accepted Answer

{ (1, 0, 0), (0, 1, 0), (0, 0, 1) }

Answer

{ (1, 0, 1), (0, 1, 1), (1, 1, 0) }

Answer

{ (1, 2, 3), (4, 5, 6), (7, 8, 9) }

Answer

{ (1, 1, 1), (2, 2, 2), (3, 3, 3) }

Question 25

What is the maximum number of vectors in a linearly independent spanning set for a vector space V of dimension n?

Accepted Answer

n

Answer

Cannot be determined

Answer

n - 1

Answer

n + 1

Question 26

Does the set of vectors {(1, 1, 0), (0, 1, 1), (1, 0, 1)} span R³?

Accepted Answer

Yes

Answer

Cannot be determined

Answer

Not applicable

Answer

No

Question 27

What is a direct consequence of the existence of a spanning set for a vector space?

Accepted Answer

Every vector in the vector space can be expressed as a linear combination of the vectors in the spanning set.

Answer

The vector space is finite-dimensional.

Question 28

Which of the following statements about spanning sets is true?

Accepted Answer

A and B can both be spanning sets for a vector space, even if A has more vectors than B.

Answer

Every spanning set for a vector space contains the zero vector.

Answer

Any set of linearly independent vectors is a spanning set.

Question 29

Consider the vector space of all continuous functions on the interval [0, 1]. Which of the following sets of functions is a spanning set?

Accepted Answer

{ 1, x, x², ..., x^n } where n is any positive integer

Answer

{ e^x, e^(-x), e^(2x) }

Answer

{ sin(x), cos(x), tan(x) }

Question 30

Which of the following sets of vectors spans the vector space R^2?

Accepted Answer

{(1, 0), (0, 1)}

Answer

{(1, 0), (0, 0)}

Answer

{(1, 1), (2, 2)}

Answer

{(1, 1), (2, -1)}

Question 31

Is the set of vectors {(1, 1, 0), (0, 1, 1), (1, 0, 1)} a spanning set for R^3?

Accepted Answer

Yes

Answer

Yes, but only for a subspace of R^3

Answer

No

Question 32

What is the minimum number of vectors required to span a vector space of dimension n?

Accepted Answer

n

Answer

n-1

Answer

n+1

Question 33

Which of the following statements about spanning sets is not true?

Accepted Answer

A spanning set can only contain the zero vector.

Answer

Two spanning sets for a vector space may have a different number of vectors.

Answer

A spanning set must contain at least one nonzero vector.

Question 34

Consider the set of vectors {(1, 2), (-1, 1), (2, 3)}. Which of the following is a linear combination of these vectors?

Accepted Answer

(0, 5)

Answer

(2, 4)

Answer

(1, 0)

Answer

(3, 6)

Question 35

Assume a vector space V is spanned by {(v1, v2, v3)}. Which of the following statements is necessarily true?

Accepted Answer

Every vector in V can be written as a linear combination of (v1, v2, v3).

Answer

V is a subspace of R^3.

Answer

(v1, v2, v3) is a linearly independent set.

Question 36

Which of the following is a consequence of a linearly independent set of vectors not spanning a vector space?

Accepted Answer

The vector space must have a higher dimension.

Answer

The vector space must be infinite-dimensional.

Answer

The vector space must be finite-dimensional.

Answer

The vector space must have a lower dimension.

Question 37

Given a vector space V with spanning set {(u1, u2)} and a vector v in V, which of the following is always possible?

Accepted Answer

Express v as a linear combination of (u1, u2).

Answer

Express v as a linear combination of (v, u1, u2).

Answer

Express u2 as a linear combination of (v, u1).

Answer

Express u1 as a linear combination of (v, u2).

Question 38

Consider the set of vectors {(a, b, c) | a + b + c = 0}. Which of the following is true about this set?

Accepted Answer

It is a spanning set for a subspace of R^3.

Answer

It is a linearly independent set.

Answer

It is a basis for R^3.

Answer

It is not a spanning set for R^3.

Question 39

Which of the following vector spaces has an infinite number of spanning sets?

Accepted Answer

The vector space of all real-valued functions on the interval [0, 1].

Answer

The vector space of all polynomials of degree less than or equal to 2.

Answer

The vector space of all matrices of order 2x2.

Question 40

Which of the following sets of vectors in R^3 forms a spanning set?

Accepted Answer

{ (1, 0, 0), (0, 1, 0), (0, 0, 1) }

Answer

{ (1, 0, 1), (0, 1, 1), (1, 1, 0) }

Answer

{ (1, 0, -1), (0, 1, 2), (1, -1, 3) }

Answer

{ (1, 2, 3), (2, 4, 6), (3, 6, 9) }

Question 41

Consider the set S = { (1, 2), (3, 4) } in R^2. Is S a spanning set for R^2?

Accepted Answer

Yes

Answer

No

Answer

S is a basis for R^2.

Answer

S is a linearly dependent set.

Question 42

Let W be the subspace of R^4 spanned by the vectors (1, 2, 3, 4) and (2, 4, 6, 8). Which of the following vectors is in W?

Accepted Answer

(3, 6, 9, 12)

Answer

(0, 0, 0, 1)

Answer

(2, 3, 5, 7)

Answer

(1, 0, 1, 0)

Question 43

True or False: A set of n linearly independent vectors in an n-dimensional vector space always forms a spanning set.

Accepted Answer

True

Answer

False

Answer

The statement is true only if the vectors are normalized.

Answer

The statement is true only if the vectors are orthogonal.

Question 44

Consider the vectors (1, 1, 1), (2, 1, 0), and (3, 1, 1) in R^3. Do these vectors form a spanning set for R^3?

Accepted Answer

Yes

Answer

No

Answer

The vectors are orthogonal.

Answer

The vectors are linearly dependent.

Question 45

Let S = { (1, 0), (0, 1), (1, 1) } in R^2. Is S a spanning set for R^2?

Accepted Answer

Yes

Answer

No

Answer

S is a basis for R^2.

Answer

S is a linearly dependent set.

Question 46

Consider the set of vectors { (1, 2, 1), (2, 1, 3), (3, 3, 5) } in R^3. Is this set a spanning set for R^3?

Accepted Answer

No

Answer

The set is linearly independent.

Answer

The set is orthogonal.

Answer

Yes

Question 47

Which of the following statements about spanning sets is true?

Accepted Answer

A spanning set for a vector space V is a set of vectors such that every vector in V can be written as a linear combination of the vectors in the set.

Answer

A spanning set for a vector space V is a set of vectors that is linearly independent.

Question 48

Consider the vector space R^n. Which of the following sets of vectors always forms a spanning set for R^n?

Accepted Answer

The set of all standard basis vectors

Answer

The set of all normalized vectors

Answer

The set of all orthogonal vectors

Answer

The set of all linearly independent vectors

Question 49

Let V be a vector space of dimension n. Which of the following statements about spanning sets in V is true?

Accepted Answer

Any set of n linearly independent vectors in V is a spanning set.

Answer

Any set of n orthogonal vectors in V is a spanning set.

Answer

Any set of n normalized vectors in V is a spanning set.