Linear Independence: Definition, Properties, and Examples

Question 1

Which of the following sets of vectors is linearly independent?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 2

If a set of vectors has more vectors than its span, what must be true?

Accepted Answer

The set must be linearly dependent.

Answer

The set must be linearly independent.

Question 3

Determine whether the set of vectors { (1, -1), (2, 1) } is linearly independent in R².

Accepted Answer

Linearly independent

Answer

Linearly dependent

Question 4

Which of the following properties is true for linearly independent sets of vectors?

Accepted Answer

The set has a unique solution for any linear combination.

Answer

The zero vector can be expressed as a linear combination of the set.

Answer

The set spans the entire vector space.

Answer

The set has an infinite number of linear combinations.

Question 5

If a set of vectors has a nontrivial linear combination that equals the zero vector, then the set is:

Accepted Answer

Linearly dependent

Answer

Linearly independent

Question 6

Consider the vectors v₁ = (1, 0), v₂ = (1, 1), and v₃ = (2, 2). Is the set {v₁, v₂, v₃} linearly independent in R²?

Accepted Answer

No

Answer

Yes

Question 7

In a 2D vector space, consider vectors v1 = (1, 0) and v2 = (0, 1). Are they linearly independent? Explain your answer geometrically.

Accepted Answer

No, because they are perpendicular.

Answer

Yes, because they are parallel.

Answer

Yes, because they are scalar multiples of each other.

Answer

No, because they are the same vector.

Question 8

Which statement about linearly dependent sets of vectors is true?

Accepted Answer

They span the same subspace as a subset of themselves.

Answer

They can be used to generate all vectors in the entire vector space.

Answer

They are always orthogonal to each other.

Answer

They have an infinite number of linearly independent subsets.

Question 9

What is the rank of a matrix whose columns form a linearly independent set of vectors?

Accepted Answer

Equal to the number of columns

Answer

0

Answer

1

Answer

Equal to the number of rows

Question 10

Which method can be used to determine if a set of vectors is linearly independent?

Accepted Answer

Gaussian elimination

Answer

Cramer's rule

Answer

Gram-Schmidt process

Answer

Eigenvalue decomposition

Question 11

If a set of vectors is linearly dependent, then at least one vector in the set can be expressed as a linear combination of the others.

Accepted Answer

True

Answer

False

Question 12

What is the dimension of the subspace spanned by a linearly independent set of vectors of size n?

Accepted Answer

n

Answer

n-1

Answer

n+1

Answer

Cannot be determined

Question 13

Which of the following is the precise definition of linear independence?

Accepted Answer

A set of vectors is linearly independent if none of them can be expressed as a linear combination of the others.

Answer

A set of vectors is linearly independent if they span the entire vector space.

Answer

A set of vectors is linearly independent if they are all perpendicular to each other.

Answer

A set of vectors is linearly independent if they have equal magnitudes.

Question 14

Determine whether the following set of vectors is linearly independent: {(1, 0, 0), (0, 1, 0), (0, 0, 1)}

Accepted Answer

Linearly independent

Answer

Linearly dependent

Question 15

Which of the following properties of linearly independent sets of vectors is incorrect?

Accepted Answer

A linearly independent set of vectors must contain the zero vector.

Answer

A linearly independent set of vectors can be extended to form a basis for the vector space.

Answer

A linearly independent set of vectors has a unique representation in terms of its basis vectors.

Answer

A linearly independent set of vectors spans the entire vector space.

Question 16

Determine the dimension of the vector space spanned by the following linearly independent set of vectors: {(1, 0, 1), (0, 1, 1), (1, 1, 0)}

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 17

Determine the truth value of the following statement: 'Any nonzero vector is linearly independent.'

Accepted Answer

False

Answer

True

Question 18

Determine the linear independence of the following set of vectors in R^2: {(1, 2), (3, 6), (5, 10)}

Accepted Answer

Linearly dependent

Answer

Linearly independent

Question 19

Given a matrix whose row vectors are linearly independent, what can be concluded about the invertibility of the matrix?

Accepted Answer

The matrix is always invertible.

Answer

The matrix is never invertible.

Answer

The invertibility depends on whether the vectors are orthogonal.

Question 20

Consider the vectors v1 = (1, 2), v2 = (3, 4), and v3 = (5, 6). Are these vectors linearly independent?

Accepted Answer

Yes

Answer

Not enough information

Answer

No

Answer

Only v1 and v2 are linearly independent

Question 21

Which of the following is a property of linearly independent vectors?

Accepted Answer

Their linear combination can only be the zero vector when all coefficients are zero.

Answer

Their cross product is always nonzero.

Answer

Their dot product is always 0.

Question 22

Which of the following is a necessary condition for a set of vectors to be linearly dependent?

Accepted Answer

At least one vector can be written as a linear combination of the others.

Answer

All vectors have the same magnitude.

Answer

The vectors span the vector space they belong to.

Question 23

What is the formal definition of linear independence for a set of vectors {v1, v2, ..., vn}?

Accepted Answer

The only solution to the equation c1*v1 + c2*v2 + ... + cn*vn = 0 is c1 = c2 = ... = cn = 0.

Answer

The vectors are orthogonal to each other.

Answer

The vectors span the entire vector space.

Question 24

Determine if the vectors (1, 0, 2), (2, 1, 0), and (0, 1, -4) are linearly independent.

Accepted Answer

Linearly independent

Answer

Cannot be determined

Answer

Linearly dependent

Question 25

If a set of vectors includes the zero vector, what can be concluded about its linear independence?

Accepted Answer

The set is linearly dependent.

Answer

It depends on the other vectors within the set.

Answer

The set remains linearly independent.

Question 26

A set of vectors is considered linearly dependent if and only if:

Accepted Answer

At least one vector can be expressed as a linear combination of the others.

Answer

The vectors form a basis for the entire vector space.

Answer

The vectors are all scalar multiples of each other.

Question 27

In R^2, what is the maximum number of linearly independent vectors?

Accepted Answer

2

Answer

3

Answer

Infinite

Answer

1

Question 28

Which method can be used to determine the linear independence of a set of vectors?

Accepted Answer

Gaussian elimination

Answer

Finding the eigenvalues

Answer

Dot product

Answer

Cross product

Question 29

If a set of vectors forms a basis for a vector space, what can be said about their linear independence?

Accepted Answer

The vectors are linearly independent.

Answer

The vectors are linearly dependent.

Answer

Linear independence cannot be determined from this information.

Answer

The vectors are orthogonal.

Question 30

If S is a linearly independent set of vectors and T is a subset of S, what can be concluded about the linear independence of T?

Accepted Answer

T is linearly independent.

Answer

The linear independence of T cannot be determined.

Answer

T is linearly dependent.

Question 31

Consider the vectors u and v. If u = 2v, what is the relationship between their linear independence?

Accepted Answer

The vectors are linearly dependent.

Answer

The linear independence cannot be determined.

Answer

The vectors are linearly independent.

Answer

The vectors are orthogonal.

Question 32

Why is the concept of linear independence significant in linear algebra?

Accepted Answer

It helps determine the dimension of a vector space and construct bases.

Answer

It helps in finding the eigenvalues of a matrix.

Answer

It simplifies the calculation of dot products.