Basis and Dimension: A Comprehensive Guide for Linear Algebra Students

Question 1

Which of the following properties does NOT hold for a basis of a vector space?

Accepted Answer

Uniqueness

Answer

Linear independence

Answer

Spanning the space

Answer

Finite number of vectors

Question 2

Consider the set of vectors {v1, v2, v3} in R^3. Which of the following conditions guarantees that {v1, v2, v3} is a basis for R^3?

Accepted Answer

v1, v2, and v3 are linearly independent.

Answer

v1 and v2 are linearly independent, and v3 can be expressed as a linear combination of v1 and v2.

Answer

v1 and v2 span R^3, and v3 is a linear combination of v1 and v2.

Answer

v1, v2, and v3 span R^3.

Question 3

Let V be a vector space with a basis B that has n vectors. What can be said about the dimension of V?

Accepted Answer

The dimension of V is n.

Answer

The dimension of V is n - 1.

Answer

The dimension of V is n + 1.

Answer

The dimension of V cannot be determined from the given information.

Question 4

Determine whether the following statement is true or false: In any vector space, every linearly independent set of vectors can be extended to form a basis for that vector space.

Accepted Answer

True

Answer

False

Question 5

What are the necessary and sufficient conditions for a set of vectors to form a basis for a vector space?

Accepted Answer

Linear independence and spanning the space

Answer

Linear dependence and spanning the space

Answer

Linear independence and finite cardinality

Answer

Linear dependence and finite cardinality

Question 6

Let V be a vector space with dimension n. Which of the following statements is true?

Accepted Answer

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

Answer

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

Answer

Any set of n + 1 linearly independent vectors in V forms a basis for V.

Answer

Any set of n - 1 linearly dependent vectors in V forms a basis for V.

Question 7

Which of the following is NOT a property of a basis for a vector space?

Accepted Answer

Contains a finite number of vectors

Answer

Linear independence of the vectors

Answer

The vectors span the vector space

Answer

Every vector in the space can be expressed uniquely as a linear combination of the basis vectors

Question 8

Which of the following sets of vectors forms a basis for R^3?

Accepted Answer

[1, 0, 0], [0, 1, 0], [0, 0, 1]

Answer

[1, 2, 3], [4, 5, 6], [7, 8, 9]

Answer

[1, 1, 1], [2, 2, 2], [3, 3, 3]

Question 9

What is the dimension of the subspace spanned by the vectors [1, 2, 3, 0] and [0, 0, 1, 2]?

Accepted Answer

2

Answer

1

Question 10

What is the dimension of the null space of the matrix A = [1 2 3; 4 5 6; 7 8 9]?

Accepted Answer

2

Answer

1

Question 11

If a vector space has a basis with 5 vectors, what is its dimension?

Accepted Answer

5

Answer

4

Question 12

Which of the following sets of vectors forms a basis for R³?

Accepted Answer

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

Answer

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

Answer

{(1, 2, 3), (2, 3, 4), (3, 4, 5)}

Answer

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

Question 13

Which of the following transformations is not a linear transformation?

Accepted Answer

Rotation of a vector

Answer

Matrix multiplication

Answer

Projection onto a subspace

Answer

Vector addition

Question 14

True or False: Every subspace of R^n is spanned by a finite set of vectors.

Accepted Answer

True

Answer

False

Question 15

Which of the following is NOT a characteristic of a basis for a vector space?

Accepted Answer

Determinant of the basis matrix is non-zero

Answer

Linear independence

Answer

Spanning

Answer

Finite cardinality

Question 16

Given a vector space V, let W be a subspace of V spanned by the set S. If S contains n linearly independent vectors, the dimension of W is:

Accepted Answer

n

Answer

n+1

Answer

n-1

Answer

Cannot be determined from the given information

Question 17

Let V be a vector space with dimension n. If W is a subspace of V with dimension m, then:

Accepted Answer

m ≤ n

Answer

m = n

Answer

m > n

Answer

The relationship between m and n cannot be determined.

Question 18

Which of the following sets of vectors is a basis for R³?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 19

In a vector space of dimension n, if you have a set of n linearly independent vectors, it automatically forms a basis.

Accepted Answer

True

Answer

False

Question 20

In a vector space V of dimension n, any linearly dependent set of vectors in V has at most n vectors.

Accepted Answer

True

Answer

False

Question 21

Which of the following statements is NOT true about a basis for a vector space?

Accepted Answer

A basis is a minimal spanning set.

Answer

The vectors in the basis are linearly independent.

Answer

The vectors in the basis span the vector space.

Answer

Every vector in the vector space can be expressed uniquely as a linear combination of the basis vectors.

Question 22

Let W be the subspace of R^4 spanned by the vectors {(1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 1, 1)}. Find a basis for W.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 23

Which of the following is a basis for the vector space R³?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 24

Which of the following sets of vectors forms an orthonormal basis for R³?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 25

Which of the following statements is true about a linearly independent set of vectors?

Accepted Answer

It can be used to generate every vector in the vector space it spans.

Answer

It has more vectors than the number of linearly independent vectors it contains.

Answer

It is not a basis for the vector space it spans.

Answer

It is orthogonal to the null space of the matrix formed by the vectors.

Question 26

Let V be a vector space with basis B = {(1, 0), (0, 1)}. What is the matrix representation of the linear transformation T: V -> V defined by T(x, y) = (2x - y, x + y) with respect to the basis B?

Accepted Answer

[[2, -1], [1, 1]]

Answer

[[1, -1], [2, 1]]

Answer

[[2, 1], [1, 1]]

Answer

[[1, 1], [2, 1]]

Question 27

Which of the following statements is true about the column space of a matrix?

Accepted Answer

It is the set of all vectors that can be generated by linear combinations of the columns of the matrix.

Answer

It is the set of all vectors that are orthogonal to the null space of the matrix.

Answer

It is the set of all vectors that are not in the null space of the matrix.

Answer

It is the set of all vectors that are orthogonal to the row space of the matrix.

Question 28

Let T: R² -> R³ be a linear transformation. If T(1, 0) = (1, 2, 3) and T(0, 1) = (4, 5, 6), what is the matrix representation of T?

Accepted Answer

[[1, 4], [2, 5], [3, 6]]

Answer

[[1, 2], [4, 5]]

Answer

[[1, 2, 3], [4, 5, 6]]

Answer

[[1, 4], [2, 5]]

Question 29

What is the dimension of the vector space of all real-valued functions that are continuous on the interval [0, 1]?

Accepted Answer

Infinite

Answer

1

Answer

2

Answer

3

Question 30

A set of vectors that spans a vector space V and is linearly independent is called a:

Accepted Answer

Basis

Answer

Subspace

Answer

Linear combination

Answer

Spanning set

Question 31

If a vector space V has dimension n, then every basis for V contains exactly:

Accepted Answer

n vectors

Answer

n+1 vectors

Answer

n-1 vectors

Answer

Any number of vectors

Question 32

The determinant of a matrix is a:

Accepted Answer

Scalar value

Answer

Vector value

Answer

Matrix value

Answer

Polynomial value

Question 33

Which of the following is an eigenvalue of the matrix A = [[2, 1], [-1, 2]]?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 34

A diagonalizable matrix is a matrix that can be expressed as a:

Accepted Answer

Product of its eigenvalues and eigenvectors

Answer

Sum of its eigenvalues and eigenvectors

Answer

Difference of its eigenvalues and eigenvectors

Answer

Quotient of its eigenvalues and eigenvectors

Question 35

Which of the following is a property of orthogonal vectors?

Accepted Answer

Their dot product is zero

Answer

Their cross product is zero

Answer

Their sum is zero

Answer

Their magnitude is zero

Question 36

The Gram-Schmidt process can be used to:

Accepted Answer

Find an orthogonal basis for a set of vectors

Answer

Find a linearly independent set of vectors

Answer

Find the eigenvalues of a matrix

Answer

Find the eigenvectors of a matrix

Question 37

Determine if the set of vectors { (1, 0, 1), (0, 1, 1), (1, 1, 1) } is linearly independent. Linear independence means that no vector in the set can be written as a linear combination of the other vectors.

Accepted Answer

Yes

Answer

Cannot be determined

Answer

No

Answer

Dependent if and only if a = 0

Question 38

Determine if the vectors { (2, 1, 3), (4, 3, 7), (6, 5, 11) } span R^3. A set of vectors spans a space if every vector in the space can be written as a linear combination of the given vectors.

Accepted Answer

Yes

Answer

Cannot be determined

Answer

No

Answer

Dependent if and only if a = 0

Question 39

What is the dimension of the vector space P₂, the space of all polynomials of degree at most 2?

Accepted Answer

3

Answer

4

Answer

1

Answer

2

Question 40

Consider the subspace of R⁴ spanned by the vectors (1, 2, 0, 1) and (0, 1, 1, 1). What is the dimension of this subspace?

Accepted Answer

2

Answer

3

Answer

4

Answer

1

Question 41

If a vector space V has a basis consisting of n vectors, then any other basis for V must also have ______ vectors.

Accepted Answer

n

Answer

n - 1

Answer

n + 1

Answer

Any number of

Question 42

Which of the following statements is TRUE about a set of vectors that spans a vector space?

Accepted Answer

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

Answer

The set must be orthogonal.

Answer

The set must be linearly independent.

Question 43

A set of vectors is linearly independent if and only if:

Accepted Answer

The only linear combination of the vectors that equals the zero vector is the trivial combination.

Answer

The set spans the entire vector space.

Answer

The vectors are orthogonal to each other.

Question 44

If a set of vectors is linearly independent and spans a vector space, then the set is called:

Accepted Answer

A basis

Answer

A spanning set

Answer

A linearly dependent set

Answer

An orthogonal set

Question 45

What is the dimension of the solution space of the homogeneous system of linear equations Ax = 0, where A is a 4x5 matrix?

Accepted Answer

At least 1

Answer

0

Answer

Exactly 5

Answer

Exactly 4

Question 46

If a vector space V has dimension n, then any set of n + 1 vectors in V must be:

Accepted Answer

Linearly dependent

Answer

A basis

Answer

Orthogonal

Answer

Linearly independent

Question 47

Which of the following is NOT a property of a basis for a vector space?

Accepted Answer

The vectors in the basis must be orthogonal to each other.

Answer

The vectors in the basis must span the entire vector space.

Answer

The vectors in the basis must be linearly independent.

Question 48

Find the dimension of the subspace of R^4 spanned by the vectors {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0)}

Accepted Answer

3

Answer

4

Answer

2

Answer

1

Question 49

Select the matrix that represents a transformation preserving the linear independence of vectors:

Accepted Answer

Matrix with a non-zero determinant

Answer

Matrix with linearly dependent rows

Answer

Matrix with a zero determinant

Answer

Matrix with linearly dependent columns

Question 50

Determine the rank of the matrix representing the linear map defined by T(x) = (2x, -x):

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 51

Find the dimension of the vector space of all real symmetric matrices of order 3:

Accepted Answer

6

Answer

9

Answer

3

Answer

12