Linear Algebra Learning Quizzes: Enhance Your Understanding

Question 1

Given a subspace W of Rⁿ, what is the dimension of its orthogonal complement W^⊥?

Accepted Answer

The dimension of W^⊥ is always equal to n - dim(W).

Answer

The dimension of W^⊥ is equal to the dimension of W.

Answer

The dimension of W^⊥ is always equal to the rank of the matrix representation of W.

Question 2

In linear algebra, a set of vectors is linearly independent if:

Accepted Answer

No vector in the set can be expressed as a linear combination of the other vectors.

Answer

They are all parallel to each other.

Answer

They span the vector space.

Answer

They have the same number of elements as the vector space's dimension.

Question 3

In linear algebra, which statement about the rank of a matrix is always accurate?

Accepted Answer

The rank equals the number of linearly independent rows.

Answer

The rank equals the number of linearly dependent columns.

Answer

The rank equals the number of non-zero rows.

Question 4

Determine if the set of vectors {(1, 1, 0), (1, 0, 1), (0, 1, 1)} in R^3 is linearly independent and spans R^3.

Accepted Answer

Linearly independent and spans R^3

Answer

Linearly independent but does not span R^3

Answer

Linearly dependent and spans R^3

Answer

Linearly dependent and does not span R^3

Question 5

Determine the dimension of the eigenspace associated with the eigenvalue λ = 2 for the matrix A = [[1, 2], [1, 1]]

Accepted Answer

1

Answer

0

Answer

2

Question 6

State whether the following statement is true or false: The eigenvectors of a symmetric matrix are always orthogonal.

Accepted Answer

True

Answer

False

Question 7

In computer graphics, which linear algebra operation is used to apply geometric transformations, like scaling and rotation, to 2D objects?

Accepted Answer

Matrix Multiplication

Answer

Eigenvalue Decomposition

Answer

Vector Addition

Question 8

What is the null space of the matrix A = [[1, 2], [3, 4]]?

Accepted Answer

The set of all vectors x such that Ax = 0.

Answer

The set of all vectors x such that Ax = 1.

Answer

The set of all vectors x such that Ax = 2.

Answer

The set of all vectors x such that Ax = 3.

Question 9

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

Answer

3

Answer

4

Question 10

Which of the following is **not** a characteristic of a linear transformation?

Accepted Answer

Preserves the dimension of the vector space

Answer

Preserves vector addition

Answer

Preserves scalar multiplication

Answer

Maps the zero vector to the zero vector

Question 11

Identify the null space of the matrix A = [1 -2; 2 -4].

Accepted Answer

The set of vectors in the form [2x; x]

Answer

The set of vectors in the form [x; 2x]

Answer

The set of vectors in the form [2x + 1; x]

Answer

The set of vectors in the form [x + 1; 2x]

Question 12

Use the Gram-Schmidt process to orthonormalize the vector set {(1, 1, 1), (0, 1, 1), (1, 0, 1)}.

Accepted Answer

{(1/√3, 1/√3, 1/√3), (0, 1/√2, 1/√2), (1/√6, -1/√6, 1/√6)}

Answer

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

Answer

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

Answer

{(√3/3, √3/3, √3/3), (0, √2/2, √2/2), (√6/6, -√6/6, √6/6)}

Question 13

Which of the following is NOT an elementary row operation?

Accepted Answer

Inverting a row

Answer

Swapping two rows

Answer

Multiplying a row by a nonzero constant

Answer

Adding a multiple of one row to another row

Question 14

Which of the following transformations is a linear transformation?

Accepted Answer

y = 2x + 3

Answer

y = x^2

Answer

y = sin(x)

Answer

y = e^x

Question 15

Which matrix below represents the identity matrix?

Accepted Answer

[1 0]
[0 1]

Answer

[2 0]
[0 2]

Answer

[0 1]
[1 0]

Answer

[1 2]
[3 4]

Question 16

Which of the following transformations is an example of a linear transformation?

Accepted Answer

T(x) = 2x

Answer

T(x) = x^2

Answer

T(x) = |x|

Answer

T(x) = sin(x)

Question 17

Which of the following is the correct formula for finding the null space of a matrix?

Accepted Answer

Null(A) = {x | Ax = 0}

Answer

Null(A) = {x | Ax = b}

Answer

Null(A) = {x | bAx = 0}

Answer

Null(A) = {x | xbA = 0}

Question 18

What is the rank of the matrix $ [[1, -2, 3], [2, 0, 1], [3, -4, 5]] $?

Accepted Answer

2

Answer

1

Answer

0

Answer

3

Question 19

Which of the following is NOT a property of eigenvalues?

Accepted Answer

They are always real numbers.

Answer

They represent the scaling factor of eigenvectors.

Answer

They are associated with eigenvectors.

Answer

They can be complex numbers.

Question 20

Which of the following is NOT a fundamental property of matrices?

Accepted Answer

Matrices are always commutative under multiplication.

Answer

Matrices can be multiplied by scalars.

Answer

Matrices can be used to represent systems of linear equations.

Answer

Matrices can be added and subtracted.

Question 21

Calculate the determinant of the matrix  [[2, 1], [3, 4]]?

Accepted Answer

5

Answer

1

Answer

-1

Answer

10

Question 22

Which of the following functions is a linear transformation?

Accepted Answer

The function f(x) = 3x

Answer

The function f(x) = |x|

Answer

The function f(x) = 2x + 1

Answer

The function f(x) = x^2

Question 23

Which of the following sets forms a basis for the vector space 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, 0, 0], [0, 1, 1]]

Answer

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

Question 24

Which of the following properties is NOT generally true for matrix multiplication?

Accepted Answer

Commutativity (AB = BA)

Answer

Associativity (A(BC) = (AB)C)

Answer

Distributivity over addition (A(B + C) = AB + AC)

Answer

Existence of an identity matrix (AI = IA = A)

Question 25

A system of linear equations is represented by Ax = b, where A is a matrix, x is a vector of unknowns, and b is a vector of constants. What condition must be met for the system to have a unique solution?

Accepted Answer

The matrix A must be invertible.

Answer

The number of equations must be greater than the number of unknowns.

Answer

The matrix A must be singular.

Answer

The vector b must be the zero vector.

Question 26

What does the span of a set of vectors represent?

Accepted Answer

The set of all possible linear combinations of the vectors.

Answer

The number of vectors in the set.

Answer

The length of the longest vector in the set.

Answer

The dimension of the vector space.

Question 27

If a linear transformation T maps a vector v to its scalar multiple k*v, what is k referred to as?

Accepted Answer

Eigenvalue

Answer

Eigenvector

Answer

Null space

Answer

Determinant

Question 28

A set of vectors is considered linearly independent if:

Accepted Answer

No vector in the set can be written as a linear combination of the other vectors.

Answer

The vectors have the same direction.

Answer

The vectors are all orthogonal to each other.

Answer

The vectors form a basis for the vector space.

Question 29

Which method can be used to find the eigenvalues of a square matrix A?

Accepted Answer

Solve the characteristic equation det(A - λI) = 0, where λ is an eigenvalue and I is the identity matrix.

Answer

Compute the dot product of the columns of A.

Answer

Calculate the inverse of the matrix A.

Answer

Find the row echelon form of the matrix A.

Question 30

What is an eigenvector corresponding to an eigenvalue of a matrix?

Accepted Answer

A non-zero vector that, when multiplied by the matrix, results in a scalar multiple of itself, where the scalar is the eigenvalue.

Answer

A vector that is orthogonal to all other eigenvectors.

Answer

The zero vector.

Answer

Any vector in the null space of the matrix.

Question 31

Which of the following properties is NOT generally true for matrix multiplication?

Accepted Answer

Commutativity (AB = BA)

Answer

Distributivity over addition (A(B+C) = AB + AC)

Answer

Existence of an identity matrix (AI = IA = A)

Answer

Associativity (A(BC) = (AB)C)

Question 32

A system of linear equations has a unique solution if and only if the determinant of its coefficient matrix is:

Accepted Answer

Non-zero

Answer

Equal to the number of unknowns

Answer

Zero

Answer

Equal to the number of equations

Question 33

What is the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [3, 6, 9]]?

Accepted Answer

1

Answer

2

Answer

3

Answer

0

Question 34

Which of the following properties defines a linear transformation T?

Accepted Answer

T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalar c

Answer

T(cu) = cT(u) for all vectors u and scalar c

Answer

T(u + v) = T(u)T(v) for all vectors u, v

Answer

T(u + v) = T(u) + T(v) for all vectors u, v

Question 35

An eigenvector of a linear transformation is a non-zero vector that, when the transformation is applied, is:

Accepted Answer

Scaled by a factor

Answer

Orthogonal to the corresponding eigenvector

Answer

Always transformed into the zero vector

Question 36

If v is an eigenvector of matrix A with eigenvalue λ, then (A - λI)v equals:

Accepted Answer

The zero vector

Answer

A

Answer

v

Answer

λv

Question 37

What is the dimension of the null space of a 3x5 matrix with a rank of 2?

Accepted Answer

3

Answer

5

Answer

2

Answer

1

Question 38

Which of the following matrices is orthogonal?

Accepted Answer

[[cosθ, -sinθ], [sinθ, cosθ]]

Answer

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

Answer

[[2, 0], [0, 3]]

Answer

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

Question 39

The dot product of two orthogonal vectors is always:

Accepted Answer

0

Answer

Equal to the product of their magnitudes

Answer

-1

Answer

1

Question 40

Which method is commonly used to find the best-fit line for a set of data points?

Accepted Answer

Least Squares Approximation

Answer

Cramer's Rule

Answer

Eigenvalue Decomposition

Answer

Gaussian Elimination

Question 41

For which of the following matrices is the product of its eigenvalues equal to its determinant?

Accepted Answer

A singular matrix

Answer

An orthogonal matrix

Answer

A Hermitian matrix

Answer

A non-singular matrix

Question 42

Which of the following is the rank of the matrix [[1, 0, 3], [0, 1, 2], [0, 0, 0]]?

Accepted Answer

2

Answer

3

Answer

0

Answer

1

Question 43

If the null space of a 5x7 matrix has dimension 3, what is the rank of the matrix?

Accepted Answer

4

Answer

5

Answer

2

Question 44

Which of the following is NOT an eigenvalue of the matrix [[2, 1], [-1, 2]]?

Accepted Answer

0

Answer

1

Answer

-1

Answer

3

Question 45

For the matrix A = [[1, 2], [0, 1]], what is the eigenvalue corresponding to the eigenvector [1, 1]^T?

Accepted Answer

1

Answer

0

Answer

3

Answer

2

Question 46

What is the dimension of the eigenspace corresponding to the eigenvalue λ of a matrix A?

Accepted Answer

The algebraic multiplicity of λ.

Answer

The number of linearly independent eigenvectors associated with λ.

Answer

The geometric multiplicity of λ.

Answer

The rank of the matrix A.

Question 47

Which of the following is a vector in the column space of the matrix A = [[1, 2], [3, 4]]?

Accepted Answer

[1, 3]^T

Answer

[1, 2, 3]^T

Answer

[2, 4]^T

Answer

[3, 2]^T

Question 48

What is the geometric interpretation of the null space of a matrix A?

Accepted Answer

The subspace of vectors that are orthogonal to the row space of A.

Answer

The subspace of vectors that are orthogonal to the eigenspace of A.

Answer

The subspace of vectors that are orthogonal to the column space of A.

Question 49

What is the determinant of the following matrix:

[[2, 1], [4, 3]]

Accepted Answer

2

Answer

6

Answer

8

Answer

10

Question 50

Find the eigenvalues of the matrix:

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

Accepted Answer

1 and 4

Answer

2 and 3

Answer

-1 and 6

Answer

0 and 5

Question 51

Given a system of linear equations, what does the rank of the coefficient matrix tell us?

Accepted Answer

The number of linearly independent equations in the system.

Answer

The number of solutions to the system.

Answer

The number of variables in the system.

Question 52

Which of the following is an example of an orthogonal matrix?

Accepted Answer

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

Answer

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

Answer

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

Answer

[[1, 2], [3, 4]]

Question 53

What is the dimension of the vector space spanned by the vectors:

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

Accepted Answer

2

Answer

0

Answer

3

Answer

1

Question 54

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 vectors are normalized.

Answer

The vectors are orthogonal.

Question 55

What is the inverse of the matrix:

[[1, 2], [3, 4]]

Accepted Answer

[[ -2, 1], [ 3/2, -1/2]]

Answer

[[ 1, 2], [ 3, 4]]

Answer

[[ 4, -2], [ -3, 1]]

Answer

[[ 1, -2], [ -3, 4]]

Question 56

Which of the following statements about eigenvectors is TRUE?

Accepted Answer

Eigenvectors are scaled by the corresponding eigenvalue when multiplied by the matrix.

Answer

Eigenvectors are always linearly dependent.

Answer

Eigenvectors are always orthogonal.

Question 57

A system of linear equations is said to be homogeneous if:

Accepted Answer

The constant terms in each equation are all zero.

Answer

The number of equations equals the number of variables.

Answer

The coefficient matrix is invertible.

Answer

The system has a unique solution.