Master Linear Algebra: Interactive Quizzes and Practice Questions

Question 1

What is the primary concept behind singular value decomposition (SVD)?

Accepted Answer

Factorizing a matrix into three matrices to uncover its structure.

Answer

Solving linear equations with Gaussian elimination.

Answer

Graphically displaying Eigenvalues and Eigenvectors.

Answer

Identifying patterns in large datasets using statistical methods.

Question 2

Find the column space and null space of the matrix:
A = 
[ 1 2 -1 ]
[ 0 1  1 ]
[ 1 1  0 ]

Accepted Answer

Column space: 2D plane through the origin; Null space: 1D line through the origin

Answer

Column space: 3D space; Null space: 0

Answer

Column space: 1D line through the origin; Null space: 2D plane through the origin

Question 3

In computer graphics, how do matrices facilitate 3D transformations and rotations?

Accepted Answer

Matrices represent transformations as linear combinations, enabling the manipulation and rotation of 3D objects.

Answer

Matrices store the coordinates of 3D objects for rendering on the screen.

Answer

Matrices are not used in computer graphics for transformations and rotations.

Answer

Matrices are only used in 2D graphics, as 3D transformations require more complex operations.

Question 4

What is a positive definite matrix? Describe its key properties and provide an example.

Accepted Answer

A square, symmetric matrix with all positive eigenvalues.

Answer

A matrix with all positive entries.

Answer

An invertible matrix.

Question 5

A vector with components (1, 2, 3) can be represented in matrix form as:

Accepted Answer

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

Answer

[[1, 2, 3]]

Answer

[[1, 2], [3]]

Answer

[[3, 2, 1]]

Question 6

Which of the following linear transformations is a reflection?

Accepted Answer

T(x, y) = (x, -y)

Answer

T(x, y) = (x + 1, y)

Answer

T(x, y) = (2x, y)

Answer

T(x, y) = (x, x + y)

Question 7

The null space of the matrix [[1, 2], [3, 4]] is:

Accepted Answer

The set of all vectors (x, y) such that x = 2y

Answer

The set of all vectors (x, y) such that x = -2y

Answer

The set of all vectors (x, y) such that x = y

Answer

The set of all vectors (x, y) such that x = 0

Question 8

Which of the following is NOT an application of linear algebra?

Accepted Answer

Thermodynamics

Answer

Computer graphics

Answer

Solving systems of equations

Answer

Data analysis

Question 9

Which of the following transformations is linear?

Accepted Answer

f(x) = 2x + 1

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = |x|

Question 10

Find the inverse of the matrix [[2, -1], [1, 2]]

Accepted Answer

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

Answer

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

Answer

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

Answer

The matrix does not have an inverse.

Question 11

Determine the null space of the matrix [[1, 0, 2], [0, 1, 1]]

Accepted Answer

{(-2t, 0, t) | t ∈ R}

Answer

{(-2t, t, t) | t ∈ R}

Answer

{(-2t, t, 0) | t ∈ R}

Answer

{(-t, 0, t) | t ∈ R}

Question 12

Which of the following represents a vector in three-dimensional space (R³)?

Accepted Answer

(1, 2, 3)

Answer

(1, 2)

Answer

[1, 2]

Answer

[1, 2, 3, 4]

Question 13

Which of the following sets of vectors in R³ is linearly independent?

Accepted Answer

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

Answer

{(1, 2, 3), (4, 5, 6)}

Answer

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

Answer

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

Question 14

Which of the following correctly represents a vector in three-dimensional space (R^3)?

Accepted Answer

(1, 2, 3)

Answer

x^2 + y^2 = 1 (equation of a circle)

Answer

y = mx + b (equation of a line)

Answer

{1, 3, 5, 7} (a set)

Question 15

Calculate the determinant of the matrix A = [2 3; 4 5].

Accepted Answer

10

Answer

-2

Answer

2

Answer

-10

Question 16

Which of the following transformations is considered a linear transformation?

Accepted Answer

h(x) = 2x + 3 (a linear function)

Answer

f(x) = x^2 (a quadratic function)

Answer

g(x) = |x| (absolute value function)

Answer

j(x) = sin(x) (a trigonometric function)

Question 17

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

Accepted Answer

{k(1, 2) | k ∈ R}

Answer

{k(1, 1) | k ∈ R}

Answer

{k(2, 1) | k ∈ R}

Answer

{0} (the zero vector)

Question 18

Find the eigenvalue decomposition of the matrix A = [2 3; -1 1].

Accepted Answer

A = PΛP^-1, where P = [1 1; -1 1], Λ = [1 0; 0 2]

Answer

A = PΛP^-1, where P = [1 -1; 1 1], Λ = [2 0; 0 1]

Answer

A = PΛP^-1, where P = [1 2; -1 1], Λ = [2 0; 0 3]

Answer

A = PΛP^-1, where P = [2 1; -1 1], Λ = [1 0; 0 3]

Question 19

Select the matrix that represents the identity matrix for a 3x3 matrix:

Accepted Answer

[1 0 0; 0 1 0; 0 0 1]

Answer

[0 1 0; 1 0 1; 0 1 0]

Answer

[1 1 0; 0 1 1; 0 0 1]

Answer

[0 0 1; 1 0 1; 0 1 1]

Question 20

If a system of linear equations has no solutions, the solution space is:

Accepted Answer

Empty

Answer

One-dimensional

Answer

Two-dimensional

Answer

Three-dimensional

Question 21

What is the concept of 'linear least squares'? Explain how it can be used to fit a line to a set of data points.

Accepted Answer

Linear least squares is a statistical technique that finds the best-fit line or curve to a set of data points by minimizing the sum of the squared vertical distances between the data points and the line or curve.

Answer

Linear least squares is a method for solving systems of linear equations with more variables than equations.

Answer

Linear least squares is a method for finding the eigenvalues of a matrix.

Answer

Linear least squares is a method for finding the eigenvectors of a matrix.

Question 22

Which of the following concepts does NOT belong to algebraic topology?

Accepted Answer

Vector space

Answer

Homology

Answer

Cohomology

Question 23

How does LU decomposition of matrix [A] benefit the solution of linear equation systems?

Accepted Answer

LU decomposition simplifies solving linear equations by converting [A] into triangular matrices (L and U).

Answer

LU decomposition enhances numerical precision in solving linear equations.

Answer

LU decomposition is applicable to both square and non-square matrices.

Answer

LU decomposition is primarily used in image compression with limited applications elsewhere.

Question 24

Financial Applications of Linear Algebra:

How do matrices assist financial analysts in their work?

Accepted Answer

Simplifying calculations and visualizing complex data

Answer

Predicting future financial trends

Answer

Ensuring the accuracy of financial statements

Question 25

Which of the following is NOT a property of a vector in linear algebra?

Accepted Answer

Determinant

Answer

Direction

Answer

Magnitude

Answer

Dot product

Question 26

The matrix obtained by adding corresponding elements of two matrices A and B is known as:

Accepted Answer

Sum matrix

Answer

Product matrix

Answer

Transpose matrix

Answer

Inverse matrix

Question 27

Which of the following is a linear transformation?

Accepted Answer

f(x) = mx + c

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = e^x

Question 28

The multiplicative inverse of a square matrix A, if it exists, is denoted as:

Accepted Answer

A^(-1)

Answer

A^T

Answer

A^-1

Answer

A-1

Question 29

Which of the following is NOT an application of linear algebra?

Accepted Answer

Biology

Answer

Computer graphics

Answer

Data analysis

Answer

Quantum mechanics

Question 30

The determinant of a square matrix A is used to find:

Accepted Answer

Rank of A

Answer

Inverse of A

Answer

Eigenvalues of A

Answer

Nullity of A

Question 31

An eigenvalue of a square matrix A is a scalar value λ such that:

Accepted Answer

Aλ = λA

Answer

Aλ = 0

Answer

λA = Aλ

Answer

λA = 0

Question 32

The dot product of two vectors is:

Accepted Answer

A scalar value

Answer

A vector value

Answer

A matrix value

Answer

An eigenvalue

Question 33

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

Accepted Answer

{(x, y) | x = -2y}

Answer

{(x, y) | x = y}

Answer

{(x, y) | x = 0}

Answer

{(x, y) | x = 2y}

Question 34

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

Accepted Answer

The function f(x) = 2x + 3

Answer

The function f(x) = sin(x)

Answer

The function f(x) = x^2

Answer

The function f(x) = |x|

Question 35

Which of the following operations is NOT valid for matrices?

Accepted Answer

Addition of matrices with different dimensions

Answer

Multiplication of a matrix by a scalar

Answer

Multiplication of a matrix by a vector

Answer

Transposition of a matrix

Question 36

Which of the following is a linear transformation?

Accepted Answer

Function that preserves vector addition and scalar multiplication

Answer

Function that only preserves vector addition

Answer

Function that only preserves scalar multiplication

Question 37

Which of the following is an orthonormal basis for the subspace spanned by the vectors [1, 1, 0] and [0, 1, 1]?

Accepted Answer

[[1/√2, 1/√2, 0], [0, 1/√2, -1/√2]]

Answer

[[1/3, 1/3, 1/3], [0, 2/3, -1/3]]

Answer

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

Answer

[[1/2, 1/2, 0], [0, 1/2, -1/2]]

Question 38

Which of the following is a diagonalizable matrix?

Accepted Answer

Matrix with distinct eigenvalues

Answer

Matrix with repeated eigenvalues

Answer

Matrix with complex eigenvalues

Answer

Matrix with zero eigenvalues

Question 39

Calculate the dot product of the vectors u = <2, -1> and v = <3, 4>.

Accepted Answer

2

Answer

-11

Answer

10

Question 40

What is the dimension of the subspace spanned by the vectors **v1** = <1, 0, 1> and **v2** = <2, 1, 3>?

Accepted Answer

2

Answer

1

Answer

4

Answer

3

Question 41

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

Accepted Answer

T(x, y) = (x^2, y)

Answer

T(x, y) = (2x, 3y)

Answer

T(x, y) = (-y, x)

Answer

T(x, y) = (x + y, x - y)

Question 42

Find the eigenvalues of the matrix B = [ 2 1 ]
                                             [ 0 3 ].

Accepted Answer

λ1 = 2, λ2 = 3

Answer

λ1 = 1, λ2 = 2

Answer

λ1 = 0, λ2 = 3

Answer

λ1 = -2, λ2 = -3

Question 43

Which of the following statements is true about orthogonal vectors?

Accepted Answer

Their dot product is zero.

Answer

They point in the same direction.

Answer

They are linearly dependent.

Answer

They have the same magnitude.

Question 44

Which of the following matrices is an identity matrix?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 45

Determine the dimensions of a matrix with 4 rows and 3 columns.

Accepted Answer

4 x 3

Answer

3 x 4

Answer

4 x 4

Answer

3 x 3

Question 46

Identify the linear transformation among the following options:

Accepted Answer

f(x) = 3x - 2

Answer

f(x) = sin(2x)

Answer

f(x) = |x + 1|

Answer

f(x) = x^3

Question 47

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

Accepted Answer

11

Answer

8

Answer

-11

Answer

7

Question 48

Describe the geometric interpretation of a matrix with eigenvalues of 3 and -2.

Accepted Answer

A scaling and a rotation

Answer

A reflection and a projection

Answer

A shear and a dilation

Answer

A translation and a rotation

Question 49

What is the sum of the vectors (2, 3) and (-1, 4)?

Accepted Answer

(1, 7)

Answer

(-2, 1)

Answer

(1, -1)

Answer

(3, 1)

Question 50

What is the transpose of the matrix  [[1, 2], [3, 4]]?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 51

A system of linear equations has infinitely many solutions. What does this tell us about the determinant of its coefficient matrix?

Accepted Answer

The determinant is zero.

Answer

The determinant is one.

Answer

The determinant is a negative number.

Answer

The determinant is a positive number.

Question 52

Which of the following vectors is an eigenvector of the matrix [[2, 1], [1, 2]]?

Accepted Answer

(1, 1)

Answer

(1, 0)

Answer

(2, 1)

Answer

(1, -1)

Question 53

What is the determinant of the matrix [[a, b], [c, d]]?

Accepted Answer

ad - bc

Answer

ab - cd

Answer

ac - bd

Answer

ad + bc

Question 54

Which of the following is a valid row operation used in Gaussian elimination?

Accepted Answer

Multiplying a row by a non-zero constant.

Answer

Subtracting a row from itself.

Answer

Adding a row to itself.

Answer

Dividing a row by zero.

Question 55

A linear transformation is called an isomorphism if it is both one-to-one and onto. What does this mean in terms of the transformation?

Accepted Answer

It preserves both the structure and the information of the vector space.

Answer

It only preserves the structure of the vector space.

Answer

It does not preserve any information or structure.

Answer

It only preserves the information of the vector space.

Question 56

What is the result of adding the following vectors: [1, 2, 3] + [4, 5, 6]?

Accepted Answer

[5, 7, 9]

Answer

[0, 0, 0]

Answer

[14, 15, 18]

Answer

[3, 3, 3]

Question 57

Which of the following matrices is the transpose of the matrix A = [[1, 2], [3, 4]]?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 58

What is the determinant of the matrix B = [[2, 1], [4, 3]]?

Accepted Answer

2

Answer

10

Answer

8

Answer

6

Question 59

Which of the following is a valid linear transformation?

Accepted Answer

T(x, y) = (2x + y, x - 3y)

Answer

T(x, y) = (x^2, y)

Answer

T(x, y) = (x + 1, y + 1)

Answer

T(x, y) = (sin(x), cos(y))

Question 60

What is the solution to the system of linear equations represented by the following augmented matrix: [[1, 2, 3 | 5], [2, 1, 1 | 2]]?

Accepted Answer

x = -1, y = 2, z = 1

Answer

x = 1, y = -2, z = 1

Answer

x = 0, y = 0, z = 0

Answer

x = 2, y = 1, z = -1

Question 61

The eigenvalues of a matrix are 2 and -3. Which of the following could be the corresponding eigenvectors?

Accepted Answer

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

Answer

[2, 0] and [0, -3]

Answer

[1, 0] and [0, 1]

Answer

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

Question 62

A system of linear equations is said to be inconsistent if it has:

Accepted Answer

No solutions

Answer

Infinitely many solutions

Answer

A solution that depends on a parameter

Answer

One unique solution

Question 63

Which of the following is NOT a property of the dot product of two vectors?

Accepted Answer

The dot product is commutative, but not associative.

Answer

The dot product of a vector with itself gives its squared magnitude.

Answer

The dot product is distributive over vector addition.

Question 64

A 3x3 matrix has a determinant of zero. What does this imply about the matrix?

Accepted Answer

The matrix is singular.

Answer

The matrix is invertible.

Answer

The matrix represents a rotation.

Answer

The matrix has three distinct eigenvalues.