Linear Algebra Quizzes: Test Your Understanding

Question 1

Which property accurately describes an orthogonal matrix?

Accepted Answer

Its columns form an orthonormal basis.

Answer

Its determinant is not equal to 1.

Answer

Its rows are not orthogonal to each other.

Answer

The product of two orthogonal matrices is not necessarily orthogonal.

Question 2

In an inner product space, what is the definition of an orthonormal basis?

Accepted Answer

Linearly independent vectors with unit length and orthogonal to each other

Answer

Linearly independent vectors with unit length

Answer

Linearly independent vectors orthogonal to each other

Answer

Linearly dependent vectors orthogonal to each other

Question 3

Which conditions must a set of vectors in R^3 satisfy to form a subspace?

Accepted Answer

Contains the zero vector

Accepted Answer

Closed under vector addition

Accepted Answer

Closed under scalar multiplication

Answer

Linearly independent

Question 4

Which of the following is NOT an application of linear algebra in engineering and physics?

Accepted Answer

Predicting weather patterns

Answer

Solving systems of linear equations

Answer

Matrix representation of physical systems

Answer

Geometric transformations

Question 5

In linear algebra, what is the definition of the determinant of a square matrix?

Accepted Answer

A scalar value calculated from the matrix's elements, providing information about its properties.

Answer

A measure of the number of linearly dependent rows in the matrix.

Answer

A positive value only for matrices that can be inverted.

Answer

A measure of the area of the parallelogram spanned by the matrix's rows.

Question 6

In linear algebra, a linear transformation T maps vectors in vector space V to vectors in vector space W. Which of the following transformations is NOT a linear transformation?

Accepted Answer

T(x) = x + 5

Answer

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

Answer

T(ax) = aT(x) for any scalar a

Answer

T(x) = Ax for a fixed matrix A

Question 7

What is the defining characteristic of an orthogonal set of vectors?

Accepted Answer

The dot product of any two vectors in the set is zero.

Answer

The vectors are linearly independent.

Answer

The set of vectors spans the vector space.

Answer

The magnitude of each vector in the set is 1.

Question 8

Identify the set that does not meet the criteria of a vector space:

Accepted Answer

Functions mapping real numbers to real numbers

Answer

Polynomials with real coefficients of degree less than or equal to 3

Answer

2x2 matrices with real entries

Answer

Vectors in R^3 with zero third component

Question 9

Identify the formula that accurately represents the null space of matrix A.

Accepted Answer

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

Answer

N(A) = {x | Ax ≠ 0}

Answer

N(A) = {x | Aᵀx = 0}

Answer

N(A) = {x | Aᵀx ≠ 0}

Question 10

Which technique minimizes the difference between data points and a linear model by iteratively adjusting model parameters?

Accepted Answer

Gradient descent

Answer

Principal component analysis

Answer

Linear programming

Question 11

Identify the linear transformation among the following functions:

Accepted Answer

f(x) = 2x - 1

Answer

f(x) = x^2

Answer

f(x, y) = x / y

Answer

f(x) = sin(x)

Question 12

Which theorem is specifically used to calculate the determinant of a triangular matrix?

Accepted Answer

Triangular matrix theorem

Answer

Cayley-Hamilton Theorem

Answer

Frobenius companion matrix theorem

Question 13

Which of the following feature is required for a set of vectors to constitute a vector space?

Accepted Answer

The set contains a unique zero vector.

Accepted Answer

The set is closed under vector addition.

Accepted Answer

The set is closed under scalar multiplication.

Answer

The set has a finite number of dimensions.

Question 14

What is a defining characteristic of an orthogonal matrix?

Accepted Answer

Its inverse equals its transpose.

Answer

Its determinant equals 1.

Answer

Its eigenvalues are all positive and real.

Answer

Its columns are linearly dependent.

Question 15

Find the eigenvalues of the matrix:

A = | 2 1 |
    | -1 2 |

Accepted Answer

2, -2

Answer

3, 1

Answer

-2, 1

Answer

2, 1

Question 16

What is the definition of the null space of a linear transformation?

Accepted Answer

The set of vectors mapped to the zero vector

Answer

The set of vectors not mapped to the zero vector

Answer

The set of vectors mapped to themselves

Question 17

Solve the following system of linear equations:

2x + y = 5
x - y = 1

Accepted Answer

(x = 1, y = 3)

Answer

(x = 2, y = 1)

Answer

(x = 3, y = 1)

Answer

(x = 0, y = 0)

Question 18

In three-dimensional space, let W be the set of all vectors (a, b, c) such that a + b + c equals zero. Is W a subspace of R^3?

Accepted Answer

Yes, W is a subspace of R^3.

Answer

No, W is not a subspace of R^3 because it is not closed under addition.

Answer

No, W is not a subspace of R^3 because it does not contain the zero vector.

Question 19

For an m x n matrix A, what is the rank of A defined as?

Accepted Answer

The number of linearly independent rows in A

Accepted Answer

The number of linearly independent columns in A

Answer

The number of rows in A

Answer

The number of columns in A

Question 20

In linear algebra, what is a linear combination of vectors?

Accepted Answer

A linear combination is a new vector formed by adding scalar multiples of the original vectors.

Answer

A linear combination is a vector that lies on the same line as one of the original vectors.

Answer

A linear combination is a vector with the same length as one of the original vectors.

Answer

A linear combination is only possible if the original vectors are perpendicular to each other.

Question 21

For a square matrix A with real-valued entries, how does the determinant of A relate to its eigenvalues?

Accepted Answer

The determinant of A is equal to the product of its eigenvalues.

Answer

The determinant of A is the sum of its eigenvalues.

Answer

The determinant of A is the product of the differences between its eigenvalues.

Answer

The inverse of A is uniquely determined by its eigenvalues.

Question 22

In linear algebra, how do you find the image of a vector v under a linear transformation T?

Accepted Answer

Multiply v by the transformation matrix of T.

Answer

Subtract v from the transformation matrix of T.

Answer

Add v to the transformation matrix of T.

Answer

Take the determinant of the transformation matrix of T and multiply by v.

Question 23

In an inner product space, two non-zero vectors are orthogonal if their dot product is what?

Accepted Answer

Zero

Answer

One

Answer

Negative one

Answer

Equal

Question 24

Given that matrices A and B are both orthogonal, demonstrate that their product AB is also orthogonal.

Accepted Answer

The product of orthogonal matrices A and B, denoted as AB, is an orthogonal matrix.

Answer

The product of orthogonal matrices AB may contain non-zero off-diagonal elements.

Answer

Multiplying two orthogonal matrices does not guarantee an orthogonal result.

Question 25

To find the multiplicative inverse, A^(-1), of an invertible matrix A using elementary row operations, which of the following methods is correct?

Accepted Answer

Adjoin the identity matrix I to A and apply row operations to transform I into A^(-1).

Answer

Subtract the identity matrix I from A and perform row operations to transform A to A^(-1).

Answer

Divide each element of A by its determinant to obtain A^(-1).

Question 26

Which of the following sets does NOT satisfy the axioms of a vector space over the field of real numbers?

Accepted Answer

The set of all functions from R to R

Answer

The set of all real numbers

Answer

The set of all 2x2 matrices

Answer

The set of all polynomials with real coefficients

Question 27

Which of the following transformations is a linear transformation?

Accepted Answer

The matrix multiplication A * X, where A is a fixed matrix

Answer

The function f(x) = x^2

Answer

The translation of a vector by (1, 2)

Answer

The rotation of a vector by 90 degrees

Question 28

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

Accepted Answer

The vectors must not be linearly related.

Answer

The set must span the vector space.

Answer

The set must contain only the zero vector.

Answer

The number of vectors in the set must be less than the dimension of the vector space.

Question 29

The determinant of a matrix is a

Accepted Answer

Scalar

Answer

Vector

Answer

Matrix

Answer

Polynomial

Question 30

The rank of a matrix is equal to the:

Accepted Answer

Number of linearly independent rows or columns

Answer

Number of linearly dependent rows or columns

Answer

Sum of the dimensions

Answer

Product of the dimensions

Question 31

The null space of a matrix is the set of vectors that:

Accepted Answer

Are mapped to the zero vector

Answer

Are orthogonal to the column space

Answer

Span the row space

Answer

Are not in the column space

Question 32

The system of linear equations Ax = b has a unique solution if:

Accepted Answer

A is invertible

Answer

b is in the column space of A

Answer

A is square

Answer

A is symmetric

Question 33

Which of the following is not a property of vector spaces?

Accepted Answer

Commutativity of multiplication

Answer

Closure under addition

Answer

Existence of zero vector

Answer

Distributivity of multiplication over addition

Question 34

The matrix product of two matrices A and B is defined if and only if:

Accepted Answer

The number of columns in A is equal to the number of rows in B

Answer

The number of rows in A is equal to the number of columns in B

Answer

The dimensions of A and B are the same

Answer

The determinants of A and B are equal

Question 35

In data compression, which linear algebra technique is used to find the directions of greatest variation in a dataset, allowing for effective dimensionality reduction?

Accepted Answer

Principal Component Analysis (PCA)

Answer

Gaussian Elimination

Answer

Matrix Inversion

Answer

LU Decomposition

Question 36

In optimization problems like linear programming and quadratic programming, how does linear algebra contribute to finding solutions?

Accepted Answer

Linear algebra helps formulate optimization problems as systems of linear equations or inequalities, enabling efficient solutions.

Answer

Linear algebra is not applicable to solving optimization problems.

Answer

Linear algebra is only useful for solving linear programming problems, not quadratic programming.

Answer

Mathematical techniques are not suitable for solving optimization problems.

Question 37

In linear regression, what's the key assumption about the connection between the dependent variable and the independent variables?

Accepted Answer

Linearity

Answer

Normality

Answer

Independence

Question 38

When a linear transformation T maps a 3D vector space V onto a 2D vector space W, what's the dimension of the kernel of T?

Accepted Answer

1

Answer

2

Answer

0

Answer

4

Question 39

Given the set of vectors in R^3: { [1, 2, 3], [2, 1, 4], [3, 4, 1] }, determine if it spans the entire R^3.

Accepted Answer

Yes, the set spans R^3

Answer

No, the set does not span R^3

Question 40

When examining systems of differential equations, which matrix characteristic primarily determines the system's behavior?

Accepted Answer

Eigenvalues

Answer

Determinant

Answer

Eigenvectors

Answer

Trace

Question 41

In linear algebra, which application employs orthogonal projections to determine the closest point within a subspace to a given vector?

Accepted Answer

Closest point in a subspace

Answer

Solving systems of linear equations

Answer

Best approximation of functions

Answer

Signal compression

Question 42

Which of the following properties holds true for eigenvalues and eigenvectors of a matrix?

Accepted Answer

Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Accepted Answer

All eigenvalues of a symmetric matrix are real.

Answer

The number of linearly independent eigenvectors of a matrix equals the number of eigenvalues.

Question 43

Which of the following statements about the rank of a matrix are true?

Accepted Answer

The rank of a matrix is equal to the number of linearly independent rows in the matrix.

Accepted Answer

The rank of a matrix is equal to the number of linearly independent columns in the matrix.

Question 44

Find the determinant of the matrix A = [[2, 1], [-1, 3]]

Accepted Answer

7

Answer

5

Answer

-7

Question 45

What is the dimension of the vector space of all 3x3 matrices?

Accepted Answer

9

Answer

2

Answer

8

Question 46

Which of the following is a property of matrix multiplication?

Accepted Answer

Associativity

Answer

None of the above

Answer

Distributivity

Answer

Commutativity

Question 47

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

Accepted Answer

5

Answer

15

Answer

1

Answer

10

Question 48

Which of the following vectors is a linear combination of [1, 2, 3] and [4, 5, 6]?

Accepted Answer

[7, 9, 11]

Answer

[0, 0, 0]

Answer

[1, 1, 1]

Answer

[8, 10, 12]

Question 49

What is the rank of the matrix [[1, 2], [3, 6]]?

Accepted Answer

1

Answer

0

Answer

2

Question 50

Which of the following is a characteristic of an orthogonal matrix?

Accepted Answer

Its inverse is equal to its transpose

Answer

Its eigenvalues are all real

Answer

Its trace is equal to 0

Answer

Its determinant is equal to 1

Question 51

What is the solution to the linear equation system Ax = b, where A is a 3x3 invertible matrix and x is a column vector?

Accepted Answer

x = A^(-1)b

Answer

x = (A^T)^(-1)b

Answer

x = Ab

Answer

x = bA^(-1)

Question 52

What is the eigenvalue of the matrix [[1, 2], [-2, 1]]?

Accepted Answer

3 or -1

Answer

1 or -3

Answer

0 or 1

Answer

2 or -2

Question 53

Which of the following is a basis for the vector space R^3?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 54

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

Accepted Answer

Span{(1, -1, 1)}

Answer

Span{(1, 0, 1)}

Answer

Span{(1, -1, 0)}

Answer

Span{(1, 1, 1)}

Question 55

Which of the following is a linear combination of the vectors [2, 1] and [3, 2]?

Accepted Answer

[6, 5]

Answer

[4, 3]

Answer

[5, 6]

Answer

[7, 9]

Question 56

Find the determinant of the matrix [[1, 2], [3, 4]]

Accepted Answer

-2

Answer

1

Answer

4

Answer

2

Question 57

If A is a 3x4 matrix and B is a 4x5 matrix, what is the size of the product AB?

Accepted Answer

3x5

Answer

4x5

Answer

3x4

Answer

4x3

Question 58

Find the null space of the matrix [[1, 2], [3, 4]]

Accepted Answer

{[t, -2t]}

Answer

{[t, -t]}

Answer

{[2t, t]}

Question 59

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

Accepted Answer

1

Answer

2

Answer

3

Answer

0

Question 60

Find the inverse of the matrix [[1, 2], [3, 4]]

Accepted Answer

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

Answer

[1, 2, -3, 4]

Answer

[2, 1, -3, 4]

Answer

[-2, 1, 3, 4]

Question 61

Which of the following is a linear transformation?

Accepted Answer

[2x + 1, y - 3]

Answer

[x^2, y^3]

Answer

[sin(x), cos(y)]

Question 62

Find the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

Accepted Answer

1

Answer

2

Answer

4

Answer

3

Question 63

Which of the following is a subspace of R^3?

Accepted Answer

{[x, y, z] | x + y + z = 0}

Answer

{[x, y, z] | x^2 + y^2 + z^2 = 1}

Answer

{[x, y, z] | x > 0, y > 0, z > 0}

Answer

{[x, y, z] | x = y = 0}

Question 64

Find the orthogonal complement of the subspace {[1, 2, 3]} in R^3

Accepted Answer

{[x, y, z] | x + 2y + 3z = 0}

Answer

{[x, y, z] | y = 0}

Answer

{[x, y, z] | x = 0}

Answer

{[x, y, z] | z = 0}