Linear Algebra Quizzes: Test Your Understanding

Question 1

For a square matrix with an odd number of rows, what condition guarantees its invertibility?

Accepted Answer

Its determinant is non-zero.

Answer

Its determinant is positive.

Answer

Its determinant is odd.

Answer

Its determinant is negative.

Question 2

Which of the following sets does NOT form a vector space?

Accepted Answer

Functions from real numbers to real numbers

Answer

Polynomials with real coefficients

Answer

2x2 matrices with real entries

Answer

Complex numbers

Question 3

In a vector space V over a field F, a subspace W is defined. How does the dimension of W relate to the dimension of V?

Accepted Answer

The dimension of W is always less than or equal to the dimension of V.

Answer

The dimension of W is always greater than or equal to the dimension of V.

Answer

The dimension of W is always equal to the dimension of V.

Answer

The dimension of W is independent of the dimension of V.

Question 4

Which condition determines if a set of vectors is linearly independent?

Accepted Answer

Non-zero determinant of the vector matrix

Answer

Equal magnitudes for all vectors

Answer

Zero dot product for all vector pairs

Answer

Vector count less than or equal to matrix row count

Question 5

Which property is not characteristic of a linear transformation?

Accepted Answer

Preserves the dot product

Answer

Preserves vector addition

Answer

Preserves scalar multiplication

Answer

Maps the zero vector to the zero vector

Question 6

Identify the set of vectors that does not form a subspace of R^3:

Accepted Answer

{(a, b, a + b)}

Answer

{(a, b, 0)}

Answer

{(a, 0, b)}

Answer

{(0, a, b)}

Question 7

Which of the following sets is NOT a subspace of R^3? Provide a clear explanation for your choice.

Accepted Answer

Set D: All vectors of the form (a, b, c) such that a + b + c = 0

Answer

Set A: All vectors of the form (a, b, 0)

Answer

Set B: All vectors of the form (a, 0, b)

Answer

Set C: All vectors of the form (0, b, c)

Question 8

Which property is essential for a linear transformation?

Accepted Answer

Preserves vector addition and scalar multiplication

Answer

Can alter the dimension of a vector space

Answer

Only defined for matrices

Question 9

Identify the set that is not a subspace of the vector space R^3:

Accepted Answer

Vectors with third component equal to one

Answer

Vectors with first component equal to zero

Answer

Vectors with second component equal to zero

Question 10

In a vector space V over a field F, what can be said about the set S containing all linear combinations of vectors in V?

Accepted Answer

S is always a subspace of V.

Answer

S always forms a linearly independent set.

Answer

S always serves as a basis for V.

Question 11

Identify the subset of R^3 that is NOT a subspace:

Accepted Answer

Vectors with odd first component

Answer

Vectors with zero first component

Answer

Vectors with equal first and second components

Answer

Vectors with first component equal to the sum of the second and third components

Question 12

Given a square matrix A with a determinant of 0, which statement about A is always true?

Accepted Answer

A has at least one eigenvalue of 0

Answer

A is invertible

Answer

A has only real eigenvalues

Answer

A is a symmetric matrix

Question 13

In the context of a linear transformation T: R^n -> R^m, which of the following statements accurately describes the null space of T?

Accepted Answer

The null space of T is always a subspace of R^n.

Accepted Answer

The null space of T is equivalent to its kernel.

Answer

The null space of T is always one-dimensional.

Answer

The null space of T is always orthogonal to the range of T.

Question 14

What distinguishes the vector space R^n from other vector spaces?

Accepted Answer

R^n is finite-dimensional, with its dimension determined by n.

Answer

R^n is an infinite-dimensional vector space.

Answer

R^n is not a vector space.

Answer

R^n is a vector space over the field of complex numbers.

Question 15

Given a square matrix A, which property applies to the matrix A^T A?

Accepted Answer

Symmetric matrix

Answer

Diagonal matrix

Answer

Orthogonal matrix

Answer

Positive definite matrix

Question 16

Which of the following properties is true about the null space of a linear transformation?

Accepted Answer

It is a subspace of the domain.

Answer

It is a subspace of the codomain.

Answer

It is the same as the range of the transformation.

Answer

It is always trivial.

Question 17

In linear algebra, consider a vector space V over a field F. Let W be a subspace of V. Which of the following properties is always true about W?

Accepted Answer

W is closed under vector addition.

Accepted Answer

W contains the zero vector.

Accepted Answer

W is itself a vector space.

Answer

W is always a proper subspace of V.

Question 18

In a vector space, which condition must be met for a set of vectors to be linearly independent?

Accepted Answer

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

Answer

The vectors are orthogonal to each other.

Answer

The set of vectors spans the vector space.

Answer

The vectors have the same length.

Question 19

Given a symmetric matrix A, what is a defining characteristic of its eigenvalues?

Accepted Answer

They must be real.

Answer

They can be complex.

Answer

They are always positive.

Answer

They are always distinct.

Question 20

When is the Jacobi iterative method guaranteed to converge in the context of solving systems of linear equations?

Accepted Answer

For any symmetric positive-definite matrix A

Answer

For any diagonally dominant matrix A

Answer

For any matrix A with distinct eigenvalues

Question 21

What is the definition of the adjoint of a linear transformation?

Accepted Answer

A linear transformation from the dual space to the vector space.

Answer

The inverse of the linear transformation.

Answer

A self-adjoint operator.

Question 22

Find the null space of the matrix [[1, 2], [3, 4]] by solving the homogeneous system of equations.

Accepted Answer

{[x, -2x]}

Answer

{[x, x]}

Answer

{[0, 0]}

Answer

{[2x, x]}

Question 23

Which of the following sets of vectors in R^3 does NOT satisfy the definition of a subspace?

Accepted Answer

The set of all vectors with zero norm

Answer

The set of all vectors with zero first component

Answer

The set of all vectors with equal first and second components

Answer

The set of all vectors with non-positive third component

Question 24

Calculate the orthogonal projection of the vector [1, 2, 3] onto the subspace spanned by the vectors [1, 0, 0] and [0, 1, 0].

Accepted Answer

[1, 2, 0]

Answer

[1, 0, 0]

Answer

[0, 2, 0]

Answer

[0, 0, 3]

Question 25

Determine the linear equation that best fits the given data using the method of least squares: (1, 2), (2, 5), (3, 8), (4, 12)

Accepted Answer

y = 3x + 1

Answer

y = x + 1

Answer

y = x^2 + 1

Answer

y = 1/x

Question 26

In $$R^n$$, which characteristic defines a linearly independent set of vectors?

Accepted Answer

No vector can be expressed as a linear combination of the others.

Answer

Vectors point in different directions.

Answer

Vectors have equal lengths.

Answer

Vectors have a scalar product of zero.

Question 27

For solving a system of linear equations with a large number of variables (n), where the equations have coefficients that form a tridiagonal matrix, which method is computationally most efficient?

Accepted Answer

Tridiagonal Matrix Algorithm (TDMA)

Answer

Gaussian Elimination

Answer

LU Decomposition

Answer

Jacobi Iteration

Question 28

For the Gauss-Seidel iterative method to converge, which condition must the coefficient matrix satisfy?

Accepted Answer

Diagonal Dominance

Answer

Symmetry

Answer

Non-zero Determinant

Answer

Positive Definiteness

Question 29

Which expression is equivalent to the determinant of a 3x3 matrix A?

Accepted Answer

Volume of the parallelepiped spanned by the three columns of A

Answer

Area of the parallelogram spanned by the first two columns of A

Answer

Product of the eigenvalues of A

Answer

Trace of A

Question 30

Determine the rank of the matrix [1 2; 3 6].

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 31

Identify an eigenvalue of the matrix A = [2 1; -1 2].

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 32

Define the null space of the matrix A = [1 2; 3 4].

Accepted Answer

Vectors satisfying Ax = 0

Answer

Vectors perpendicular to [1 2]

Answer

Vectors perpendicular to [3 4]

Answer

Vectors spanning the column space of A

Question 33

Determine an orthonormal basis for R^2.

Accepted Answer

[(1/√2, 1/√2), (-1/√2, 1/√2)]

Answer

[(1, 0), (0, 1)]

Answer

[(1, 1), (0, 1)]

Answer

[(1, 0), (1, 1)]

Question 34

Which characteristic is NOT a property of matrices?

Accepted Answer

Matrices are commutative during multiplication.

Answer

Matrices with equal dimensions can be added and subtracted.

Answer

Matrices can be multiplied by scalars.

Answer

The transpose of a matrix can be calculated.

Question 35

Which transformation does NOT qualify as a linear transformation?

Accepted Answer

Translation

Answer

Rotation

Answer

Projection

Answer

Scaling

Question 36

Determine the eigenvalues of the matrix B = [1 2; -2 1].

Accepted Answer

1, -2

Answer

1, 2

Answer

-1, -2

Answer

Additional information is required to determine the eigenvalues.

Question 37

Identify which of the following functions is a linear transformation by checking if it satisfies the properties of linearity: f(x + y) = f(x) + f(y) and f(cx) = c f(x).

Accepted Answer

f(x) = 2x + 1

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 38

Define the null space of the matrix A = [[1, 2], [3, 4]] as the set of all vectors x such that Ax = 0.

Accepted Answer

The set of all vectors in the kernel of A

Answer

The set of all vectors orthogonal to the row space of A

Answer

The set of all vectors orthogonal to the column space of A

Answer

The set of all vectors in the image of A

Question 39

Find the orthogonal complement of the subspace S = span({[1, 0, 1], [0, 1, 1]}) in R^3 by finding a set of vectors orthogonal to every vector in S.

Accepted Answer

The set of all vectors orthogonal to both [1, 0, 1] and [0, 1, 1]

Answer

The set of all vectors in R^3

Answer

The set of all vectors with zero third component

Answer

The set of all vectors with unit length

Question 40

In image processing, which practical application utilizes matrix operations to execute image transformations like rotations and translations?

Accepted Answer

Performing image transformations (rotations, translations) using matrix operations

Answer

Adjusting image brightness and contrast by solving systems of linear equations

Answer

Performing dimensionality reduction by calculating eigenvalues and eigenvectors of a covariance matrix

Answer

Compressing images by finding the singular value decomposition

Question 41

To prove that f is a linear transformation, demonstrate that f(x + y + z) = f(x) + f(y) + f(z) for all x, y, z in R^n.

Accepted Answer

Prove that f(x + y + z) = f(x) + f(y) + f(z) for all x, y, z in R^n.

Answer

Show that f(0) = 0.

Answer

Show that f is injective.

Answer

Prove that f is surjective.

Question 42

Find the determinant of the matrix A = [[2, 1], [3, 4]] using the rule of Sarrus (or any other method).

Accepted Answer

5

Answer

10

Answer

-5

Answer

11

Question 43

Which of the following is the characteristic equation of the matrix A = [[2, 1], [-1, 2]]?

Accepted Answer

(λ - 1)(λ - 3) = 0

Answer

(λ - 2)(λ - 4) = 0

Answer

(λ + 1)(λ + 3) = 0

Answer

(λ - 1)² = 0

Question 44

Which of the following linear transformations is NOT invertible?

Accepted Answer

Projection of R^n onto a subspace of dimension n-1

Answer

Rotation of R^2 by 90 degrees

Answer

Scaling of R^n by a nonzero scalar

Question 45

Which of the following is **not** a vector space?

Accepted Answer

The set of all matrices

Answer

The set of all real numbers

Answer

The set of all functions

Answer

The set of all polynomials

Question 46

What is the dimension of the vector space R^n?

Accepted Answer

n

Answer

2n

Answer

n/2

Answer

n^2

Question 47

What is the determinant of the matrix?

Accepted Answer

ad - bc

Answer

b + c

Answer

a + d

Answer

a - d

Question 48

What are the eigenvalues of the matrix?

Accepted Answer

The roots of the characteristic polynomial

Answer

The elements of the inverse

Answer

The coefficients of the characteristic polynomial

Answer

The elements of the diagonal

Question 49

What is the column space of the matrix?

Accepted Answer

The set of all linear combinations of the columns of the matrix

Answer

The set of all linear combinations of the rows of the matrix

Answer

The set of all vectors that are orthogonal to the row space

Question 50

What is the rank of the matrix?

Accepted Answer

The number of linearly independent rows or columns

Answer

The number of rows

Answer

The determinant of the matrix

Answer

The number of columns

Question 51

What is the trace of the matrix?

Accepted Answer

The sum of the diagonal elements

Answer

The determinant of the matrix

Answer

The rank of the matrix

Answer

The nullity of the matrix

Question 52

What is the inverse of the matrix?

Accepted Answer

The matrix that, when multiplied by the original matrix, results in the identity matrix

Answer

The matrix that, when added to the original matrix, results in the identity matrix

Answer

The matrix that, when subtracted from the original matrix, results in the identity matrix

Question 53

What is the characteristic polynomial of the matrix?

Accepted Answer

The polynomial whose roots are the eigenvalues of the matrix

Answer

The polynomial whose roots are the coefficients of the matrix

Answer

The polynomial whose roots are the diagonal elements of the matrix

Question 54

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

Accepted Answer

5

Answer

1

Answer

0

Answer

-5

Question 55

Which of the following is a linear transformation?

Accepted Answer

The function f(x) = 2x + 1

Answer

The function f(x) = 1/x

Answer

The function f(x) = x^2

Answer

The function f(x) = |x|

Question 56

Which of the following is a linear transformation?

Accepted Answer

f(x) = 2x + 1

Answer

k(x) = |x|

Answer

h(x) = 1 / x

Answer

g(x) = x^2

Question 57

What is the trace of the matrix A = [[2, 1], [-1, 2]]?

Accepted Answer

4

Answer

6

Answer

2

Question 58

Which of the following is a diagonalizable matrix?

Accepted Answer

[[1, 0], [0, 2]]

Answer

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

Answer

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

Answer

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

Question 59

Which of the following is a linear transformation?

Accepted Answer

f(x) = Ax where A is a matrix

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = e^x

Question 60

What is the kernel of the linear transformation f(x) = Ax?

Accepted Answer

{x | Ax = 0}

Answer

{x | x = Ax}

Answer

{x | f(x) = 0}

Answer

{x | f(x) = x}

Question 61

Under which condition is the determinant of a matrix equal to the product of its eigenvalues?

Accepted Answer

The matrix is diagonalizable

Answer

The matrix is invertible

Answer

The matrix is symmetric

Answer

The matrix is positive definite

Question 62

The null space of a matrix A refers to the set of vectors that, when multiplied by A, result in:

Accepted Answer

The zero vector

Answer

A vector orthogonal to the row space of A

Answer

A vector in the column space of A

Answer

A scaled version of the vector

Question 63

The rank of a matrix is equivalent to the number of:

Accepted Answer

Linearly independent rows or columns

Answer

Non-zero rows or columns

Answer

Eigenvalues

Answer

Total rows or columns

Question 64

The inverse of a matrix exists if and only if its determinant is:

Accepted Answer

Non-zero

Answer

Zero

Answer

Positive

Answer

Negative