Linear Algebra I: Master the Fundamentals of Vector Spaces and Matrices

Question 1

Determine which of the following is not a subspace of R^3:

Accepted Answer

Vectors with strictly positive second coordinate

Answer

Vectors with zero first coordinate

Answer

Vectors with third coordinate equal to 0

Answer

Vectors that are multiples of (1, 1, 1)

Question 2

Consider a finite-dimensional vector space V over a field F. Which of the following statements is true for any set of n linearly independent vectors in V, where n is the dimension of V?

Accepted Answer

The set of vectors forms a basis for V.

Accepted Answer

The set of vectors spans V.

Answer

The set of vectors is linearly dependent.

Answer

The dimension of V is greater than n.

Question 3

Which of the following accurately describes a linear transformation?

Accepted Answer

(a) It maintains both vector addition and scalar multiplication.

Answer

(b) It maintains only vector addition.

Answer

(c) It maintains only scalar multiplication.

Answer

(d) It maintains neither vector addition nor scalar multiplication.

Question 4

Which property is NOT characteristic of a vector space?

Accepted Answer

Commutativity of multiplication

Answer

Closure under addition

Answer

Closure under scalar multiplication

Answer

Associativity of addition

Question 5

Which of the following is not a fundamental concept of linear algebra?

Accepted Answer

Groups

Answer

Vector spaces

Answer

Matrices

Answer

Eigenvalues

Question 6

In linear algebra, what is the main purpose of the Gram-Schmidt process?

Accepted Answer

To generate a set of orthonormal vectors that span a subspace.

Answer

To find the eigenvalues and eigenvectors of a matrix.

Answer

To solve systems of linear equations using matrices.

Answer

To calculate the null space of a linear transformation.

Question 7

Which of the following accurately describes the Gram-Schmidt process?

Accepted Answer

Produces an orthonormal basis for the span of the given vectors.

Answer

Can only be applied to linearly dependent sets of vectors.

Answer

Is computationally efficient for large systems of equations.

Answer

Requires the vectors to be normalized before application.

Question 8

Which type of matrix cannot be transformed into a diagonal matrix?

Accepted Answer

Matrix with a repeated eigenvalue

Answer

Matrix with distinct eigenvalues

Answer

Symmetric matrix

Question 9

For a real symmetric matrix A, the dimension of the eigenspace corresponding to an eigenvalue of A is equal to:

Accepted Answer

The number of linearly independent eigenvectors of A corresponding to that eigenvalue

Answer

The rank of A

Answer

The trace of A

Answer

The determinant of A

Question 10

In an inner product space over the real numbers, what is the inner product of two orthogonal vectors?

Accepted Answer

0

Answer

1

Answer

A positive value

Answer

A negative value

Question 11

Is the linear independence of a set of vectors affected by the choice of coordinate system?

Accepted Answer

No, linear independence is independent of the coordinate system.

Answer

Yes, linear independence depends on the coordinate system.

Answer

Linear independence may or may not depend on the coordinate system.

Answer

Linear independence always depends on the coordinate system.

Question 12

Within a vector space, which statement accurately describes subspaces?

Accepted Answer

Every subspace inherently possesses the properties of a vector space.

Answer

Any subset of a vector space is automatically considered a subspace.

Question 13

Within a vector space V, what property uniquely defines a linear subspace W?

Accepted Answer

W is a subset of V that remains closed under both vector addition and scalar multiplication.

Answer

W is a subset of V that remains closed under vector addition only.

Answer

W is a subset of V that remains closed under scalar multiplication only.

Question 14

In a vector space V over a field F, which property is always true about the zero vector 0?

Accepted Answer

The zero vector is unique.

Answer

The zero vector has infinite length.

Answer

The zero vector is linearly independent.

Answer

The zero vector is an eigenvector of any linear transformation.

Question 15

Which property is inherent to all linear transformations?

Accepted Answer

Maps the zero vector to itself

Answer

Preserves vector magnitudes

Answer

Applicable only within a single vector space

Question 16

Identify the transformation that is NOT linear:

Accepted Answer

T(x) = x^2

Answer

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

Answer

T(cx) = cT(x)

Question 17

Identify the subspace of R^3:

Accepted Answer

Vectors with zero first component

Accepted Answer

Vectors with equal second and third components

Answer

Vectors with positive third component

Question 18

Under the linear transformation T(x, y) = (x + 2y, 3x - y), what shape does the unit circle transform into?

Accepted Answer

Ellipse centered at the origin

Answer

Unit circle

Answer

Parabola

Answer

Hyperbola

Question 19

What is the dimension of the vector space R³?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 20

What is the geometric multiplicity of the eigenvalue λ = 1 of the matrix A = [[1, 2], [3, 4]]?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 21

What is the kernel of the linear transformation T(x, y) = (2x + y, x - y)?

Accepted Answer

{(a, b) | a + b = 0}

Answer

{(a, b) | a - b = 0}

Answer

{(a, b) | 2a + b = 0}

Answer

{(a, b) | a - 2b = 0}

Question 22

Identify a basis for the vector space R³.

Accepted Answer

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

Answer

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

Answer

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

Question 23

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

Accepted Answer

The set of all vectors (x, y) such that x + 2y = 0

Answer

The set of all vectors (x, y) such that 3x + 4y = 0

Answer

The set of all vectors (x, y) such that x - 2y = 0

Answer

The set of all vectors (x, y) such that 3x - 4y = 0

Question 24

Determine the rank of the matrix A = [[1, 2, 3], [4, 5, 6]]

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 25

What is the dimension of the vector space R^3?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 26

Which of the following transformations is a linear transformation?

Accepted Answer

f(x) = 2x + 1

Answer

f(x) = x^2

Answer

f(x, y) = xy

Answer

f(x) = sin(x)

Question 27

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

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

Question 28

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

Accepted Answer

5

Answer

-5

Answer

10

Answer

-10

Question 29

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

Answer

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

Answer

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

Question 30

What is the dimension of the null space of the matrix A = \begin{bmatrix} 1 & 2 \ -2 & 1 \ end{bmatrix}?

Accepted Answer

1

Answer

0

Answer

2

Answer

3

Question 31

Which of the following is NOT a defining criterion for a subspace of R³?

Accepted Answer

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

Answer

Contains the zero vector

Answer

Closed under vector addition

Answer

Closed under scalar multiplication

Question 32

Which of the following operations is NOT applicable to vectors?

Accepted Answer

Matrix multiplication

Answer

Vector addition

Answer

Scalar multiplication

Answer

Vector multiplication

Question 33

Which of the following matrices has an inverse?

Accepted Answer

B = [[0, 1], [1, 0]]

Answer

A = [[1, 2], [3, 4]]

Answer

C = [[1, 2], [2, 4]]

Answer

D = [[0, 0], [0, 0]]

Question 34

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

Accepted Answer

5

Answer

7

Answer

-5

Answer

-7

Question 35

How can a linearly independent set of vectors in a vector space be extended to form a basis for the entire space?

Accepted Answer

Add linearly independent vectors until the set spans the entire space.

Answer

Remove vectors until the set becomes linearly dependent.

Answer

Multiply each vector by a non-zero constant.

Answer

Transpose the set of vectors.

Question 36

In an inner product space, what characterizes an orthonormal basis?

Accepted Answer

Unit length for all vectors in the basis

Accepted Answer

Orthogonality between all pairs of vectors in the basis

Accepted Answer

Linear independence of the basis

Answer

Finite number of vectors in the basis

Question 37

If A is a 3x3 matrix and det(A) = 0, then which of the following is true?

Accepted Answer

A is singular

Answer

A is invertible

Answer

A has a unique solution

Answer

A is diagonalizable

Question 38

Which of the following is true about symmetric matrices in vector spaces over the field of real numbers?

Accepted Answer

Symmetric matrices form a vector space.

Answer

All matrices form a vector space.

Answer

Any set of vectors is a vector space.

Answer

Real numbers are irrelevant to vector spaces.

Question 39

Let A be a square matrix. The determinant of A is defined as:

Accepted Answer

The alternating sum of the products of the elements in each row (or column) of A and their corresponding cofactors.

Answer

The product of the elements in the main diagonal of A.

Answer

The difference between the sum of the elements in the rows and the sum of the elements in the columns of A.

Question 40

What is the dimension of the vector space R^n (where n is a positive integer)?

Accepted Answer

n

Answer

n + 1

Answer

n^2

Answer

n - 1

Question 41

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

Accepted Answer

ad - bc

Answer

a - b

Answer

a + b

Answer

cd

Question 42

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

Accepted Answer

The set of all vectors of the form [x, 2x]

Answer

The set of all vectors such that x >= 1

Answer

The set of all vectors such that y <= 0

Answer

The set of all vectors of the form [x, x]

Question 43

Which of the following is an example of a vector space over the field of real numbers?

Accepted Answer

The set of all 3x3 matrices with real entries

Answer

The set of all polynomials with degree greater than 2

Answer

The set of all odd integers

Answer

The set of all functions from the real numbers to the complex numbers

Question 44

Which of the following matrices is invertible?

Accepted Answer

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

Answer

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

Answer

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

Question 45

Is the set of all vectors of the form [x, y, 0] a subspace of R^3?

Accepted Answer

Yes

Answer

No, it's not closed under vector addition.

Answer

No, it doesn't contain the zero vector.

Answer

No, it's not closed under scalar multiplication.

Question 46

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

Accepted Answer

3

Answer

1

Answer

2

Answer

0

Question 47

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

Accepted Answer

The set of all vectors of the form [-2x, x]

Answer

The set of all vectors of the form [0, x]

Answer

The set of all vectors of the form [x, 0]

Answer

The set of all vectors of the form [x, y], where x + y = 0

Question 48

Is the function f: R^2 -> R^2 defined by f(x, y) = (x + y, x - y) a linear transformation?

Accepted Answer

Yes

Answer

No, it is not surjective.

Answer

No, it is not homogeneous.

Answer

No, it is not additive.

Question 49

Identify the property that does NOT belong to a vector space:

Accepted Answer

The set of all scalar multiples of a vector is not closed under vector addition.

Answer

The zero vector is unique.

Answer

The set of all linear combinations of vectors is a subspace.

Question 50

The determinant of a matrix A represents:

Accepted Answer

The signed area or volume of the parallelepiped spanned by its row (or column) vectors.

Answer

The product of its eigenvalues

Answer

The trace of the matrix

Question 51

If a square matrix has distinct eigenvalues, what can we conclude about the matrix?

Accepted Answer

It is diagonalizable.

Answer

It is symmetric.

Answer

It is invertible.

Answer

It is positive definite.

Question 52

What does the rank of a matrix represent?

Accepted Answer

The number of linearly independent rows (or columns) of the matrix.

Answer

The sum of the number of rows and columns of the matrix.

Answer

The number of rows of the matrix.

Answer

The number of columns of the matrix.

Question 53

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

Accepted Answer

The set of vectors does not have a zero vector.

Answer

Vector addition is associative.

Answer

The set of vectors is closed under vector addition.

Answer

Scalar multiplication is distributive over vector addition.

Question 54

Which of the following sets, equipped with the usual addition and scalar multiplication, forms a vector space?

Accepted Answer

The set of all polynomials of degree at most 3.

Answer

The set of all vectors in R^3 whose first component is positive.

Answer

The set of all 2x2 matrices with determinant equal to 1.

Answer

The set of all polynomials of degree exactly 3.

Question 55

Given a 3x5 matrix A, what is the largest possible dimension of its row space?

Accepted Answer

3

Answer

5

Answer

2

Answer

Depends on the specific entries of A.

Question 56

Consider the linear transformation T: R^3 -> R^2 defined by T(x, y, z) = (x + 2y, y - z). What is the matrix representation of T with respect to the standard bases?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 57

A linear transformation T: V -> W is one-to-one if and only if:

Accepted Answer

The kernel of T is {0}.

Answer

T is invertible.

Answer

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

Answer

The range of T is W.

Question 58

Calculate the determinant of the matrix: [[2, 1, 0], [0, -1, 3], [1, 0, 2]]

Accepted Answer

-7

Answer

1

Answer

9

Answer

5

Question 59

The matrix  [[2, 1], [1, 2]]  is diagonalizable. Which of the following is a diagonal matrix similar to it?

Accepted Answer

[[3, 0], [0, 1]]

Answer

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

Answer

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

Answer

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

Question 60

Which of the following is an orthogonal basis for the subspace of R^3 spanned by the vectors (1, 1, 0) and (1, -1, 0)?

Accepted Answer

{(1/sqrt(2), 1/sqrt(2), 0), (1/sqrt(2), -1/sqrt(2), 0)}

Answer

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

Answer

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

Question 61

Consider the system of equations:  x + y = 2, 2x + 2y = 4.  Which statement best describes the solution set?

Accepted Answer

The system has infinitely many solutions along a line.

Answer

The system has a unique solution.

Answer

The system has no solution.