Fields and Vector Spaces: Master the Fundamentals of Algebra

Question 1

Identify the conditions that must be satisfied for a subset of a vector space to be considered a subspace.

Accepted Answer

Closure under both vector addition and scalar multiplication, and containing the zero vector

Answer

Closure under vector addition and containing the zero vector

Answer

Closure under scalar multiplication and containing the zero vector

Answer

Closure under linear combinations

Question 2

What is the dimension of the vector space consisting of all 2x2 matrices with real entries?

Accepted Answer

4

Answer

1

Answer

2

Answer

8

Question 3

Which of the following is NOT a field?

Accepted Answer

2x2 matrices with real entries under addition and matrix multiplication

Answer

Rational numbers under addition and multiplication

Answer

Complex numbers under addition and multiplication

Answer

Polynomials with real coefficients under addition and multiplication

Question 4

For a vector space V over a field F, what constitutes a basis for V?

Accepted Answer

A set of linearly independent vectors that span V

Answer

A set of linearly dependent vectors that span V

Answer

A set of vectors that form a subspace of V

Answer

A set of vectors that are orthogonal to each other

Question 5

Identify the characteristic of a linear map:

Accepted Answer

It preserves vector addition and scalar multiplication

Answer

It preserves vector subtraction and scalar division

Answer

It preserves vector norms

Answer

It preserves vector orientations

Question 6

Which condition is not necessary for the existence of an inverse linear map?

Accepted Answer

The linear map is a projection

Answer

The linear map is one-to-one

Answer

The linear map is onto

Answer

The domain and codomain of the linear map are finite-dimensional

Question 7

Which of the following is a linear map from R^2 to R^3?

Accepted Answer

f(x, y) = (x + y, x - y, 2x)

Answer

f(x, y) = (x^2, y^2, z^2)

Answer

f(x, y) = (x, y/z, z)

Answer

f(x, y) = (x, y, z^2)

Question 8

What is the dimension of the vector space of all symmetric matrices of order 3 over the field of real numbers?

Accepted Answer

6

Answer

3

Answer

9

Answer

12

Question 9

Which of the following is an example of a finite cyclic group?

Accepted Answer

The group of integers under addition modulo 7

Answer

The group of real numbers under addition

Answer

The group of complex numbers under multiplication

Answer

The group of invertible matrices of order 2 over the field of real numbers

Question 10

Let V be a vector space and T: V -> V be a linear map. Which of the following conditions is equivalent to T being invertible?

Accepted Answer

T is one-to-one and onto

Answer

T is one-to-one

Answer

T is onto

Answer

T has a left inverse and a right inverse

Question 11

Which of the following is an application of field theory in number theory?

Accepted Answer

Fermat's Last Theorem

Answer

Goldbach's Conjecture

Answer

P versus NP problem

Answer

Traveling Salesman Problem

Question 12

Let V be a vector space over a field F. Which of the following is a property of a linearly dependent set of vectors in V?

Accepted Answer

One vector in the set can be expressed as a linear combination of the other vectors in the set

Answer

The set contains a zero vector

Answer

The set spans V

Answer

The set is a basis for V

Question 13

In the context of vector spaces, which set forms a basis, meaning it is a linearly independent and spans the entire space?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 14

Which property does NOT belong to a field?

Accepted Answer

Distributivity of multiplication over addition and subtraction

Answer

Closure under addition and multiplication

Answer

Commutativity of addition and multiplication

Answer

Existence of additive and multiplicative identities

Question 15

Consider the set of all complex numbers with modulus 1. Does this set form a vector space over the field of real numbers?

Accepted Answer

No, it is not a vector space because it is not closed under scalar multiplication.

Answer

Yes, it is a vector space.

Answer

No, it is not a vector space because it is not closed under addition.

Answer

No, it is not a vector space because it does not contain the zero vector.

Question 16

Which function is an isomorphism between the vector spaces R² and C?

Accepted Answer

f(x, y) = (x, iy)

Answer

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

Answer

f(x, y) = (x, x + y)

Answer

f(x, y) = (y, x)

Question 17

The dimension of the vector space ℝ³ is

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 18

If V is a finite-dimensional vector space and U is a subspace of V, then the dimension of U is

Accepted Answer

Less than or equal to the dimension of V

Answer

Equal to the dimension of V

Answer

Greater than or equal to the dimension of V

Answer

Not related to the dimension of V

Question 19

The kernel of a linear transformation is

Accepted Answer

The set of all vectors that are mapped to the zero vector

Answer

The set of all vectors that are not mapped to the zero vector

Answer

The set of all vectors that are mapped to the identity element

Answer

The set of all vectors that are not mapped to the identity element

Question 20

Which of the following is an isomorphism between the vector spaces ℝ² and ℝ³?

Accepted Answer

f(x, y) = (x, y, 0)

Answer

f(x, y) = (x + y, x - y, 0)

Answer

f(x, y) = (x, y, x * y)

Answer

No such isomorphism exists

Question 21

Which of the following transformations is a linear transformation?

Accepted Answer

A function that preserves both addition and multiplication

Answer

A function that preserves addition but not multiplication

Answer

A function that preserves neither addition nor multiplication

Answer

A function that preserves multiplication but not addition

Question 22

Which of the following 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 23

Define a field and provide an example.

Accepted Answer

A set with two operations, addition and multiplication, that satisfy specific axioms. Example: The set of real numbers.

Answer

A set with one operation, addition, that satisfies specific axioms. Example: The set of integers.

Answer

A set with two operations, addition and multiplication, that satisfy specific axioms. Example: The set of complex numbers.

Answer

A set with one operation, multiplication, that satisfies specific axioms. Example: The set of rational numbers.

Question 24

Identify the basis for the vector space of polynomials of degree less than or equal to 2 over a field F.

Accepted Answer

{1, x, x^2}

Answer

{1, x, x^3}

Answer

{1, x^2, x^3}

Answer

{1, x, x^2, x^3}

Question 25

Which of the following is the definition of a field?

Accepted Answer

An algebraic structure with two operations, addition and multiplication, which satisfy certain axioms

Answer

A set of numbers that can be added, subtracted, multiplied, and divided without leaving the set

Answer

A collection of vectors that span a vector space

Answer

A transformation that maps vectors in one vector space to vectors in another vector space

Question 26

Which of the following statements about linear transformations is true?

Accepted Answer

They preserve the linear structure of a vector space

Answer

They are always invertible

Answer

They are always one-to-one

Answer

They are always onto

Question 27

Which of the following is NOT a property of a basis for a vector space?

Accepted Answer

It is a unique set of vectors

Answer

It is a linearly independent set of vectors

Answer

It spans the vector space

Answer

It is a minimal generating set for the vector space

Question 28

Which of the following is a necessary and sufficient condition for a set of vectors to be a basis for a vector space?

Accepted Answer

The set is linearly independent and spans the vector space

Answer

The set is linearly dependent and spans the vector space

Answer

The set is linearly independent but does not span the vector space

Answer

The set is linearly dependent but spans the vector space

Question 29

Which of the following is NOT a field?

Accepted Answer

The set of integers

Answer

The set of real numbers

Answer

The set of complex numbers

Answer

The set of rational numbers

Question 30

Determine the subspace of R^3 that consists of all vectors with the first component equal to zero.

Accepted Answer

The y-z plane

Answer

The x-axis

Answer

The set of all vectors in R^3

Answer

The origin

Question 31

Which of the following transformations from R^2 to R^2 is linear?

Accepted Answer

f(x, y) = 2x + 3y

Answer

f(x, y) = x^2 + y^2

Answer

f(x, y) = |x - y|

Answer

f(x, y) = sin(x) + cos(y)