Fields and Vector Spaces: Master the Fundamentals of Abstract Algebra

Question 1

Which set of vectors forms 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, 0), (0, 1, 0)}

Answer

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

Answer

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

Question 2

The vector space P(x) consisting of all polynomials in x has a dimension of:

Accepted Answer

Infinite

Answer

Finite

Answer

Dependent on the degree of the polynomial

Answer

Zero

Question 3

Determine the set that is not a subspace of the vector space V.

Accepted Answer

The set of vectors perpendicular to a given vector

Answer

The set of linear combinations of a given set of vectors

Answer

The set of all vectors with a particular component equal to zero

Answer

The set of vectors that form a line through the origin

Question 4

Identify the characteristic that defines a linear transformation between vector spaces.

Accepted Answer

It maintains both vector addition and scalar multiplication

Answer

It maintains vector addition

Answer

It maintains scalar multiplication

Answer

It maps the zero vector to the zero vector

Question 5

Calculate the dimension of the vector space consisting of 3x3 matrices.

Accepted Answer

9

Answer

3

Answer

6

Answer

18

Question 6

Identify the set that forms a basis for the vector space of polynomials of degree less than or equal to 2.

Accepted Answer

{1, x, x^2}

Answer

{-1, 1, x}

Answer

{0, x, x^2}

Answer

{x, x^2, x^3}

Question 7

Identify the characteristic of a Euclidean vector space.

Accepted Answer

It possesses an inner product that induces a norm

Answer

It is a finite-dimensional vector space

Answer

It is a vector space over the complex numbers

Answer

It is a subspace of a normed vector space

Question 8

Identify the condition that does not constitute an eigenvalue of the linear transformation T: V -> V.

Accepted Answer

A real number with <T(v), v> = 0 for some non-zero vector v

Answer

The root of the characteristic polynomial of T

Answer

A complex number with <T(v), v> = 0 for some non-zero vector v

Answer

A scalar with <T(v) - s*v, v> = 0 for some non-zero vector v

Question 9

Determine the type of matrix that can be diagonalized.

Accepted Answer

A matrix with distinct eigenvalues

Answer

A matrix with repeated eigenvalues

Answer

A matrix with complex eigenvalues

Answer

A matrix with all eigenvalues equal to 0

Question 10

Which of the following is not a field?

Accepted Answer

Z (integers)

Answer

Q (rational numbers)

Answer

R (real numbers)

Answer

C (complex numbers)

Question 11

Let F be a field and V be a vector space over F. Which of the following is a linear combination of vectors v1, v2, ..., vn in V?

Accepted Answer

c1v1 + c2v2 + ... + cnvn, where c1, c2, ..., cn are scalars in F

Answer

Any vector in V

Question 12

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

Accepted Answer

A set of vectors that contains the zero vector

Answer

Any set of vectors that spans V

Question 13

Let F be a field and V be a vector space over F. Which of the following is a basis for V?

Accepted Answer

A linearly independent set of vectors that spans V

Answer

Any set of vectors that spans V

Question 14

Let F be a field and V be a vector space over F. Which of the following is a homomorphism from V to V?

Accepted Answer

A linear map from V to V

Answer

Any map from V to V

Question 15

Let F be a field and V be a vector space over F. Which of the following is a quotient space of V?

Accepted Answer

V modulo a subspace W

Answer

Any subspace of V

Question 16

Let F be a field and V be a vector space over F. Which of the following is a direct sum of V?

Accepted Answer

A vector space W such that V is isomorphic to W x W

Answer

Any vector space that is a subspace of V

Question 17

Which of the following is NOT a field?

Accepted Answer

The set of integers with ordinary addition and multiplication operations

Answer

The set of rational numbers with ordinary addition and multiplication operations

Answer

The set of complex numbers with ordinary addition and multiplication operations

Question 18

Which of the following is a basis for the vector space R³?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 19

Which of the following is an inner product on R²?

Accepted Answer

<(a, b), (c, d)> = a₁c₁ + a₂c₂

Answer

<(a, b), (c, d)> = a₁² + a₂²

Answer

<(a, b), (c, d)> = |a₁| |c₁|

Answer

<(a, b), (c, d)> = a₁c₁ - a₂c₂

Question 20

Which of the following is an orthonormal basis for R²?

Accepted Answer

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

Answer

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

Answer

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

Question 21

Which of the following is a linear combination of the vectors (1, 2) and (3, 4)?

Accepted Answer

(5, 8)

Answer

(2, 4)

Answer

(6, 8)

Answer

(4, 6)

Question 22

Which of the following is a basis for the vector space R^2?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 23

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

Accepted Answer

{(x, y) | x + 2y = 0}

Answer

{(x, y) | x - y = 0}

Answer

{(x, y) | 2x + y = 0}

Answer

{(x, y) | x - 2y = 0}

Question 24

Which of the following is the determinant of the matrix A = [1 2; 3 4]?

Accepted Answer

-2

Answer

4

Answer

0

Answer

2