Vector Spaces: Definition, Examples, and Applications

Question 1

Define a vector space.

Accepted Answer

A set of elements with vector addition and scalar multiplication operations that satisfy specific axioms.

Answer

A set of points in Euclidean space.

Answer

A set of functions that meet certain conditions.

Question 2

In the context of vector spaces, define linear independence and provide clear examples.

Accepted Answer

Linear independence means that no vector in a set can be written as a linear combination of the other vectors in the set.

Answer

Linear independence means that a set of vectors can be expressed as linear combinations of one another.

Answer

A set of vectors is linearly independent if they span the entire vector space.

Answer

Linear independence is irrelevant in the study of vector spaces.

Question 3

Consider the set V of polynomials of degree at most 2. Determine if V forms a vector space under the standard operations of polynomial addition and scalar multiplication.

Accepted Answer

Yes, V forms a vector space because it satisfies all the vector space axioms.

Answer

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

Answer

No, V does not form a vector space because it does not have an additive inverse.

Question 4

What is the definition of a basis for a vector space, and why is it important for understanding the structure of the space?

Accepted Answer

A set of linearly independent vectors that span the space

Answer

A set of linearly dependent vectors that span the space

Answer

A set of orthogonal vectors that span the space

Answer

A set of normalized vectors that span the space

Question 5

In vector space V over field F, what property does the zero vector 0 possess?

Accepted Answer

Vector addition of 0 to any vector v yields v.

Answer

0 is equivalent to all vectors in V.

Answer

Vector subtraction of v from 0 yields v.

Answer

Scalar multiplication of 0 by any vector v results in 0.

Question 6

For R³ (the set of ordered triples of real numbers), which operation defines vector addition?

Accepted Answer

Element-wise addition: (x1, x2, x3) + (y1, y2, y3) = (x1 + y1, x2 + y2, x3 + y3)

Answer

Element-wise multiplication: (x1, x2, x3) + (y1, y2, y3) = (x1 * y1, x2 * y2, x3 * y3)

Answer

Element-wise subtraction: (x1, x2, x3) + (y1, y2, y3) = (x1 - y1, x2 - y2, x3 - y3)

Answer

Identity operation: (x1, x2, x3) + (y1, y2, y3) = (x1, x2, x3)

Question 7

Regarding the intersection of a subspace W and its parent vector space V, which statement holds true?

Accepted Answer

The intersection is a subspace of V.

Answer

The intersection is a vector space.

Answer

The intersection is a subspace of W.

Answer

The intersection is always the zero subspace.

Question 8

Which operations on the set of all polynomials with real coefficients define a vector space structure?

Accepted Answer

Addition and scalar multiplication

Answer

Multiplication and division

Answer

Differentiation and integration

Answer

Composition and evaluation

Question 9

Given vectors v1 = (1, 2, 3) and v2 = (4, 5, 6), determine if the set {v1, v2} is linearly independent in the vector space R^3?

Accepted Answer

Linearly independent

Answer

Linearly dependent

Answer

Cannot be determined

Question 10

Consider the matrix A = [[1, 2], [3, 4]]. Determine the null space of the linear transformation T(x) = Ax?

Accepted Answer

The set of all vectors of the form (x, -2x)

Answer

The set of all vectors perpendicular to (1, 2)

Answer

The set of all vectors parallel to (3, 4)

Answer

The set of all vectors of the form (-3x, x)

Question 11

Which of the following vector spaces form a field?

Accepted Answer

C

Answer

R^2

Answer

R x R

Answer

P3(R)

Question 12

Consider the vector space of continuous functions on the interval [0, 1]. Which of the following is a basis for this vector space?

Accepted Answer

{1, x, x^2}

Answer

{sin(x), cos(x), e^x}

Answer

{1, e^x, e^(2x)}

Answer

{x, x^2, x^3, ...}

Question 13

Why are vector spaces essential mathematical structures?

Accepted Answer

They provide a framework for studying linear equations and transformations.

Accepted Answer

They enable the representation of geometric objects and transformations.

Accepted Answer

They facilitate the analysis of data and optimization problems.

Question 14

Which of the following is a basis for the vector space P2 (the set of polynomials of degree at most 2)?

Accepted Answer

{ 1, x, x^2 }

Answer

{ 1, x^2, x^4 }

Answer

{ 1, x, x^3 }

Answer

{ 1, x^2, x^5 }

Question 15

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

Accepted Answer

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

Answer

T(x, y) = (x, x^2, y)

Answer

T(x, y) = (x, x^2, xy)

Answer

T(x, y) = (x^3, y^3, 0)

Question 16

Let V be a vector space over a field F. Which of the following is true about the subspace spanned by a set of vectors?

Accepted Answer

It is the set of all linear combinations of that set of vectors.

Answer

It is the smallest subspace containing that set of vectors.

Answer

It is the largest subspace contained in the set of vectors.

Answer

It is the set of all multiples of that set of vectors.

Question 17

Which of the following is an example of an infinite-dimensional vector space?

Accepted Answer

The set of all functions from R to R

Answer

R^n (where n is any positive integer)

Answer

The set of all polynomials with real coefficients

Answer

The set of all matrices with finitely many nonzero entries

Question 18

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

Accepted Answer

Closure under vector multiplication

Answer

Closure under vector addition

Answer

Associativity of vector addition

Answer

Distributivity of scalar multiplication over vector addition

Question 19

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

Accepted Answer

Commutativity of vector addition

Answer

Associativity of scalar multiplication (multiplication of a vector by a scalar)

Answer

Existence of a zero vector (vector with all components zero)

Answer

Existence of additive inverses (for each vector, there exists another one such that their sum is the zero vector)

Question 20

What is the dimension of the vector space R^n (the n-dimensional Euclidean space)?

Accepted Answer

n

Answer

n^2

Answer

2n

Answer

n-1

Question 21

The vector space V has dimension n. How many vectors are in any basis for V?

Accepted Answer

n

Answer

n-1

Answer

n+1

Answer

2n

Question 22

Which property is NOT characteristic of vector spaces?

Accepted Answer

Vector spaces can have an infinite number of elements

Answer

Closure under vector addition

Answer

Closure under scalar multiplication

Answer

Associativity of vector addition

Question 23

Which of the following operations is not a vector space operation?

Accepted Answer

Vector multiplication

Answer

Taking the zero vector

Answer

Vector addition

Answer

Scalar multiplication

Question 24

Which of the following is a property of the null space of a linear transformation?

Accepted Answer

It is a subspace of the domain

Answer

It is linearly independent

Answer

It is the same as the range

Answer

It is always a vector space

Question 25

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

Accepted Answer

The product of two vectors in the vector space is always a scalar

Answer

The zero vector is unique

Answer

For every vector v and scalar c, cv is a vector in the vector space

Question 26

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

Accepted Answer

The set of all vectors in V that are multiples of a fixed vector v

Answer

The set of all vectors in V that are perpendicular to a fixed vector v

Answer

The set of all vectors in V that have a zero component

Question 27

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

Accepted Answer

A set of linearly independent vectors that spans V

Answer

A set of vectors that is linearly independent but does not span V

Answer

A set of vectors that is linearly dependent

Answer

A set of vectors that spans V but is not linearly independent

Question 28

Which of the following is a linear transformation from one vector space to another?

Accepted Answer

A function that preserves vector addition and scalar multiplication

Answer

A function that preserves only vector addition

Answer

A function that preserves only scalar multiplication

Question 29

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

Accepted Answer

n

Answer

n+1

Answer

0

Answer

n-1

Question 30

What is the null space of a linear transformation?

Accepted Answer

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

Answer

The set of all vectors that are mapped to a nonzero vector

Answer

The set of all vectors that are linearly independent

Question 31

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

Accepted Answer

The number of vectors exceeds the dimension of the vector space

Answer

The vectors are all orthogonal

Answer

The vectors are all parallel

Question 32

Consider the set of all polynomials of degree less than or equal to 3. Which of the following operations satisfies closure under vector addition?

Accepted Answer

Adding two polynomials of degree less than or equal to 3

Answer

Integrating a polynomial of degree less than or equal to 3

Answer

Multiplying two polynomials of degree less than or equal to 3

Answer

Taking the derivative of a polynomial of degree less than or equal to 3

Question 33

Let V be a vector space and let S be a subset of V. Which of the following conditions is necessary and sufficient to ensure that S is a subspace of V?

Accepted Answer

S is non-empty, closed under vector addition, and closed under scalar multiplication

Answer

S is non-empty and closed under vector addition

Answer

S is non-empty and spans V

Answer

S is closed under vector addition and closed under scalar multiplication

Question 34

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

Accepted Answer

n

Answer

n-1

Answer

n+1

Answer

Cannot be determined without more information

Question 35

Consider the vector space P2 of polynomials of degree less than or equal to 2. What is a basis for P2?

Accepted Answer

{1, x, x^2}

Answer

{1, x, x^3}

Answer

{1, 2x, x^2}

Answer

{0, 1, x}

Question 36

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

Accepted Answer

The number of vectors is greater than the dimension of the vector space

Answer

The vectors are all nonzero

Answer

The vectors are all orthogonal

Answer

The vectors are all parallel

Question 37

Let V be a finite-dimensional vector space. Which of the following statements is true?

Accepted Answer

Every linearly independent set in V is a basis for V

Answer

Every spanning set in V is linearly independent

Answer

Every subset of V is a subspace of V

Answer

Every subspace of V has the same dimension

Question 38

Which of the following sets is NOT a vector space over the field of real numbers?

Accepted Answer

The set of all polynomials with real coefficients of degree exactly 3

Answer

The set of all real-valued functions defined on the interval [0, 1]

Answer

The set of all 2x2 matrices with real entries

Answer

The set of all real numbers

Question 39

Consider the set S = {(1, 0, 1), (0, 1, 1), (1, 1, 0)} in R^3. What is the span of S?

Accepted Answer

R^3

Answer

A plane in R^3

Answer

A line in R^3

Answer

The empty set

Question 40

Let V be a vector space over the field F. Which of the following is a necessary and sufficient condition for a set S to be a basis for V?

Accepted Answer

S is linearly independent and spans V.

Answer

S is linearly independent and contains the zero vector.

Answer

S spans V and is closed under addition.

Answer

S spans V and contains the zero vector.

Question 41

What is the dimension of the vector space of all polynomials with real coefficients of degree at most 2?

Accepted Answer

3

Answer

4

Answer

Infinite

Answer

2

Question 42

Which of the following sets is a subspace of the vector space of all real-valued functions defined on the real line?

Accepted Answer

The set of all even functions

Answer

The set of all functions that are strictly increasing

Answer

The set of all functions that take on only rational values

Answer

The set of all functions that are not differentiable at x = 0

Question 43

Which of the following is NOT a property of a linear transformation T: V -> W?

Accepted Answer

T(u + v) = T(u) + T(v) + T(u) * T(v)

Answer

T(u + v) = T(u) + T(v)

Answer

T(cu) = cT(u) for any scalar c

Answer

T(0_V) = 0_W

Question 44

Let V and W be vector spaces over the same field. What is the dimension of the vector space of all linear transformations from V to W?

Accepted Answer

dim(V) * dim(W)

Answer

dim(V) - dim(W)

Answer

dim(V) / dim(W)

Answer

dim(V) + dim(W)

Question 45

Which of the following statements about inner product spaces is TRUE?

Accepted Answer

The norm of a vector is always non-negative.

Answer

The Cauchy-Schwarz inequality holds only for real vector spaces.

Answer

Any two orthogonal vectors in an inner product space are linearly dependent.

Answer

The inner product of any two vectors is always positive.

Question 46

Which of the following sets, with the standard operations of addition and scalar multiplication, is NOT a vector space over the real numbers?

Accepted Answer

The set of all polynomials with real coefficients, where scalar multiplication is defined as k * p(x) = p(kx) for any real number k and polynomial p(x).

Answer

The set of all 2x2 matrices with real entries.

Answer

The set of all real numbers.

Answer

The set of all vectors of the form (a, b, 0), where a and b are real numbers.

Question 47

Let V be a vector space over the real numbers. Which of the following statements is ALWAYS true about V?

Accepted Answer

The zero vector is unique.

Answer

The product of two vectors in V is always another vector in V.

Answer

The scalar multiplication of a vector in V by a real number always results in a vector in V.

Answer

The sum of two vectors in V is always another vector in V.

Question 48

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

Accepted Answer

The set of all vectors of the form (a, b, 0), where a and b are real numbers.

Answer

The set of all vectors of the form (a, a + 1, 0), where a is a real number.

Answer

The set of all vectors of the form (1, b, c), where b and c are real numbers.

Answer

The set of all vectors of the form (a, b, c), where a, b, and c are positive real numbers.

Question 49

Let S = {(1, 0, 1), (0, 1, 1)} be a subset of R^3. Which of the following sets represents the span of S?

Accepted Answer

The set of all vectors of the form (a, b, a + b), where a and b are real numbers.

Answer

The set of all vectors of the form (a, b, 0), where a and b are real numbers.

Answer

The set of all vectors of the form (a, 0, a), where a is a real number.

Answer

The set of all vectors of the form (a, b, c), where a, b, and c are real numbers.

Question 50

Which of the following sets is linearly independent in the vector space R^2?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 51

Let V be a vector space and let S be a subset of V. Which of the following statements is NOT necessarily true if S is a basis for V?

Accepted Answer

S contains all the vectors in V.

Answer

S is a minimal spanning set for V.

Answer

S spans V.

Answer

S is linearly independent.

Question 52

What is the dimension of the vector space P_3, the space of all polynomials of degree at most 3 with real coefficients?

Accepted Answer

4

Answer

2

Answer

1

Answer

3

Question 53

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 54

Which of the following is an example of an inner product space?

Accepted Answer

The space of all continuous real-valued functions defined on the interval [0, 1], with the inner product defined as <f, g> = ∫[0, 1] f(x)g(x) dx.

Answer

The set of all polynomials with real coefficients, with the inner product defined as <p, q> = p(0)q(0).

Answer

The set of all 2x2 matrices with real entries, with the inner product defined as <A, B> = det(AB).

Answer

The set of all real numbers, with the inner product defined as <x, y> = x + y.

Question 55

What is the orthogonal complement of the subspace W = {(x, y, 0) | x, y ∈ R} in the inner product space R^3, with the standard inner product?

Accepted Answer

{(0, 0, z) | z ∈ R}

Answer

{(x, 0, z) | x, z ∈ R}

Answer

R^3

Answer

{(x, y, 0) | x, y ∈ R}