Vector Spaces: Definition and Properties - Linear Algebra Quiz

Question 1

Which condition is not required for a set to form a vector space?

Accepted Answer

Existence of a multiplicative identity

Answer

Closure under vector addition

Answer

Closure under scalar multiplication

Answer

Associativity of scalar multiplication

Question 2

In the context of vector spaces, which operation is not applicable?

Accepted Answer

Multiplication of vectors

Answer

Addition of vectors

Answer

Multiplication of a vector by a scalar

Answer

Subtraction of vectors

Question 3

What is the definition of the zero vector in a vector space?

Accepted Answer

A vector with all components equal to zero

Answer

A vector with all components equal to one

Answer

A vector with the same magnitude as every other vector

Answer

A vector that is perpendicular to every other vector

Question 4

A vector space V has dimension 4. How many linearly independent vectors form a basis for V?

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 5

Consider a set of all polynomials with real coefficients. Determine if it forms a vector space under the usual operations of polynomial addition and scalar multiplication.

Accepted Answer

Yes, it is a vector space.

Answer

No, it does not satisfy closure under vector addition.

Answer

No, it does not satisfy closure under scalar multiplication.

Answer

No, it does not have an additive identity.

Question 6

Which property is NOT essential for defining a vector space?

Accepted Answer

Commutativity of vector addition

Answer

Closure under addition

Answer

Associativity of scalar multiplication

Answer

Existence of a zero vector

Question 7

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

Accepted Answer

(6, 10)

Answer

(5, 8)

Answer

(2, 4)

Answer

(9, 12)

Question 8

Determine the dimension of the vector space R^3.

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 9

Which set of vectors forms a basis for P2, the space of polynomials with degree less than or equal to 2?

Accepted Answer

{1, x, x^2}

Answer

{1, x}

Answer

{1, x^2}

Answer

{x, x^2}

Question 10

Identify a linear transformation from R^3 to R^2.

Accepted Answer

A mapping that takes (x, y, z) to (x, y).

Answer

A mapping that takes (x, y, z) to (x + y, y + z).

Answer

A mapping that takes (x, y, z) to (x, y, 0).

Answer

A mapping that takes (x, y, z) to (x^2 + y^2 + z^2).

Question 11

Which property is NOT required for a set of vectors to form a vector space?

Accepted Answer

Commutativity of scalar multiplication

Answer

Closure under vector addition

Answer

Closure under scalar multiplication

Answer

Existence of an additive identity vector

Question 12

Which statement is NOT a defining property of a vector space?

Accepted Answer

All vector spaces have a basis.

Answer

The zero vector is unique.

Answer

Vector addition is associative.

Answer

Scalar multiplication distributes over vector addition.

Question 13

Which of the following statements most accurately defines a vector space?

Accepted Answer

A collection or set (V) that possesses two fundamental binary operations, vector addition denoted as (oplus), and scalar multiplication denoted as (odot) that uphold certain axioms.

Answer

A collection of vectors that uphold the properties of solely magnitude and direction

Answer

A collection of points residing in a (n)-dimensional space

Question 14

Which operation below is not defined within a vector space?

Accepted Answer

Vector division, as the operation of dividing one vector by another is typically not defined or supported within the context of a vector space.

Answer

Scalar multiplication, an operation that involves multiplying a vector by a scalar quantity.

Answer

Vector addition, an operation that combines two vectors to yield a new vector.

Answer

Linear transformation, an operation that maps one vector space to another vector space while preserving vector addition and scalar multiplication.

Question 15

Which option presents the necessary and sufficient conditions for a non-empty set V to qualify as a vector space over a field F?

Accepted Answer

V forms an abelian group under vector addition and a module over F, implying both vector addition and scalar multiplication operations.

Answer

V is closed under vector addition and scalar multiplication operations.

Answer

V possesses a unique zero vector.

Question 16

Consider the set V consisting of all ordered pairs of real numbers (x, y). Vector addition and scalar multiplication operations are defined as (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) and c(x, y) = (cx, cy), respectively. Determine if V qualifies as a vector space over the field of real numbers.

Accepted Answer

Yes, V satisfies all properties of a vector space under the given operations.

Answer

No, V fails closure under scalar multiplication.

Answer

No, V lacks a unique zero vector.

Answer

No, V fails closure under vector addition.

Question 17

In the vector space V of continuous functions on the interval [0, 1], which operation is NOT a vector space operation?

Accepted Answer

Vector multiplication, defined as (f, g) -> f(x)g(x).

Answer

Scalar multiplication, defined as (c, f) -> cf(x).

Answer

Vector addition, defined as (f, g) -> f(x) + g(x).

Question 18

Determine the dimension of the vector space P2, consisting of all polynomials of degree 2 or less, over the field of real numbers.

Accepted Answer

3

Answer

4

Answer

2

Answer

1

Question 19

State whether the following statement is true or false: The kernel of a linear transformation T: V -> W is always a subspace of V.

Accepted Answer

True

Answer

False, the kernel is always a subset of W.

Answer

True, but only if V is finite-dimensional.

Answer

False, the kernel is always a superset of V.

Question 20

Which of the following is the formal definition of a vector space?

Accepted Answer

A non-empty set V together with two operations: addition of vectors and scalar multiplication of vectors

Answer

A set of vectors with specified magnitude and direction

Question 21

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

Accepted Answer

Multiplication is associative

Answer

There exists a zero vector

Answer

Vector addition is commutative

Answer

Scalar multiplication distributes over vector addition

Question 22

Consider the set of all polynomials of degree less than or equal to 3. Which of the following operations would NOT make this set a vector space?

Accepted Answer

Vector addition defined as polynomial subtraction

Answer

Zero vector defined as the zero polynomial

Answer

Vector addition defined as polynomial addition

Answer

Scalar multiplication defined as multiplication by a real number

Question 23

Consider the set of all polynomials of degree less than or equal to 2. Which of the following subsets is a vector subspace?

Accepted Answer

Set of all polynomials with a constant term of zero

Answer

Set of all polynomials with a leading coefficient of 1

Answer

Set of all polynomials with an even degree

Question 24

Consider the set of all 3x3 matrices with real entries. Which of the following subsets is a vector subspace?

Accepted Answer

Set of all symmetric matrices

Answer

Set of all matrices with an odd number of nonzero entries

Answer

Set of all matrices with a positive trace

Answer

Set of all matrices with a determinant of 1