Closure under scalar multiplication

Oct 13, 2025 by miodeya.com 90 views

Determine if the Set Z = 1, -2, -1, 0, 1, 2, •••} is a Vector Space Over R

In mathematics, a vector space is a fundamental concept that provides a framework for understanding and working with linear transformations, matrices, and other algebraic structures. A vector space is a set of vectors that can be added together and scaled by real numbers, while satisfying certain properties such as commutativity, associativity, and distributivity. In this article, we will determine whether the set Z = 1, -2, -1, 0, 1, 2, •••} is a vector space over R.

A vector space is a set of vectors that can be added together and scaled by real numbers, while satisfying the following properties:

Closure under addition: The sum of any two vectors in the set is also in the set.
Closure under scalar multiplication: The product of any vector in the set and a real number is also in the set.
Commutativity of addition: The order of addition does not matter.
Associativity of addition: The order in which vectors are added does not matter.
Distributivity of scalar multiplication over vector addition: The product of a real number and the sum of two vectors is equal to the sum of the products of the real number and each vector.
Distributivity of scalar multiplication over scalar addition: The product of the sum of two real numbers and a vector is equal to the sum of the products of each real number and the vector.
Existence of additive identity: There exists a vector in the set that, when added to any other vector, leaves the other vector unchanged.
Existence of additive inverse: For each vector in the set, there exists another vector in the set that, when added to the first vector, results in the additive identity.

To determine whether the set Z = 1, -2, -1, 0, 1, 2, •••} is a vector space over R, we need to check whether it satisfies all the properties of a vector space.

Closure under addition

The set Z is closed under addition because the sum of any two integers is also an integer. For example, 1 + 2 = 3, which is in the set Z.

Closure under scalar multiplication

The set Z is closed under scalar multiplication because the product of any integer and a real number is also an integer. For example, 2 × 3 = 6, which is in the set Z.

Commutativity of addition

The set Z satisfies the commutativity of addition because the order of addition does not matter. For example, 1 + 2 = 2 + 1 = 3.

Associativity of addition

The set Z satisfies the associativity of addition because the order in which vectors are added does not matter. For example, (1 + 2) + 3 = 1 + (2 + 3) = 6.

Distributivity of scalar multiplication over vector addition

The set Z satisfies the distributivity of scalar multiplication over vector addition because the product of a real number and the sum of two vectors is equal to the sum of the products of the real number and each vector. For example, 2 × (1 + 2) = 2 × 1 + 2 × 2 = 6.

Distributivity of scalar multiplication over scalar addition

The set Z satisfies the distributivity of scalar multiplication over scalar addition because the product of the sum of two real numbers and a vector is equal to the sum of the products of each real number and the vector. For example, (2 + 3) × 1 = 2 × 1 + 3 × 1 = 5.

Existence of additive identity

The set Z has an additive identity, which is 0. When 0 is added to any other vector in the set, the result is the same vector.

Existence of additive inverse

For each vector in the set Z, there exists another vector in the set that, when added to the first vector, results in the additive identity. For example, the additive inverse of 1 is -1, because 1 + (-1) = 0.

Based on the properties of a vector space, we can conclude that the set Z = 1, -2, -1, 0, 1, 2, •••} is a vector space over R. It satisfies all the properties of a vector space, including closure under addition and scalar multiplication, commutativity and associativity of addition, distributivity of scalar multiplication over vector addition and scalar addition, existence of additive identity, and existence of additive inverse.

Hoffman, K., & Kunze, R. (1971). Linear Algebra**. Prentice Hall._
Strang, G. (1988). Linear Algebra and Its Applications**. Academic Press._
Lang, S. (1993). Linear Algebra**. Springer-Verlag._

Note: The references provided are classic textbooks on linear algebra, which provide a comprehensive introduction to the subject.
Q&A: Determining if the Set Z is a Vector Space Over R

In our previous article, we determined that the set Z = 1, -2, -1, 0, 1, 2, •••} is a vector space over R. However, we understand that some readers may still have questions about this topic. In this article, we will address some of the most frequently asked questions about determining if the set Z is a vector space over R.

Q: What is the difference between a vector space and a set?

A: A vector space is a set of vectors that can be added together and scaled by real numbers, while satisfying certain properties such as commutativity, associativity, and distributivity. A set, on the other hand, is a collection of objects, without any specific properties or operations defined on it.

Q: Why is it important to determine if the set Z is a vector space over R?

A: Determining if the set Z is a vector space over R is important because it helps us understand the properties of the set and how it can be used in mathematical operations. Vector spaces are a fundamental concept in linear algebra, and understanding their properties is essential for solving problems in mathematics, physics, engineering, and other fields.

Q: What are the properties of a vector space?

A: The properties of a vector space include:

Closure under addition: The sum of any two vectors in the set is also in the set.
Closure under scalar multiplication: The product of any vector in the set and a real number is also in the set.
Commutativity of addition: The order of addition does not matter.
Associativity of addition: The order in which vectors are added does not matter.
Distributivity of scalar multiplication over vector addition: The product of a real number and the sum of two vectors is equal to the sum of the products of the real number and each vector.
Distributivity of scalar multiplication over scalar addition: The product of the sum of two real numbers and a vector is equal to the sum of the products of each real number and the vector.
Existence of additive identity: There exists a vector in the set that, when added to any other vector, leaves the other vector unchanged.
Existence of additive inverse: For each vector in the set, there exists another vector in the set that, when added to the first vector, results in the additive identity.

Q: How do I determine if a set is a vector space over R?

A: To determine if a set is a vector space over R, you need to check whether it satisfies all the properties of a vector space. This involves checking whether the set is closed under addition and scalar multiplication, whether it satisfies the commutativity and associativity of addition, and whether it satisfies the distributivity of scalar multiplication over vector addition and scalar addition.

Q: What are some common mistakes to avoid when determining if a set is a vector space over R?

A: Some common mistakes to avoid when determining if a set is a vector space over R include:

Failing to check all the properties of a vector space: Make sure to check all the properties of a vector space, including closure under addition and scalar multiplication, commutativity and associativity of addition, and distributivity of scalar multiplication over vector addition and scalar addition.
Assuming that a set is a vector space without checking its properties: Do not assume that a set is a vector space without checking its properties. Instead, carefully check whether the set satisfies all the properties of a vector space.
Using incorrect or incomplete definitions of vector space properties: Make sure to use correct and complete definitions of vector space properties. Incomplete or incorrect definitions can lead to incorrect conclusions.

In this article, we have addressed some of the most frequently asked questions about determining if the set Z is a vector space over R. We have provided a brief overview of the properties of a vector space and how to determine if a set is a vector space over R. We hope that this article has been helpful in clarifying any confusion about this topic.

Hoffman, K., & Kunze, R. (1971). Linear Algebra**. Prentice Hall._
Strang, G. (1988). Linear Algebra and Its Applications**. Academic Press._
Lang, S. (1993). Linear Algebra**. Springer-Verlag._