### Vector Space Properties EvaluationDetermine whether the following set is a real vector space (and select what fails if it is not).#### The Set:The set of all integers $ \left( \ldots, -1, 0, 1, 2, \ldots \right) $#### Properties:**Check all the properties below to verify if the set of all integers forms a real vector space:**- [ ] **Closure under Addition**- [ ] **Closure under Scalar Multiplication**- [ ] **Commutativity of Addition**- [ ] **Associativity of Addition**- [ ] **Existence of Zero Vector**- [ ] **Existence of Additive Inverses**- [ ] **Multiplicative Identity**- [ ] **Associativity of Scalar Multiplication**- [ ] **Distributivity over Vector Addition**- [ ] **Distributivity over Scalar Addition**- [ ] **The set is a real vector space.**Evaluate each property to verify if the set of all integers qualifies as a real vector space. If any property fails, identify it explicitly.---#### Explanation:To determine if the set of all integers qualifies as a real vector space, each of the listed properties must hold true. Properties such as closure under addition and scalar multiplication, commutativity and associativity of addition, the existence of additive inverses and the zero vector, and the distributive properties are essential checks for verifying whether a set adheres to the criteria of a real vector space. If even one of these properties does not hold, the set cannot be considered a real vector space.

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Determine whether the following set is a real vector space (and select what fails if it is not). The set of all integers (..., -1, 0, 1, 2, ...) O closure under Addition Closure under Scalar Multiplication O Commutativity of Addition Associativity of Addition Existence of Zero Vector Existence of Additive Inverses O Multiplicative Identity Associativity of Scalar Multiplication Distributivity over Vector Addition Distributivity over Scalar Addition O The set is a real vector space.

Determine whether the following set is a real vector space (and select what fails if it is not). The set of all integers (..., -1, 0, 1, 2, ...) O closure under Addition Closure under Scalar Multiplication O Commutativity of Addition Associativity of Addition Existence of Zero Vector Existence of Additive Inverses O Multiplicative Identity Associativity of Scalar Multiplication Distributivity over Vector Addition Distributivity over Scalar Addition O The set is a real vector space.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

### Vector Space Properties Evaluation

Determine whether the following set is a real vector space (and select what fails if it is not).

#### The Set:
The set of all integers $ \left( \ldots, -1, 0, 1, 2, \ldots \right) $

#### Properties:
**Check all the properties below to verify if the set of all integers forms a real vector space:**

- [ ] **Closure under Addition**
- [ ] **Closure under Scalar Multiplication**
- [ ] **Commutativity of Addition**
- [ ] **Associativity of Addition**
- [ ] **Existence of Zero Vector**
- [ ] **Existence of Additive Inverses**
- [ ] **Multiplicative Identity**
- [ ] **Associativity of Scalar Multiplication**
- [ ] **Distributivity over Vector Addition**
- [ ] **Distributivity over Scalar Addition**

- [ ] **The set is a real vector space.**

Evaluate each property to verify if the set of all integers qualifies as a real vector space. If any property fails, identify it explicitly.

---

#### Explanation:
To determine if the set of all integers qualifies as a real vector space, each of the listed properties must hold true. Properties such as closure under addition and scalar multiplication, commutativity and associativity of addition, the existence of additive inverses and the zero vector, and the distributive properties are essential checks for verifying whether a set adheres to the criteria of a real vector space. If even one of these properties does not hold, the set cannot be considered a real vector space.

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