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
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F022ad49b-1aed-40ba-8a0b-57b9d135ec5b%2Fb7d55021-9b4b-497f-b19d-60308b8d6a5a%2F4zu45ol_processed.png&w=3840&q=75)
Transcribed Image Text:### 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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