Difference between a vector space and a field

In summary, a vector space is defined as a set of objects that can undergo the laws of algebra "over" the field of scalars. While the laws of algebra also hold in a field, a field is not necessarily a vector space. This does not make the definition of a vector space meaningless or circular, as a field can still be defined as a vector space over itself. The key distinction is that vector spaces allow for the vectors to be different from the underlying field, while this is not a requirement in a field.
  • #1
torquerotates
207
0
One book defined a vector space as a set of objects that can undergo the laws of algebra "over" the field of scalars. But doesn't the laws of algebra also hold in a field? If so, wouldn't a field be a vector space also? Wouldn't that make the definition of a vector space meaningless as it uses circular logic?
 
Physics news on Phys.org
  • #2
That's a good observation. Every field is in fact a vector space over itself or over any of its subfields. But of course the converse isn't true, i.e. there are vector spaces which aren't fields. The reason being that while in a field you must be able to "multiply" the elements together (and not just multiply by scalars), there is no such requirement in a vector space.
 
  • #3
torquerotates said:
One book defined a vector space as a set of objects that can undergo the laws of algebra "over" the field of scalars. But doesn't the laws of algebra also hold in a field?

Presumably, but it is slightly ambiguous phrase.

If so, wouldn't a field be a vector space also? Wouldn't that make the definition of a vector space meaningless as it uses circular logic?

No, there is nothing circular, and it certainly isn't circular logic. At worst it would be 'redundant', but since vector spaces are not fields it isn't.
 
  • #4
Given a field, F, you certainly can define F to be a vector space over itself. That's not often done because you don't learn anything new. The dimension would, of course, be 1.

However, since the definition of "vector space" allows for the vectors to be different from the underlying field, no, the definition is not circular.
 

What is a vector space?

A vector space is a set of objects, called vectors, that can be added together and multiplied by numbers, called scalars. This set of objects must follow certain rules, such as closure under addition and scalar multiplication, to be considered a vector space.

What is a field?

A field is a set of numbers, called scalars, that can be added, subtracted, multiplied, and divided. It follows specific rules, such as the existence of additive and multiplicative identities and inverses, to be considered a field.

What is the difference between a vector space and a field?

The main difference between a vector space and a field is that a vector space is a set of objects while a field is a set of numbers. Vector spaces require both addition and scalar multiplication operations, while fields require addition, subtraction, multiplication, and division operations.

Can a vector space be a field?

No, a vector space cannot be a field because a vector space does not necessarily have all the properties of a field. For example, a vector space may not have multiplicative inverses, which is a required property of a field.

Why is it important to understand the difference between a vector space and a field?

Understanding the difference between a vector space and a field is important because they are fundamental concepts in mathematics and are used in various fields, such as physics, engineering, and computer science. Knowing the properties of vector spaces and fields can help in solving mathematical problems and understanding more complex concepts.

Similar threads

  • Linear and Abstract Algebra
Replies
8
Views
731
  • Linear and Abstract Algebra
Replies
7
Views
875
  • Linear and Abstract Algebra
Replies
3
Views
144
  • Linear and Abstract Algebra
Replies
1
Views
774
  • Linear and Abstract Algebra
2
Replies
38
Views
5K
Replies
15
Views
4K
  • Linear and Abstract Algebra
Replies
8
Views
1K
  • Linear and Abstract Algebra
Replies
7
Views
1K
  • Calculus
Replies
4
Views
341
  • Linear and Abstract Algebra
Replies
16
Views
2K
Back
Top