Differences Between Free Basis and Basis: A Comprehensive Guide

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In summary, the conversation discusses the differences between a basis and an ordered basis in linear algebra. The speaker believes that there is no difference between the two and provides evidence from their studies and research on the topic. They also mention that in the context of modules, all modules for fields are free, so the two terms agree in this case.
  • #1
pivoxa15
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Homework Statement


What are the differences between the two?
 
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  • #2
Do you mean ordered basis?
 
  • #3
Don't think so.
 
  • #4
Just surfing wiki, I get the impression that there is none. Do you have a reason to believe that there is a difference?
 
  • #5
quasar987 said:
Just surfing wiki, I get the impression that there is none. Do you have a reason to believe that there is a difference?

Yeah, looks the same to me. This is what I studied in my linear algebra course: there's something called a basis and then something called an ordered basis. A basis with an ordering of elements (somewhat like an indexing set that has been fixed which maps to elements of your basis) is called an ordered basis.

I think when you say free basis, you probably mean a basis without ordering...but that's just a wild guess. I haven't come across this term (free basis) in my reading of any standard textbook on Linear Algebra (cf Hoffman/Kunze).
 
  • #6
I am talking about it in the sense of modules.
 
  • #7
If you look here

http://en.wikipedia.org/wiki/Free_module

they define a free module as a module having a free basis. But then they give the definition of a free basis and it is an exact analogy to what is simply called a 'basis' in linear algebra:

http://en.wikipedia.org/wiki/Basis_(linear_algebra)#Definition

Another reason why I suspect that there is no difference btw the two is the following sentence in the article about modules:

"However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules,[...] "
 
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  • #8
Vector spaces are modules for fields. All modules for fields are free, so in this case the two notions agree.
 

1. What is a free basis?

A free basis, also known as a basis of a vector space, is a set of linearly independent vectors that span the entire vector space. In other words, any vector in the space can be expressed as a linear combination of the vectors in the basis.

2. What is the difference between a free basis and a basis?

The terms "free basis" and "basis" are often used interchangeably, but there is a subtle difference. A free basis refers specifically to a basis of a vector space, while a basis can also refer to a basis of a module or other algebraic structure.

3. How do you determine if a set of vectors is a free basis?

To determine if a set of vectors is a free basis, you can use the following two criteria: 1) the vectors must be linearly independent, meaning that no vector in the set can be expressed as a linear combination of the other vectors, and 2) the vectors must span the entire vector space, meaning that every vector in the space can be expressed as a linear combination of the vectors in the set.

4. Can a vector space have more than one free basis?

Yes, a vector space can have multiple free bases. This is because a vector space can have infinitely many linearly independent sets of vectors that span the space. However, all free bases of a given vector space will have the same number of vectors, known as the dimension of the space.

5. How are free bases used in linear algebra?

Free bases are an important concept in linear algebra because they allow us to express any vector in a vector space in terms of a set of basis vectors. This allows us to perform operations such as vector addition, scalar multiplication, and matrix multiplication in a more efficient way. Free bases are also used in solving systems of linear equations and finding the inverse of a matrix.

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