Differences Between Free Basis and Basis: A Comprehensive Guide

[pivoxa15]

Homework Statement

What are the differences between the two?

 

[maverick280857]

Do you mean ordered basis?

 

[pivoxa15]

Don't think so.

 

[quasar987]

Just surfing wiki, I get the impression that there is none. Do you have a reason to believe that there is a difference?

 

[maverick280857]

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).

 

[pivoxa15]

I am talking about it in the sense of modules.

 

[quasar987]

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,[...] "

 

[matt grime]

Vector spaces are modules for fields. All modules for fields are free, so in this case the two notions agree.