Free Vector Space Explained: Geroch & Matheinste

[matheinste]

Hello everyone.

I came across the term free vector space in a book on mathematical physics by Geroch but cannot find them in any other of my books. Can someone give me an explanation of how a free vector space differs from a standard vector space. Geroch says that any set can be made into a free vector space but is this not true for a standard vector space by considering the set members as vectors and defining appropriate vector addition and scalar multiplication laws and a zero vector ?

Thanks Matheinste

 

[matt grime]

You can define a free vector space starting from the underlying set of any vector space, but it will of course have no relation to the one you started with.

If you start with a set of cardinality c, then the free vector space over it will have _dimension_ c.

 

[mathwonk]

the adjective free, in matts disc ussion, and in your question, is part of a phrase, "free on a given set".

i.e. there is no such thing as a free vector space, or rather all vector spaces are free so the term is uncommon, but the phrase this vector space is "free on the set S", just means the set S is the basis.

so every vector space is free on some set, namely on any basis. and what matt is saying is that conversely, given any set S, there is a vector space of dimension equal to the cardinality of S, which is free on S.contrariwise, given any set S and any commutative ring R with 1, there is amodule which is free on S, but now it is no longer true that all R modules are free on some set, since some R modules have non trivial annihilator ideals. such as R/I where I is a nontrivial ideal.

 

[matheinste]

Thankyou both for your answers which more or less confirmed what I thought was the case. This has cleared the problem up nicely.

Thanks again Matheinste.

 

[mathwonk]

put another way, the forgetful functor F from k-vector spaces to sets has an adjoint functor Fr, the free functor from sets to k-vector spaces. this free functor has the property that every k-vector space is isomorphic to something in the image of Fr.

this is an in your face way of saying that every vector space has a basis, every set is a basis of some vector space, and that linear transformations are determined by what they do to a basis, and you can have that be anything you like.

 

[matheinste]

hello all.

It would help my understanding if i knew the motivation for the construction of free vector spaces over a set and also the uses which i have been asssured they have.

Any help would be gratefully received.

Matheinste.

 

[matt grime]

Every vector spaces is a free vector space, so they're automatically interesting. Plus this formal sum idea gives you group algebras, for example. It is a way of adding more structure to something - turning a group into an algebra.

 

[matheinste]

Thankyou for your reply Matt

I have read some more sources and what i originally thought seems to be correct ie all sets can be made into vector spaces with the required definitions of vector additon etc. What threw me was the seemingly complicated manner in which the sources i read derived these free vector spaces. i understand that some things that are intuitively obvious are not always correct hence the need for ( sometimes complicated ) rigorous definition. Its the necessary rigour that i can't always follow.i hope this will come with experience.

Matheinste.

 

[matheinste]

Hello all.

I have attached an extract from A Course in Modern Mathematical Physics by Szekeres. My question is why can we not just use the given set as a basis as it stands without the other "operations".

View attachment FreeVectorSpace.pdf

Matheinste.

 

[Hurkyl]

I don't really understand the question...

I will note that, for their definitions, the elements of S are not even vectors in F(S).

 

[matheinste]

Thanks Hurky.

I see what you mean. for the "intuitive" first part of the definition an arbitrary set is used and for the "rigorous" second part a set of functions is used.

I think i need to stop and think exactly what i want to know before i can expect others to be able to help me.

I will go backwards a few paces and ask a very basic question. What i have learned from this thread is that all vector spaces are free vector spaces (on some set?). If i am given a vector space can i call it, as it stands, without doing anything to it, a free vector space on a set.

Matheinste.

 

[mathwonk]

but perhaps we have not yet made fully clear the importance of that statement. in my response above i mentioned that a linear map on a free space on the set S is determined by what it does to that set. that is the importance of freeness.

i.e. the main job in mathematics is to define maps, and that job is as easy as possible only for functions with no structure to preserve. free spaces transform the job of defining structure preserving maps on the space Fr(S) into arbitrary functiions on the basis S. thus they solve the problem of defining linear maps in as simple a way as possible.

the general concept of adjoint functors describes this process of equating maps of one kind with maps of another kind.

i.e. the adjointness of the free and forgetful functors Free and For says that

linear maps from Free(S) to W, where Free(S) is the vector space free on the basis S, are the same as arbitrary functions from S to the underlying set For(W) of the vector space W.