Existence of spanning set for every vector space

[Mr Davis 97]

I know that the span of any subset of vectors in a vector space is also a vector space (subspace), but is it true that every vector space has a generating set? That is, the moment that we define a vector space, does there necessarily exist a spanning set consisting of its vectors?

 

[Stephen Tashi]

That is, the moment that we define a vector space, does there necessarily exist a spanning set consisting of its vectors?

Trivially, you can take all vectors in the vector space as a spanning set. To make the question interesting, you must put more requirements on the kind of spanning set that you want.

Some textbooks declare that by the phrase "vector space", they will mean a "finite dimensional vector space". How do your course materials use the phrase "vector space"?

 

[pwsnafu]

Are we assuming axiom of choice?

 

[Mr Davis 97]

A vector space can be finite or infinite dimensional, and we are assuming the axiom of choice.

 

[Stephen Tashi]

A vector space can be finite or infinite dimensional, and we are assuming the axiom of choice.

The set consisting of all vectors in the vector space spans the vector space. The question becomes more complicated if you demand a set that spans the space and has special properties. For example, if there are an uncountably infinite number of vectors in the vector space then does there exist a countably infinite set of vectors that spans the space?

 

[pwsnafu]

The statement "every vector space has a basis" is equivalent to AC. So just use that.