Undergrad Existence of spanning set for every vector space

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SUMMARY

Every vector space, whether finite or infinite dimensional, has a spanning set consisting of its vectors. The span of any subset of vectors in a vector space is a subspace, and the set of all vectors in the vector space trivially spans it. However, the complexity arises when additional properties are required for the spanning set, such as countability. The existence of a basis for every vector space is equivalent to the Axiom of Choice, which is a critical consideration in this context.

PREREQUISITES
  • Understanding of vector spaces and their properties
  • Familiarity with the concept of spanning sets
  • Knowledge of the Axiom of Choice in set theory
  • Basic comprehension of finite and infinite dimensional spaces
NEXT STEPS
  • Explore the implications of the Axiom of Choice on vector spaces
  • Study the concept of bases in vector spaces and their properties
  • Investigate the differences between finite and infinite dimensional vector spaces
  • Learn about countable versus uncountable sets in the context of linear algebra
USEFUL FOR

Mathematics students, educators, and researchers interested in linear algebra, particularly those focusing on vector space theory and the implications of the Axiom of Choice.

Mr Davis 97
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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?
 
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Mr Davis 97 said:
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"?
 
Are we assuming axiom of choice?
 
A vector space can be finite or infinite dimensional, and we are assuming the axiom of choice.
 
Mr Davis 97 said:
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?
 
The statement "every vector space has a basis" is equivalent to AC. So just use that.
 

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