Existence of spanning set for every vector space

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Discussion Overview

The discussion revolves around the existence of a spanning set for every vector space, exploring whether such a set must consist of the vectors in the space itself and the implications of dimensionality and the axiom of choice.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that the span of any subset of vectors in a vector space is a subspace, leading to the question of whether every vector space has a generating set.
  • One participant notes that while all vectors in a vector space can trivially form a spanning set, the question becomes more interesting with additional requirements on the spanning set.
  • There is a mention of the distinction between finite and infinite dimensional vector spaces, with some participants indicating that the axiom of choice is assumed in the discussion.
  • Another participant raises the complexity of the question when considering the existence of a countably infinite set of vectors that spans a space with uncountably infinite vectors.
  • One participant states that the assertion "every vector space has a basis" is equivalent to the axiom of choice, suggesting a reliance on this principle in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the nature of spanning sets, particularly regarding the implications of dimensionality and the axiom of choice. The discussion remains unresolved with multiple competing perspectives on the requirements for a spanning set.

Contextual Notes

The discussion highlights the dependence on definitions of vector spaces, particularly regarding dimensionality and the assumptions made about the axiom of choice. There are unresolved questions about the properties that a spanning set must have.

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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