Is the Empty Set a Valid Vector Space? A Closer Look at the Ten Axioms

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Discussion Overview

The discussion revolves around whether the empty set can be considered a valid vector space, examining its compliance with the ten axioms of vector spaces. Participants explore theoretical implications, definitions, and the nature of vector spaces in relation to the empty set.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants argue that a vector space must contain vectors, suggesting that the empty set does not qualify as a vector space.
  • Others propose that the empty set can be considered a vector space because it is a subset of every set, and thus can satisfy the axioms through vacuous truth.
  • One participant notes that the empty set lacks an additive identity, which is essential for a vector space.
  • Another viewpoint suggests that the span of the empty set is the zero vector, indicating that it can still relate to vector spaces in this way.
  • Some participants discuss the implications of the empty set being a subset of every set, debating the definitions and properties associated with subsets.
  • There are mentions of the summation of an empty set leading to the zero vector, highlighting the nuances in linear combinations.

Areas of Agreement / Disagreement

Participants express differing opinions on whether the empty set can be classified as a vector space. While some assert it cannot due to the absence of a zero vector, others argue that it can be considered a vector space under certain definitions. The discussion remains unresolved with multiple competing views.

Contextual Notes

Participants reference definitions and properties of vector spaces, but there are unresolved assumptions about the implications of the empty set's properties and its relationship to the axioms of vector spaces.

  • #31
Thanks for that, I was quite incorrectly equating {0} with { }.
 
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  • #32
Adriadne said:
Oh dear, I hope as a neophyte, I'm not making a fool of myself.
OK look. Grant me that, if the empty set is a trivial subset of V, then it must, by the definition, contain the identity. So, is there no sense, in the case that the set V is a vector space, that the zero vector can be identified with the identity?

No, I won't grant you that! Saying the empty set is a subset is not the same as saying it is a subspace!
 
  • #33
Adriadne said:
Thanks for that, I was quite incorrectly equating {0} with { }.
You're welcome.

Have fun here at PF!
 
  • #34
phoenixthoth said:
Have fun here at PF!
Ha! Be careful what you say, I have a zillion half-assed questions up my sleeve!
 

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