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

Click For Summary
SUMMARY

The discussion centers on whether the empty set can be considered a valid vector space. Participants assert that while the empty set is a subset of every set, it cannot satisfy the requirements of a vector space due to the absence of a zero vector, which serves as the additive identity. The axioms of vector spaces, particularly those requiring the existence of specific elements, cannot be fulfilled by the empty set. Thus, the consensus is that the empty set does not qualify as a vector space.

PREREQUISITES
  • Understanding of vector space axioms
  • Familiarity with set theory concepts, particularly subsets
  • Knowledge of linear algebra, specifically the definition of span
  • Basic logic principles, including vacuous truth
NEXT STEPS
  • Study the axioms of vector spaces in detail
  • Learn about the concept of span in linear algebra
  • Explore the implications of vacuous truth in mathematical logic
  • Investigate the properties of subspaces in vector spaces
USEFUL FOR

Mathematics students, educators in linear algebra, and anyone interested in the foundational concepts of vector spaces and set theory.

  • #31
Thanks for that, I was quite incorrectly equating {0} with { }.
 
Physics news on Phys.org
  • #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!
 

Similar threads

Undergrad About the definition of vector space of infinite dimension
  • · Replies 18 ·
Replies
18
Views
4K
MHB The axioms of a vector space are satisfied
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Proving inequality about dimension of quotient vector spaces
  • · Replies 25 ·
Replies
25
Views
3K
MHB Subsets of $\mathbb{R}^2$ Satisfying S2 and S3 but Not S1: Empty Set
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Is the Empty Set a Vector Space?
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad How Can a Set Fail the Scalar Identity Axiom in Vector Spaces?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad QM Qubit state space representation by Projective Hilbert space
  • · Replies 18 ·
Replies
18
Views
1K
MHB Determine vec {{x},{y},{3x+2y}} in R^3 form a vec space
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad A few questions about proving vector spaces
  • · Replies 10 ·
Replies
10
Views
3K
MHB Is $\Phi|_{U_2}$ a Vector Space Isomorphism?
  • · Replies 1 ·
Replies
1
Views
2K