Could it be argued that the set containing only the origin is a "vector space"?

Question

From the axioms I remember about vector spaces that are included on wiki, it seems that the origin itself could be called a vector space. I think traditionally its thought of as the smallest subspace (and a rather boring one) but I'm curious to know if I can claim that its a vector space strictly based off the definition of vector space.

Qiaochu Yuan · Accepted Answer

Yes, this is the zero vector space. It satisfies all of the axioms, as you can check for yourself. A nice exercise is to compute its dimension and in particular to verify explicitly that all of its bases have the same size. (This will require being very clear about the definitions of linear independence and of spanning.)

Correct answer by Qiaochu Yuan on November 26, 2020

Could it be argued that the set containing only the origin is a "vector space"?

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