Is the Empty Set a Vector Space?

[bonfire09]

In the book it states that the span of the empty set is the trivial set because a linear combination of no vectors is said to be the 0 vector. I really don't know how they came up with at? Is it just defined to be like that?

After doing some research, I figured that since the empty set is a subset of every set and that the zero vector is a subspace of every vector space that means that the span({})={0}?

 

[mfb]

Is it just defined to be like that?

Yes. It is a convenient consequence from the definition of spans.

 

[VantagePoint72]

The zero vector is in the span of any well-defined set of vectors over any field, since zero (which must be in any field) times any vector is the zero vector. Since in set theory it's useful to have the empty set as a well-defined set, it's necessary that the zero vector be in its span. Clearly nothing else is, so the span is {0}.

 

[HallsofIvy]

Perhaps the difficulty is the misunderstanding reflected in your title, "empty set as vector space?". We, and your quote, are not saying that the empty set is a vector space, we are saying that it spans a vector space containing only the single vector, 0.