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.

 

[blue_raver22]

Yes, the span of the empty set is defined to be the trivial set, which consists of only the zero vector. This is because the definition of the span of a set of vectors is the set of all possible linear combinations of those vectors. Since there are no vectors in the empty set, there are no possible linear combinations and thus the span is just the zero vector.

As for why the zero vector is a subspace of every vector space, it is because it satisfies the properties of a vector space, such as closure under addition and scalar multiplication. This means that any vector added to the zero vector or multiplied by a scalar will still result in the zero vector. Therefore, the span of the empty set, which is defined as the set of all possible linear combinations, will always include the zero vector.

In summary, the empty set can be considered a vector space, but it is simply a trivial one with only the zero vector. This may seem counterintuitive, but it is a consequence of the definitions and properties of vector spaces.