Span of an empty set is the zero vector

Question

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone".

Is Nering defining the span of the empty set to be the set containing the zero vector or is this something you can prove from the definition of span? I sense it is the latter, but the proof seems a bit tricky since you would be saying that {0} = Span of the indexed set of vectors in the empty set. But since the empty set has no vectors, it is not clear to me what its span would be.

Answer 1

Depending on how you define the span, this is either a definition or it follows from the definition of span (and judging by the wording it is probably the former). What's Nering's definition of span?

(One definition of span is the following: the span of a collection of vectors is the intersection of all subspaces containing them. The span of no vectors is therefore the intersection of all subspaces, which is $\{ 0 \}$.)

Answer 2

It is because Dim({0}) = 0.

Dimension of a vector space refers to the minimum number of basis vector that spans this vector space.

Since Dim({0}) is defined as 0, from the definition of dimension we conclude {0} can be spanned by 0 basis vectors; that is, we must define the span of the empty set as {0} for our definition of dimension to work.

"In the context of vector spaces, the span of an empty set is defined to be the vector space consisting of just the zero vector. This definition is sometimes needed for technical reasons to simplify exposition in certain proofs."