Span of an empty set is the zero vector

Solution 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 $\{ 0 \}$.)

Solution 2:

The span of a set D is the smallest subspace containing the elements of D. Now, every subspace contains 0. Thus if D is a null set the span of D can only be the subspace containing 0.

Solution 3:

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."