11
$\begingroup$

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.

  • 0
    At the risk of pushing up an already answered question, I'd like to add that since linear span of a set of vectors $S$ is always a vector space and since every vector space contains the $\mathbf{0}$ vector, the linear span of $\mathbf{\phi}$ has to contain $\mathbf{0}$. To prove that it contains only $\mathbf{0}$, it is enough to show it can't contain any other element since those elements aren't present in the set $S$.2017-02-05

4 Answers 4

20

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 $\{ 0 \}$.)

  • 0
    Could you please point me a book that has this definition you just gave?2014-12-02
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.

0

linear span of an empty set i.e L(0) is taken as the set (O),this is confusing because L(0) is the set of all linear combinations of the elements of 0 but to make a linear combination we need to have at least one vector of the set and empty set contains no vectors in it. Thus it it should have been 0 and not (O).

On the other hand, if possible, let L(0) be a set other than (O). Then then it* either contains at least one non zero element (i.e a vector of V) which is a linear combination of the elements of 0.This leads us to a contradiction that 0 is not empty.Hence the possibility of L(0) to be other than (O) is ruled out.

  • or is it self 0.Now if this possibility can be ruled out then the proof becomes complete. reader may please comment.**
0

Looking at the problem from a programatic perspective, if ever I'd like to represent (or generate maybe?) the span of a set of vectors using the linear combinations of vectors definition. I'd always start with a sum variable equating to zero and then iteratively keep on adding to the sum.

If the set would be empty, the script would return the zero vector.

So, Span of an empty set is the zero vector, makes some sense.