Is the Span of an Empty Set Always Empty? Logic Homework Problem

[annoymage]

Homework Statement

State that is either true or false

The span of an empty set is empty set

Homework Equations

n/a

The Attempt at a Solution

from definition, when you span a subset of vector space, the subset must be non-empty.
so, how should i answer this question? should i answer "true"? since the premises is undefined, so either the consequence is true or false, the statement is still true. is that correct?

 

[Raskolnikov]

No, the span of a set of vectors is the smallest subspace containing those vectors. However, the empty set is not a subspace. Since the empty set contains no vectors, it'd make sense for it to span the smallest subspace as well. What's the smallest subspace you can think of?

 

[foxjwill]

I guess you might want to say false, since, according to the definition you were given, the span of the empty set is undefined, and undefined is not the same thing as the empty set. However, I'd bring that up with your teacher.

If, however, the definition was along the lines of "The span of a subset S of a vector space V is the set of all linear combinations of elements of S", then you'd have to say True. To see why this is, notice that you can rewrite the definition as "A vector v is in span(S) if and only if v is a linear combination of elements of S". But if S is the empty set, there aren't any such vectors! Thus, span(S) must be the empty set.

 

[Office_Shredder]

It might help to observe that the "empty summation", i.e. the summation involving no terms, is often considered to default to 0

 

[foxjwill]

However, the empty set is not a subspace.

Shoot! you're right!