Table of Contents

## Why is the span of the empty set zero?

By definition, the span of a set of vectors is the set of all linear combinations of those vectors. The only possible linear combination of vectors in the empty set is the empty sum, which gives you the zero vector.

### Is an empty set equivalent to 0?

No. The empty set is empty. It doesn’t contain anything. Nothing and zero are not the same thing.

#### What can you say about the linear span of the empty set?

The span of the empty set is the set containing just the zero vector. Theorem: If S is any subset of V , the span of S is the smallest linear subspace of V containing S.

**What is the span of zero vector?**

List of some linear combinations. Let’s list some vector combinations : Zero Vector: span(0) = 0 . One vector: span(v) = a line .

**Why is the span of an empty set zero?**

It has to be 0, because when you add an empty sum to s, you want to get s. An empty operation is always the neutral element for this operation, like an empty product is 1. So here, Span(∅) is the set of all possible empty sums, which is {0}.

## Why should the span of an empty list be 0?

What is the smallest vector space containing the empty set? It’s the smallest vector space since all vector spaces contain the empty set. Therefore, the span of the empty set is the zero vector space.

### Is ø a subset of 0?

Be careful not to confuse Ø with 0. Zero is a number, but Ø is not a number; it is a set that contains NO numbers (or anything else, for that matter).

#### Can a span set be empty?

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.

**What is the span of 0?**

It’s 0 = 0v1 + ··· + 0vn. Moreover, an empty sum, that is, the sum of no vectors, is usually defined to be 0, and with that definition 0 is a linear combination of any set of vectors, empty or not. b. The span of the empty set ∅ is ∅.

**Is 0 an element of a span?**

Yes. Depending on your definition of span, it is either the smallest subspace containing a set of vectors (and hence 0 belongs to it because 0 is a member of any subspace) or it is the set of all linear combinations in which case the empty sum convention kicks in.