## Does a subspace contain the zero vector?

Every vector space, and hence, every subspace of a vector space, contains the zero vector (by definition), and every subspace therefore has at least one subspace: …

It is closed under vector addition (with itself), and it is closed under scalar multiplication: any scalar times the zero vector is the zero vector..

## What is the span of a subspace?

In linear algebra, the linear span (also called the linear hull or just span) of a set S of vectors in a vector space is the smallest linear subspace that contains the set. It can be characterized either as the intersection of all linear subspaces that contain S, or as the set of linear combinations of elements of S.

## Can a subspace be empty?

2 Answers. Vector spaces can’t be empty, because they have to contain additive identity and therefore at least 1 element! The empty set isn’t (vector spaces must contain 0). However, {0} is indeed a subspace of every vector space.

## Can 0 be a basis?

No. A basis is a linearly in-dependent set. And the set consisting of the zero vector is de-pendent, since there is a nontrivial solution to c→0=→0. If a space only contains the zero vector, the empty set is a basis for it.