[Math] Does every vector space contain a zero vector

Question

If this is true, then every vector space must always have at least one subspace, the one consisting of only the zero vector, correct?

Thanks!

Answer 1

Yes, and yes, you are correct.

The existence of a zero vector is in fact part of the definition of what a vector space is.

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:

The subspace containing only the zero vector vacuously satisfies all the properties required of a 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.