Linear algebra 2

Question

Suppose that you can see that a subset of a vector space does not have the zero vector in it. Why is it a trivial consequence of this that the subset is not a subspace?

Answer 1

A vector subspace is a subset of a vector space that meets the properties of a vector space.  That includes having an additive identity, i.e. a zero vector in the subspace, call it 0, such that for any vector A in the the subspace,

 

A + 0 = 0 + A = A

 

0 must be in the subspace, or else you do not have a subspace.  It is a matter of definition.