How can a vector space not contain the 0 vector?

I know a subspace must be a vector space that contains the zero vector and is closed under addition and scalars, but I can't think of a vector space that does not contain the 0 vector.|||The span of vectors { v1, v2, ... , vn } is the set of all vectors of the form:





      a1 v1 + a2 v2 + ... + an vn





where a1 ... an are all scalars.





However, no matter what your set is, you can choose a1=a2=...=an=0.





That gives:





      a1 v1 + a2 v2 + ... + an vn = 0 v1 + 0 v2 + ... + 0 vn = 0 + 0 + ... + 0 = 0.





That means that 0 is always in the span of any set of vectors.|||having a " zero" vector is one of the criteria of a vector space !!





but note that the "zero" vector may not even have the 'symbol' 0 in it...depends upon the given two operations for the space.|||Try thinking of a plane in R3 that doesn't pass through the origin. This won't contain the zero vector.