Is a subspace still valid without the zero vector?

[jeffreylze]

If a set of vectors does not contain the zero vector is it still a subspace?

 

[DorianG]

No, because the subspace will have negatives of elements,
i.e., for all v an element of V, (-1)v or -v will be an element.
For the subspace to be closed under addition (a necessary requirement)
v + (-v) = 0 must be an element which implies the zero vector must be in a subspace of vectors.

 

[HallsofIvy]

Some textbooks include "contains the 0 vector" as part of the definition of "subspace".
Others just say "is non-empty". As DorianG pointed out, if some vector, v, is in the subspace, then so is -v (a subspace is "closed under scalar multiplication") and so is v+ (-v)= 0 (a subspace is "close under addition").