Question

Can someone explain me the difference between vector spaces and subspaces?

Answer 1

this is what wiki says:
A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0-vector of V) is called a subspace of V. Subspaces of V are vector spaces (over the same field) in their own right.

http://en.wikipedia.org/wiki/Vector_space#Subspaces_and_quotient_spaces

Answer 2

In simple terms, the difference is that a subspace occupies a certain space within a vector space.