Why would it not be intuitive? A subspace, U, of vector space V must be closed under addition and scalar multiplication. If v is in subspace U, the (-1)v = -v is also. Then v+ (-v)= 0 is in U. That is, the zero vector of V, 0, is in U. But it is easy to show that the zero vector of a unique. Since 0 is in U and, of course, v+ 0= v for all v in U, there cannot be another zero vector in U.
If each subspace has its own zero vector, then combine these subspaces in order to get a bigger subspace or even the whole space. We will get bunch of different zeros and the whole space will very entertaining, suddenly disappearing elements and discontinuities...
Also note that these are the rules of the game that are required to have, rather than anticipating their existence based on intuition.
by definition a subspace have to be a vector space, and then all the other peoples arguments holds. What you are sugesting is just a simple subset, but that is not so interresting i linear algebra because it don't have the vector space properties.
by the way: The space in subspace means vectorspace, so it should really say subvectorspace. But a subset isn't a space so thats why there is no ambiguity.