Question

definition of vector space

Answer 1

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

Answer 2

Just in case its not clear enough, on a vector space linear operations you do, such as adding 2 vectors, scaling them, etc. will still ensure you stay on that vector space.

For example, a plane in R^n passing through the origin is a vector space. For any vectors on the plane, adding, scaling etc etc will still leave you on the plane. This shows you why the plane must pass through the origin. Suppose the plane doesn't pass through the origin. Now take some vector v, v + (-v) = 0, which somehow shows you have jumped out of the vector space. A contradiction.