My understanding is that a vector space is defined by the span of two linearly independent vectors. Any two linearly independent vectors that can define the vector space can be said to be the basis for the vector space. For example the vector (1,0) and (0,1) are the basis for the Euclidean plane but using (2,0) and (0,2) would be just as valid a basis for the Euclidean plane, just maybe less convenient. Is this the correct conceptual relationship?
If yes, then what is the difference between span and vector space?