>>11771885If all you have is one vector, then you can immediately construct a one-dimensional vector space of scalar multiples. If it's a vector, it's an element of a vector space. However, the same object doesn't have to be a vector if it "lives" in a different space. If you have some object, maybe the function f(x)=x^2 for instance, or some ordered string of numbers like (1,2,3), then calling it a vector is information about the space it lives in.
The space of ordered strings of three real numbers (a,b,c) is a vector space. The space of ordered strings of three real numbers (a,b,c) and a single ordered string (1,2,3,4) isn't. Even though (1,2,3) is in both sets, it doesn't make sense to call it a vector in the latter.
