Question

is it possible for a vector space to have more than one basis?

Answer 1

Yes.

Answer 2

sure, for example
[1 0] and [0 1]
[1 0] and [1 1]
are both bases for 2-d space

Answer 3

ok then tell me if S={u1,u2,.....,un} and S'={v1,v2,....,vn} be two bases far a vectorspace V then v1=au1+bu2+......+zun ie every element of V(it can be v1 which is also the element of 2nd basis) can be written as a linear combination of basis vectors u1,u2,....un.
how is it possible that a linearly independent vector i.e v1 in this case can b written as a linear combination of other linearly independent vectors. how???

Answer 4

vectors are independent relative to all others in a set of vectors.
In pictures: