A basis for a vector space has three properties:
1) It spans the vector space.
2) The vectors in it are independent.
3) The number of vectors in any basis for the same vector space have the same number of vectors.
Further, you can show that if any two of those is true, the third is also.
A vector space is defined to be finite dimensional if it has a finite basis and, in that case, the dimension of the vector space is defined to be the number of vectors in a basis.
I have no idea what you mean by "elements/components" if not the vectors themselves.