Final answer:
A surjective linear transformation between two vector spaces with the same dimension is an isomorphism because it is both injective and surjective, preserving the structure of the vector space.
Step-by-step explanation:
A surjective linear transformation between two vector spaces that have the same dimension is an isomorphism.
For a linear transformation to be surjective, every element of the codomain is the image of at least one element of the domain. Since the vector spaces have the same dimension and the linear transformation is surjective, this also implies that the transformation is injective, meaning each element of the domain maps to a unique element in the codomain.
An injective and surjective linear transformation is both a homomorphism (a structure-preserving map) and an isomorphism (a bijective homomorphism).
An isomorphism between vector spaces indicates a one-to-one correspondence that preserves the vector space structure, including addition and scalar multiplication. Therefore, the two vector spaces are not only algebraically equivalent but also geometrically identical; they are essentially the same vector space in different forms or 'in different clothes.'
Therefore answer is (a) Isomorphism.