How are vector spaces viewed as universal algebras?

Are C and R^2 isomorphic as vector spaces?

• I feel the answer is clearly "yes." But my analysis teacher disagrees; I can't fully speak for him, but he says the question has no meaning, as (I believe) he sees C as a one-dimensional vector space over the field C, while he does not see C as a two-dimensional vector space over the field R. See: http://en.wikipedia.org/wiki/Complex_numbers#Real_vector_space He also says: "Check out how they define isomorphism:" http://en.wikipedia.org/wiki/Vector_space#Linear_transformations I want to understand him, but I'm having trouble doing so. Can anyone help?

• Answer:

  If one considers C as a 2-dimensional vector space over R (this is the complex plane representation of complex numbers), then it is trivially isomorphic to R^2 (the real plane) by mapping the complex number (a, b) to the real pair (a, b). The vector space structure is trivially preserved (with respect to real-linear morphisms). If it was not isomorphic, there would be no complex plane representation and we'd be stuck working with an exotic 1-dimensional structure. See http://mathworld.wolfram.com/ComplexPlane.html and http://en.wikipedia.org/wiki/Complex_numbers#Real_vector_space .