I'm going through the text "Linear Algebra Done Right" 2nd edition by Axler. Made it to chapter 4 with one problem I'm unable to understand fully. The theory that two vector spaces are isomorphic if and only if they have the same dimension. I can see this easily in one direction, that is, isomorphic vector spaces will have the same dimension, but it seems I can imagine vector spaces having the same dimension but for which they are not isomorphic. Perhaps one vector space would be 2-tuples over the integers but the with modular operators limiting the number of elements in the space and another vector space of integer 2-tuples without modular operators. The latter space would have more element in it so how would any map from the former to the latter be surjective? And yet do they not have the same dimension? I'm guessing he is making assumptions about the operations on the vector spaces and they are over R or C. However, this assumption is made on many of the theories this particular theory doesn't state V explicitly and so I'm not sure what part of this theory the assumptions fall under. Any clarification on this would be appreciated.