What does "isomorphic" mean in linear algebra?

Question

My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have the same vector structure. I'm not sure what that means, so I was hoping anyone could explain its meaning to me, using knowledge from elementary linear algebra only. He started discussing it in the current section of our textbook: General Vector Spaces. I've also heard that this is an abstract algebra term, so I'm not sure if isomorphic means the same thing in both subjects, but I know absolutely no abstract algebra, so in your definition if you keep either keep abstract algebra out completely, or use very basic abstract algebra knowledge, that would be appreciated.

Answer 1

An isomorphism refers to a one-to-one and onto mapping. That is, every element in the domain is mapped to one and only one element in the target (one-to-one) and every element in the codomain is in the range.

In linear algebra, this observation is enough to say, for example, that the image under an isomorphism of a linear subspace is a linear subspace. This is an important example of the preservation of structure of an isomorphism.

Hope this has helped!