Proof that solutions can be "translated" between two isomorphic spaces

Question

When we face an analytic problem, It is generally possible to represent it on the Cartesian plane, solve it, and than convert the solution back to the related algebric solution.

I guess that we can do this because there exists an isomorphism between real numbers and the points on a plane. Does a general proof exist for this fact? That is to say, is it always safe to solve a problem in an isomorphic space, and than claim that this represents the solution also in the original space?

EDIT

I'll state the question for my specific case, hoping to be more clear. If I represent an analytical problem on the Cartesian plane (converting the numbers to points on the plane, as usual) and then solve it graphically, how can I be sure that this solution holds true when I convert it back to the analytical?

Answer 1

The concept of vector space isomorphism (for finite dimensional vector spaces) is purely dependent on their dimension. So if you are considering the vector space $\mathbb R$ over the field $\mathbb R$, and the vector space $\mathbb R^2$ over the field $\mathbb R$, then they are not isomorphic: the first one has dimension equal to $1$ and the second has dimension equal to $2$.

I am also not quite sure what you are trying to ask. If it helps you can think of real line in $\mathbb R^2$ as the elements of the type $(x,0)$, which would be a subspace of $\mathbb R^2$, but they still wouldn't be isomorphic.

Answer 2

is it always safe to solve a problem in an isomorphic space, and than claim that this represents the solution also in the original space?

In a certain/some sense(s), the only 'meaningful' problems one can ask of a (class of) space(s)/structure(s)/whatnot(s) are precisely those whose solutions can be translated between them by the relevant isomorphisms

To try to keep it in the context you're interested: the only meaningful problems one can ask of $\mathbb{C}$ as a real vector space are those that can also be asked of $\mathbb{R}^2$, and whose solutions are the same for them both. It doesn't make much sense, in this context, to speak of, say, algebraic closedness and other properties of $\mathbb{C}$ as a ring or field, for $\mathbb{R}^2$ is not even a ring/field to begin with! Or, for a topological example, it's not always meaningful to ask or speak of boundedness among metric spaces, as $(0,1)$ and $\mathbb{R}$ are 'continuously isomorphic', but only one of them is bounded: it only makes sense when the notion of isomorphism is changed/restricted to 'isometry'

This may be a bit distressing at first, as it may seem that one has to know the answer to a problem before even asking it (!), but this is not really the case, and, with some amount of time and practice, one develops a sense for recognizing contexts and asking the appropriate kind of problems