Why is isomorphism of vector spaces a linear bijection (not just bijection)?

Question

I know that isomorphism of groups is a bijection between two groups, and we can think of isomorphism as a relabeling of elements in a group. With that in mind, why is isomorphism of vector spaces defined as a linear bijection between two vector spaces? If isomorphism is just a relabeling, then it seems like that we don't need the bijection to be linear at all.

Answer 1

You are wrong from the start, when you write that “isomorphism of groups is a bijection between two groups”. It isn't. It's a bijection which is also a group homomorphism. Why should it be different with vector spaces?

Answer 2

If we didn't require isomorphism to be linear, $\mathbb R$ and $\mathbb R^2$ would be isomorphic. And any $\mathbb R^n$, for that matter. Without respecting the linear structure, the study of vector spaces would collapse to the study of underlying sets. There would be no linear algebra, just set theory.

You say that isomorphisms are relabeling. But it's not just that we are giving different names to the elements of vector space, these relabeled elements have to behave the same way as the originals, that is, if $f\colon V\to W$ is bijection, we need that $f(x)+f(y) = f(x+y)$ and $\alpha f(x) = f(\alpha x)$, otherwise we wouldn't have really relabeled vectors, we would've just scrambled them and lost the structure in the process.

Answer 3

In every category (think in mathematical structures) an isomorphism is a morphism that have an inverse that is also a morfism. In the case of linear spaces for example, you only need linearity and a biyective correspondence. (Note that this is no true in every category, for example in topological spaces, a morfism is a continuous function, and if it is biyective, it is not necessary an homeomorphism (isomorphism of topological spaces)