Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar $\alpha$ of R.
There are thoughts about theorem of axiomatizability: let К is some class of model of some signature. Class $К$ is axiomatizable <=> it closed under isomorphism, elementary submodels and ultraproduct.
That is, we can construct isomorphism between models of one-dimensional and two-dimensional vector space over R, although two-dimensional vector space isn't in our class.
The class of one-dimensional vector spaces over the reals is not an elementary class in your language — it is the not the collection of models of any theory — because it does not respect the Löwenheim-Skolem theorem. The collection of models of any fixed theory with an infinite model contains models of arbitrarily large cardinality, but all one-dimensional vector spaces over $\mathbb{R}$ have size continuum.