# List of operations on a set to make it a 2-dimensional vector space

This question might be very silly. I was working on two Examples of Friedberg- Insel-Spence's Linear Algebra. In example $$6$$, in $$\mathbb{R}^2$$(not 2-dimensional real vector space, consider it as the set $$\mathbb{R} \times \mathbb{R})$$ scalar multiplication was defined as usual, but vector addition was defined as the following:

$$(a_1, b_1) +(a_2, b_2)= (a_1+b_2, a_1- b_2)$$ for any $$(a_1, b_1), (a_2, b_2) \in \mathbb{R}^2.$$ The set $$\mathbb{R}^2$$ is closed addition and scalar multiplication, but it's not a vector space over $$\mathbb{R}$$ because $$\mathbb{R}^2$$ is not an abelian group under addition, for instance, this operation is neither commutative nor associative. Moreover, there is an issue with the distribution. Is there any complete list of ways(addition + scalar multiplication) such that the set $$\mathbb{R}^2$$ is a 2-dimensional vector space?

Let $$\pi:\mathbb{R}^2\to\mathbb{R}^2$$ be any $$1-1$$ and onto map. (Any map, not necessarily linear or even additive.)
Now define on $$\mathbb{R}^2$$ a new addition and scalar multiplication as follows: $$a\oplus b:=\pi^{-1}(\pi(a)+\pi(b)), \ \lambda\odot a:=\pi^{-1}(\lambda\cdot\pi(a)).$$
Then $$\mathbb{R}^2$$ is an $$\mathbb{R}$$ vector space of dimension 2 with respect to these operations. You can check the axioms, but as we have just re-labelled all the vectors it is probably "clear" that this is the case.
• Thanks so much. There are uncountably many such $\pi$'s. Commented Jun 17, 2020 at 9:02