I was working on two Examples of Friedberg- Insel-Spence's Linear Algebra. In example 6, in (not 2-dimensional real vector space, consider it as the set scalar multiplication was defined as usual, but vector addition was defined as the following:
for any The set is closed addition and scalar multiplication, but it's not a vector space over because 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 is a 2-dimensional vector space?