Commutative axiom in vector spaces

Question

With $$u, \space v \in (E,+,\cdot); \space a \in R$$ where E is a vector space in the real numbers, addition is defined as: $$u+v = u-v$$ and the multiplication as $$a\cdot u = (-a) \cdot u$$ How can I prove the commutative axiom $$u+v = v+u$$ in addition?

Answer 1

You can't: $u-v\neq v-u$, unless $u=v$.