Vector space and axioms

Question

How would i go around proving a(u+v)+ u= (a+1)u+av using axioms? I started with distributivity with respect to vector addition, associativity of addition, commutativity of addition, distributivity with respect to field addition. I think i am missing one axiom.

Answer 1

distributive law says $(a+1)\mathbf{u}=a\mathbf{u}+1\cdot \mathbf{u}$. Identity law says that $1\cdot \mathbf{u}=\mathbf{u}$.

As such you see that

$$(a+1)\mathbf{u}+a\mathbf{v}=(a\mathbf{u}+\mathbf{u})+a\mathbf{v}$$ $$=a\mathbf{u}+(\mathbf{u}+a\mathbf{v}) = a\mathbf{u}+(a\mathbf{v}+\mathbf{u})$$ $$=(a\mathbf{u}+a\mathbf{v})+\mathbf{u} =a(\mathbf{u}+\mathbf{v})+\mathbf{u}$$