If the set $V$ is defined by the points that go through the origin in $\mathbb{R}^2$ that satisfy the equation $ax+by=0$ then show $V$ is a vector space.
Resolution:Proving that $V$ is closed under addition and scalar multiplication, I know how to do this. How do I prove that addition is commutative $A+B=B+A$ , what resolution is correct?
Define $A=(x_1,y_1), B=(x_2,y_2)$
$1) \: A+B=(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)=(x_2+x_1,y_2+y_1)=(x_2,y_2)+(x_1,y_1)=B+A$
$2) \: A+B=(x_1+x_2,y_1+y_2)$, subs in to the line equation $a(x_1+x_2)+b(y_1+y_2)=a(x_2+x_1)+b(y_2+y_1)$
$ B+A=(x_2+x_1,y_2+y_1)$ sub that in to the line equation gives $a(x_2+x_1)+b(y_2+y_1)$ and that is what we have above so
A+B=B+A
Note: to prove that $v$ is closed under addition and scalar multiplication the second method is used.
What the right way to prove this?
(As an aside do I have to prove to each of the sets to prove that are vector spaces besides proving closed under addition and scalar multiplication do i have to prove 8 more ,like the one above and i.e. $c(u+v)=cu+cv$ being $u,v$ vectors and $c$ a scalar) (And hence memorinzing 8 more axioms)