The midpoint $M$ of points $A$ and $B$ is given by the equation $M - A = B - M$. Prove that $M = \frac{1}{2} (A + B)$ using only the vector space axioms.
I'm not sure how to start to get the ball rolling. I have tried starting with $$M = M + 0 = M + A + (-A) = B - M + A $$ But I don't know how I'm going to end up getting rid of the M on the right side. What trick am I missing here?
$A$ and $B$ and $M$ are points from the vector space. So, adding $M$ to both side of the equation $M-A=B-M$ gives, $M-A+M=B-M+M$, or, $2M-A=B$ (as, $-M$ is the inverse of $M$, $-M+M=0$). Again, adding $A$ to both side you will get $2M-A+A=B-A+A$, by same argument, $2M=B+A$. As the associative set is a field in case of vector space, we can take multiplicative inverse of scalar. Hence, $\frac{1}{2}\cdot 2M=\frac{1}{2}(B+A)$, or, $$M=\frac{1}{2}\big(A+B\big)$$We can write $B+A=A+B$, as, addition of vectors is commutative in a vector space.