How can I prove the associative law (multiplication) using vectors:
m(na) = (mn)a where m and n are real numbers, and a is a vector?
There are three cases to prove:
m>0, n<0 and m<0, n>0
I have no idea where to start.
I'm going to change your scalars to k and l, since n is normally used to signify the length of a vector.
Since you don't know the dimension the vector is in, you have to use infinite sequence notation:
a = <a0, a1, .... an>
k(la) = k(<la0, la1, .... lan>) = <kla0, kla1, ... klan>
l(ka) = l(<ka0, ka1, ... kan>) = <kla0, kla1, ... klan>
And are thus, equal.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."