prove v and u ortogonal using coprime polynomials
Can't figure out the direction to solve the following question:
1. T normal transformation in finitly created vector space F with inner dot product defined on it.
2. M1(t) and M2(t) two coprime polynomials.
3. let u,v be two vectors in F such that M1(T)u=0 and M2(T)v=0 (T is substituted for t)
prove that u and v are ortogonal (to each other)
So far i tried to use a result saying that for given 2 coprime polynomials M1 and M2 there exists 2 polynomials Q1 and Q2 such that M1(t)Q1(t)+M2(t)Q2(t)=1
Using this i can show that:
1. Q2(T)M2(T)u = u
2. Q1(T)M1(T)v = v
I don't know how to continue with this. I assumed i need to use somehow the fact that T is normal that is TT* = T*T.
My direction was to start from (u , v) = (Q2(T)M2(T)u , v) = (u, [Q2(T)M2(T)]*v ) = the idea is some how to reach to expression with ...M2(T)v... which is 0...
Can't figure out how to show this or even if this is a good plan...
Appreciate giving me hints on this