1)A)T is a normal in V inner product space

there is $\displaystyle M_{1}(t),M_{2}(t)$ different polinomial $\displaystyle u,v\in V$

prove that if $\displaystyle M_{1}(T)u=0$ and $\displaystyle M_{2}(T)v=0$ then u orthogonal to v?

how i tried:

i have defined $\displaystyle Q_{1}$ and $\displaystyle Q_{2}$and proved that v is eigen vector of $\displaystyle M_{2}(t)Q_{2}(t)$ of eigenvalue 0.

and u is eigen vector of $\displaystyle M_{2}(t)Q_{2}(t)$ of eigenvalue 1.

so if $\displaystyle M_{2}(t)Q_{2}(t)$ is normal then u is orthogonal to v.

but how to prove that M_{2}(t)Q_{2}(t) is normal?

i cant show here that TT*=T*T

MQ(MQ)*=MQQ*M*

(MQ)*MQ=Q*M*MQ

those two are not the same

??