The last step in todays lectures proof of the cayley hamilton theorem doesn't make sense to me. No problems up until the point where we showed that the characteristic polynomial we are trying to prove is the zero function is equal to: q(T)*p(T) where p(T) is the zero function.
Our professor then finishes the proof with the following steps: q(T)*p(T)(v)= q(T) performed on p(T)(v) which is equal to zero for any v since p(T)(v) is zero and q(T) beeing a linear function maps zero to zero. What I dont understand is the step q(T)*p(T)(v)= q(T) performed on p(T)(v). Wouldnt it be correct to finsh the proof saying q(T)*p(T)(v)= q(T)(v)*p(T)(v)=q(T)(v)*0=0?