I'm just learning a bit about the "minimal polynomial" today but there was a section from the book which I didn't understand. This is the section, and i've circled the bit i'm having trouble with. http://img15.imageshack.us/img15/1825/97503873.jpg [Broken] (sorry, it won't let me post an image for some reason??) Firstly it's a bit unclear to me what they mean by p(T)(v). Would this mean that you take the linear transformation T (or equivalently its matrix), stick it in the polynomial p to obtain a new linear transformation p(T), then perform this transformation on v? Okay, assuming that's correct I can understand that p(T)=0 <=> p(T)(v)=0. But then how does this imply that the minimal polynomial is the least common multiple of all those other ones?! They say it like it's completely obvious! Thanks.