An linear map $T: P_2 \to P_2$ is defined: $T(p(x)) = p(x+1)$. If $a = \operatorname{rank}T$ and $b = \operatorname{rank}(T- I)$ where $I$ denotes the identity transformation, then find $(a,b)$.
This is my problem in my textbook. In my thinking, $\operatorname{rank}T$ is equal to rank of $(p(x)) = 3$. But, I think it 's wrong, because I haven't used $p(x+1)$ yet. At b, I really don't know how to find.
Thanks :)