Find $ \text{rank}(T) $ and $ \text{nullity}(T) $. If $ T: P_{2} \to P_{1} $ is defined by
$$
T(p(x)) \stackrel{\text{df}}{=} p'(x) + p''(x),
$$
find $ \text{rank}(T) $ and $ \text{nullity}(T) $.

Note that $p'(x)$ for $p\in P_2$ is either linear or constant and $p''(x)$ for $p\in P_2$ is always constant. Now let's consider what $T$ does to the basis. Since $T(1)=0$, $T(x)=1$, and $T(x^2)=2x+2$ the rank of $T$ is maximal (why?). 
The rest is provable by the rank nullity theorem (i.e. $2+N=3$).
