For example: $T: P_2 \to P_5$ given by $T(p(x)) = x^3(p(x))$
How would I find the kernel and range for this?
All I know is that $\dim(\ker T)+ \dim(\operatorname{image}T)= \dim(p_2)$.
Therefore, $\dim(\ker T)+ \dim(\operatorname{image}T)=3$?
Also, how do I represent T by a matrix with respect to bases for the domain and codomain?
Any help at all is appreciated, as I really have absolutely no idea what I am doing.
Thanks in advance!