Are any of these in the kernel? Let $T:P_2 \rightarrow P_3$ be the linear transformation with rule $T(p)(t) = tp(t)$. Which of the following (if any) are in the kernel of $T$?


*

*$p_1(t) = t^2$

*$p_2(t) = 0$

*$p_3(t) = 1+ t$


Here is my question. To be in the Kernel of $T$, $T(u) = 0$. Would that mean only the second function would be in the Kernel, since it is the only function listed which results in $T(u) = 0$. Or does this mean that they are all in the kernel, because they all can result in $0$, if $t = 0$.

No, only 2. tells you an element of the kernel. You do not worry about evaluations (inserting a value for $t$ and see what you get). That is what linear maps $P_n\to \Bbb Z$ are for.
The relation $T(u)=0$ is an equality of polynomials in one variable(or vectors, if you will), not an equation with one unknown.
