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 $t = 0$.

The equation $T(u)=0$ is to be read in a way that the right hand side refers to the zero polynomial function, i.e. $f:\mathbb R \rightarrow \mathbb R, t\mapsto 0$, which is equal to zero for all $t$.
Thus, the first interpretation is correct, the result has to be equal to zero for all $t$, not just a particular choice. Thus, only $p_2$ is in the kernel.
To take an analogue with a vector $x \in  \mathbb R^n$: In order to be in the kernel of a matrix $A$, it does not suffice that one of the components of $Ax$ is zero, but the whole vector is.
