Answer
$dim[Ker(T)] \in \{2,3,4,5\}$
Work Step by Step
Since $T$ is an isomorphism, we can obtain:
$Rng(T)\ne P_3(R) \rightarrow Rng(T) \subset P_3(R)$
then $dim[Rng(T)] \lt dim[P_3(R)]=4$
According to Rank-Nullity Theorem:
$dim[Ker(T)]+dim[Rng(T)]=dim[P_4(R)] \\
dim[Ker(T)]+dim[Rng(T)]=5\\
dim[Ker(T)]=5-(\lt 4) \gt 5-4=1 $
(1)
We also have $Ker(T) \subseteq P_4(R)$ $\rightarrow dim[Ker(T)] \leq [dimP_4(R)] \\
\rightarrow dim[Ker(T)] \leq 5$ (2)
From (1) and (2), $dim[Ker(T)] \in \{2,3,4,5\}$
