Let $T: P_2\to P_3$ be a linear transformation defined by $T(at^{2} +bt+c) = (a-b+c)t^{3} + (-a+3b-2c)t^{2} + (-a-b)t + (2b-c).$
Find a basis of $\operatorname{range}(T)$.
Would the basis of $\operatorname{range}(T)$ be {$t^{3}$, $t^{2}$, $t$, $1$}? Since they span $T$ ?