Question: given $T: P_2 →P_3$ be linear transformation given by,
$T(p(x))= (x+1)p(x)$
Find the range(T) and basis for it.
My attempt: since $B =\{1, x, x^2\}$ form basis for $P_2$
So $Range(T)= span\{T(1), T(x), T(x^2)\}$
$= span\{x+1, x^2+ x , x^3 + x^2\}$
$ = \{a(x+1)+ b(x^2 +x)+ c(x^3 + x^2) |a, b, c ∈R \}$
and hence
$Basis$ for $range(T) = \{x+1, x^2+ x, x^3+ x^2\}$
Is my attempt/answer is correct? (but it does not matches with key. May be key is wrong!)