Answer
$T$ is a linear transformation.
Work Step by Step
A transformation is said to be linear if $T(av+w)=aT(v)+T(w),$ for all $v,w$ in the given domain and for all scalars $a$.
Let $a_0+a_1x+a_2x^2,b_0+b_1x+b_2x^2$ be arbitrary elements of the domain $P_2$ and $k$ be any arbitrary scalar. Then:
$\begin{array}{l}
T(k({a_0} + {a_1}x + {a_2}{x^2}) + ({b_0} + {b_1}x + {b_2}{x^2}))\\
= T((k{a_0} + k{a_1}x + k{a_2}{x^2}) + ({b_0} + {b_1}x + {b_2}{x^2}))\\
= T((k{a_0} + {b_0}) + (k{a_1} + {b_1})x + (k{a_2} + {b_2}){x^2})\\
= (k{a_1} + {b_1}) + 2(k{a_2} + {b_2})x\\
= (k{a_1} + 2k{a_2}x) + ({b_1} + 2{b_2}x)\\
= k({a_1} + 2{a_2}x) + ({b_1} +2 {b_2}x)\\
= kT({a_0} + {a_1}x + {a_2}{x^2}) + T({b_0} + {b_1}x + {b_2}{x^2}) \end{array}$
Hence, $T$ is a linear transformation.
