Practice FT3 Q11 - Looking for a detailed, worked solution with theoretical explanation

Let P3 = P3 (IR) be the space of real polynomials of degree at most 3. Lucas's favourite linear mapping T : Pg -> IRS is defined by (P(-1) T(P) = P(-2) P(-2) a) Select all statements below that are true. 0 x4+2x3 -712 - 20x - 12 6 ker T 0 23 - 3x - 2 6 ker T 0 x3 +5x2 +8x+46 ker T 0 x2 + 3x + 2 6 ker T 0 0 E ker T 0 x3 +3x2 -4x - 12 6 ker T b) By calculating explicitly ker 7' (or otherwise), find the nullity and rank of " and type them in the boxes below. nullity of T = rank of T = c) Evelyn's favourite mapping A : P3 -> IR3 is given by P(-946) A(P) = P(129) P(-21) Lucas thinks that it will be too hard to determine the nullity and rank of A, but Evelyn thinks they can easily be found. Who do you agree with? Provide all reasoning for your answer. Insert judgement and explanation