PLEASE HELP

Determine whether the following maps are linear transformations. If the map is a linear transformation, provide a proof that it is linear transformation (verify that (LT1) and (LT2) hold). if the map is a not linear transformation, state one of the properties of a linear transformation that does not hold (either (LT1) or (LT2)) and give a counterexample showing that the property fails. (a) Let P be an invertible n x n matrix, fixed, but unknown. T: MAR) > Mum) A i> P'lAP (b) T I M2(R) > R A +> rank(A) (C) T : .9? + M2(R) f [ (1"(1))2 f(1)f(2) f(1)f(2) (f(2))2 (d) T I .93 > 92 a+b$+cm2+d$3 I> b+(2aC)$+(4C3b)$2 (LT1) Tutv = T(u+T(v) for all u, v EV, and (LT2) T(cv) = cT(v) for all v E V and all scalars c