Problem 5 Consider A = (1) Compute T^([])? = -17 TA ? Let us denote ])? Is there any relation between T^([]) and [1]? (2) Compute TA([3] ])? Is there any relation between TA([]) and []? Let us denote √₂ = (3) Compute TA([])? Is there any relation between TA([]) and | [}]? ? Let us denote √3 = (4) Show that R3 = Span(1, 2, 3). (5) Decompose ₁ = [] in terms of 1, 2, 3. (6) Give an expression of TAK (1) in terms of k, 1, 2, 3 (and some scalars). hint: one can use Problem 4 (3) and (4) of Homework 5. (7) Define P= -I 0 (8) Let D= 0-10 . Compute P-1 Check that AP.D. P-1. (9) Give the general form of DD.D....D for n ≥ 1. In times (10) Show that for any n≥ 1, A" P.D" P-1. hint: try to compute A2, which is just (P. D. P-1) (P. D. P-1) (11) Write down the general form of A" for n ≥ 1.