For the following problem, we recall the following facts: (1) For a real number 0 1, limk-+ too q* = too. Tak (3) We say that a sequence of vectors (TK = bE )KEN converges if all the components converge (i.e. ak, bk and ck converge when k - too). (4) We say that a sequence of vectors (TK = bE )KEN goes to oo (or is unbounded), if the absolute value of at least one of its component goes to too i.e. either lak| +k-+too too or 10 *1 -*+ too too or | CK/ * + too +00 . (5) The zeroes of a quadratic polynomial ar + bx + care , + , vb2 - 4ac and ; - V62 - 4ac. (6) Theorem 0.1. For an xn matrix M, det (M) =0 if and only if TM is not injective (equivalently, if and only if TM is not bijective). Equivalently, det(M) # 0 if and only if TM is bijective (i.e. M is invertible). For this problem, one can freely use the (almost) row echelon forms of the system with the right choice for a, b, c 75 10x2+ 13 = a 4 (A) 3 2 2 + -3 = b C3 = C 75 1+ 10x2+ T3 = a corresponding (almost) row echelon form 3 b C2 - -X3 = 2 4 T3 = 7 and the solution is {(a - 2b- 15c, -46 + ec, =b + =c)}Problem 2 Let A = 0 15 ) (1) Compute XA = det(A - x13). (2) Show that the only zeroes of XA (i.e. the eigenvalues of A) are A1 = 1, 12 = ?, 13 = 3. (3) Let S1 = Ker(TA - IdR3) (which is also Ker(TA-13)). (a) Find a basis B1 of S1. Denote the vector(s) of B1, v1, . . ., Udim($1). (b) Show that for any TE S1, TA(U) = U. (c) Compute for such v, TAz(U) = TA(TA(U)) in terms of v. What then is TAK (U) for any k 2 1? (4) Let S3 = Ker(TA - , IdR3) (which is also Ker(TA-;13)). (a) Find a basis By of S3. Denote the vector(s) of B;, ul, . .., Uaim(S;). (b) Show that for any u e S;, TA(u) = qu. c) For such u compute TAZ(u) = TA(TA(u)) in terms of u. What then is TAX (u) for any k > 1? (5) Let S1 = Ker(TA - } IdRs) (which is also Ker(TA-; 13)). (a) Show that w1 = is a basis of S; . (b) Show that for any w E S1, TA(w) = 3w. (c) For a w E S1, express TAK (W) in terms of k and w. (6) Show that Ker(TA + 5Idp3) = {0} can be done without solving a system (7) (a) Show that there is a basis B of R's made up of a vector v1 E S1, a vector u] E S; and the vector wi E S;. (b) Deduce that R3 = S1 + S; + S; (8) Given a To E R3, we define (TK) to be the sequence satisfying Ek+1 = TA(Ik). for any k 2 0. (a) Express Ek in terms of A, k and To only. hint: express 1 in terms of To, then 12 in terms of To and then 's in terms of To and guess the formula for any k 2 1 (b) Let To = i . (i) Find the coordinates of To relative to the basis B. ii) Show that (Ek) KEN is unbounded (or goes to co). (c) Let zo = (i) Find the coordinates of To relative to the basis B. ii) Show that (TK) k has a finite limit. Compute the limit. (d) (i) Show that there is a subspace F C RS of dimension 2 such that for any To EW, the associated sequence (TK) k has a finite limit. (ii) Find a Cartesian equation of F. (9) Compute D = Mat (TA, B, B). 10) Let P be the matrix whose columns are v1, u1, w1. (a) Compute P-1. (b) (Bonus) Compute A = P . D . P-1. (c) (Bonus) Compute D" for any n 2 1. (d) (Bonus) Show that An = P . Dn . P-1. Compute An