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Su10 Math 270 FINAL Name: _____________________ Show all necessary work clearly, neatly, systematically and understandably. Any incorrect and/or understatement may be penalized. Failure to use correct notation may result in point deduction. Provide clear flow of thought in the presentation of proofing. 1 2 3 1. (9:5,4) Let A 2 5 3 . 1 0 8 a. Decompose A into LU. 2. Let A be an m x n matrix. Prove: a. The ATA and AAT are symmetric. (9:4,5) b. If ATA = A, then A is symmetric and A = A2. b. Find A-1 3. (6) Prove: det[adj(A)] = [det(A)]n-1 Let L: R3 R3 be a linear operator that first rotates a vector counterclockwise about the zaxis through an angle of 30° and then reflects the resulting vector about the yz-plane. Find the matrix A representing L. 4. (6) 6. (8) Prove Fundamental Subspaces Theorem: Let A be an m x n matrix. Then N(A) C (AT ) 5. (6) Prove: (A-1+B-1)-1 = A(A+B)-1B. Hint: AA-1 = A-1A = I = BB-1 = B-1B 7. (10) Let S be a subspace of C[a,b], S = span{ex,xex,x2ex}. Let D be a differentiation operator of S. Find the matrix representing D2 with respect to [ex,xex,x2ex]. Note that D2 represents the second differentiation operator of S. 8. Let plane P have the equation x – 3y + 2z = 6 and point Q(2,-3,1). Find the point on plane P that is closest to Q. (7) 1 1 0 9. (8) Let b1 1 , b2 0 and b3 1 and L: R3 R2. L x1 e1 x 2 e2 x 3 e3 x 2 x1 e1 x 3 x 2 e2 . 0 1 1 1 1 IF u1 , u2 , then find matrix representing L with respect to b1 , b2 , b3 to u1 ,u2 2 2 10. (8:3,5) The stock market performance on one day is either positive or negative. According to the statistics: If it is positive on the previous day, then it is positive again today at 80% probability. If it is negative on the previous day, then it is negative again today at 70% probability. a. Construct the transition matrix. b. In the long run, find the probability that it will be positive. 11. (18:4,6,4,4) Let L: P3 P3 where Lp(x) p' (x) p(3) . a. Show that L is a linear transformation. b. Find the matrix A representing L with respect to [x2,x,1]. c. Find Ker(L) d. Find L(P2). Note: P2 is a subspace of P3. y 1 ' 3y 1 y 2 2y 3 12. (12) Solve y 2 ' 2y 1 2y 3 for initial value y0 (1,6,5)T y 3 ' 2y 1 y 2 y 3