Practice Final (Vojta) - Math 54 Sample Final Exam In this exam the following formulas were given eax(a sin bx b cos bx C a2 b2 eax(a cos bx b sin bx ax

# Practice Final (Vojta) - Math 54 Sample Final Exam In this...

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Math 54 Sample Final Exam In this exam, the following formulas were given: Z e ax sin bx dx = e ax ( a sin bx - b cos bx ) a 2 + b 2 + C Z e ax cos bx dx = e ax ( a cos bx + b sin bx ) a 2 + b 2 + C x X n =1 2( - 1) n +1 n sin nx , - π < x < π or 0 < x < π x π 2 - 4 π X n =1 1 (2 n - 1) 2 cos nx , 0 < x < π 1. (12 points) Find the inverse of the matrix A = 7 10 - 9 1 2 - 3 - 1 1 - 6 , if it exists. Use the algorithm from the book (or from class). 2. (20 points) Let A be the 2 × 2 matrix 0 1 x 0 , where x is a real number. (a). For which values of x is A similar to a (real) diagonal matrix? (Do not diagonalize the matrix.) (b). For which values of x is A orthogonally diagonalizable? 3. (20 points) Each of the following parts gives vector spaces V and W , bases B for V and C for W , and a linear transformation T : V W . In each case find the matrix for T relative to B and C . (a). V = W = R 2 , B = { (1 , 1) , ( - 1 , 1) } , C = { (1 , 0) , (0 , 1) } , and T is counterclockwise rotation by 90 degrees. (b). V = W = Span { sin x, cos x } ⊆ C [0 , 2 π ], B = C = { sin x, cos x } , and T is the linear transformation taking a function to its derivative.