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Unformatted text preview: MATH 304505 Sample problems for the final exam Spring 2011 Any problem may be altered or replaced by a different one! Problem 1 (15 pts.) Find a quadratic polynomial p(x) such that p(1) = p(3) = 6 and p (2) = p(1). Problem 2 (20 pts.) Let v1 = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 1). Let L : R3 R3 be a linear operator on R3 such that L(v1 ) = v2 , L(v2 ) = v3 , L(v3 ) = v1 . (i) Show that the vectors v1 , v2 , v3 form a basis for R3 . (ii) Find the matrix of the operator L relative to the basis v1 , v2 , v3 . (iii) Find the matrix of the operator L relative to the standard basis. 1 1 Problem 3 (20 pts.) Let A = 0 2 1 1 1 3 0 0 1 1 . 0 1 0 0 (i) Evaluate the determinant of the matrix A. (ii) Find the inverse matrix A1 . (i) Find all eigenvalues of the matrix B. (ii) Find a basis for R3 consisting of eigenvectors of B. (iii) Find an orthonormal basis for R3 consisting of eigenvectors of B. (iv) Find a diagonal matrix X and an invertible matrix U such that B = U XU 1 . 1 1 1 Problem 4 (25 pts.) Let B = 1 1 1 . 1 1 1 Problem 5 (20 pts.) Let V be a subspace of R4 spanned by vectors x1 = (1, 1, 0, 0), x2 = (2, 0, 1, 1), and x3 = (0, 1, 1, 0). (i) Find the distance from the point y = (0, 0, 0, 4) to the subspace V . (ii) Find the distance from the point y to the orthogonal complement V . 1 Bonus Problem 6 (15 pts.) (i) Find a matrix exponential exp(tC), where C = and t R. (ii) Solve a system of differential equations dx = 3x + y, dt dy = 3y dt 3 1 0 3 subject to the initial conditions x(0) = y(0) = 1. Bonus Problem 7 (15 pts.) Consider a linear operator K 4 7 1 K(x) = Dx, where D = 1 4 9 8 4 The operator K is a rotation about an axis. (i) Find the axis of rotation. (ii) Find the angle of rotation. : R3 R3 given by 4 8 . 1 2 ...
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 Spring '08
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