MATH 304–504 Fall 2007Sample problems for the final examAny 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 andp′(2) = p(1).Problem 2 (20 pts.) Consider a linear operator L : R3→ R3given byL(x) = (x · v1)v2, where v1= (1, 2, −1), v2= (2, 2, 1).(i) Find the matrix of the operator L.(ii) Find the dimensions of the range and the kernel of L.(iii) Find bases for the range and the kernel of L.Problem 3 (20 pts.) Let A =1 1 0 01 1 1 −10 1 0 12 3 0 0.(i) Evaluate the determinant of the matrix A.(ii) Find the inverse matrix A−1.Problem 4 (25 pts.) Let B =1 1 11 1 11 1 1.(i) Find all eigenvalues of the matrix B.(ii) Find a basis for R3consisting of eigenvectors of B.(iii) Find an orthonormal basis for R3consisting of eigenvectors of B.(iv) Find a diagonal matrix X and an invertible matrix U such that B = UXU−1.Problem 5 (20 pts.) Let V be a subspace of R4spanned 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⊥.1Bonus Problem 6 (20 pts.) (i) Find a matrix exponential exp(tC), where C =3 10 3and t ∈ R.(ii) Solve a system of differential equationsdxdt= 3x + y,dydt= 3ysubject to the initial conditions x(0) = y(0) = 1.Bonus Problem 7 (20 pts.) Consider a linear operator K : R3→ R3given byK(x) = Dx, where D =19−4 7 41 −4 88 4 1.The operator K is a rotation about an axis.(i) Find the axis of rotation.(ii) Find the angle of
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