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Math 54M - Spring 2001 - Vojta - Final

# Math 54M - Spring 2001 - Vojta - Final - SUN 12:34 FAX...

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Unformatted text preview: 11/11/2001 SUN 12:34 FAX 6434330 MOFFITT LIBRARY Math 54M Final Exam P. Vojta Spring 2001, Page 1/2 0 5 0 1. (10 points) Write the matrix 1 0 2 as a product of elementary matrices. —1 3 —1 2. (15 points) Deﬁne subspaces V and W in R4 by V = Span{(1, 2, 3, 4), (U, 4, 1,*1)}, W = Span{(4, 3, 1, ~6),(2,1, 2, 1)} . Find a nonzero vector in V D W. 3. (10 points) Let {u,v} be a basis for a vector space W. Under what conditions on the scalars a, b, c, d is the set {au + bv, cu + dv} also a basis for W? Explain why. 4. (20 points) Find the least-squares solution to the following system of linear equations: n+3y+5z=—3 2:17 —2z=5 y+2z=0 m—y—3zz7. If anything unusual occurs, explain it. 5. (20 points) Find the determinant. Caution: Instruction #6 on the front page of the exam will be taken more seriously this time. 2 0 ~3 4 1 —1 0 1 i —3 5 4 7 _ -5 1 11 —3 1 1 l 6. (15 points) The matrix A : 1 1 —1 has eigenvalues —1, 2, 2. Find an orthogonal 1 —1 1 matrix Q and a diagonal matrix A such that Q”1AQ : A. 7. (10 points) Find the angle between the vector (1, 2, 3) and the m—axis. 001 11/11/2001 SUN 12:34 FAX 6434330 MOFFITT LIBRARY Math 54M Final Exam P. Vojta Spring 2001, Page 2/2 8. (15 points) Determine the longest interval in which the initial—value problem (tant)y” + (t — 1)y' 4— 33} = tan2 t, y(1/2) = 0, y'(1/2) =1 is certain to have a unique twice-differentiable solution. Do not attempt to ﬁnd the solution. X—sox’ ﬁnd the fundamental matrix @(t) satisfying (13(0) 2 I. 9. (20 points) For the equation 7 10. (15 points) For the initial—value problem H2 it» met]: describe the behavior of the solution as t —> 00. (You do not need to solve the system.) 11. (20 points) The differential equation 3 —4 1 x’ = —1 0 -1 x —2 2 0 6621': + 6—3 has a solution x : e_t . Find its general solution. _662t 1, 05:551/2 12. (25 points) Find the Fourier cosine series for the function f(:c) : . 0, 1/2 < a,“ 5 1 13. (30 points) (a). Use the method of separation of variables to replace the partial diﬁerential equation u” A 2a,; + an = 0, u(0,t) = u(l,t) = 0 with a pair of ordinary differential equations. (b). Find a set of fundamental solutions for the above PDE. You are allowed (and in fact encouraged) to use remembered facts about the heat equation. 002 ...
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