sample_midterms - EE-441 Dr Golomb 2:00 p m FIRST MIDTERM...

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- - EE-441 Dr. Golomb 2:00 p. m. FIRST MIDTERM September 28, 2006 Put your name, and all answers to be graded, on the Answer Sheet. BE ACCURATE! ( 332 ) A. Let M = 1 1 - 2 , a 3 x 3 real matrix. 310 Find each of the following: 1. The determinant of M, IMI. 2. The trace of M, Tr(M). 3. The characteristic polynomial of M,Pm(>"')' 4. The characteristic roots (eigenvalues) of M, {AI, A2,A3}' 5. Linearly independent eigenvectors {aI, a2, a3} corresponding to these eigenvalues. 6. A nonsingular matrix P such that p-I M P = A, a diagonal matrix. 7. The diagonal matrix A of problem 6. 8. The inverse, p-I, of the matrix P in problem 6. ( 112 ) B. Let A = 2 2 4 , a 3 x 3 real matrix. 336 9. What is the domain of A? 10. What is the null-space of A? 11. What is the range-space of A? 12. What is the order of A? 13. What is the rank of A? 14. What is the nullity of A? (For problems 9., 10., and 11., you can describe the spaces involved by exhibiting a basis for each one.) C. Let F7 = {O,1,2,3, -1, -2, -3} be the field of seven elements, and let V = F.f. Find the number of k-dimensional subspaces of V, for 15. k = 1 16. k = 2 17. k = 3 18. k = 4 (Give your answers as expressions, and as specific numbers.) (over)
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-- D. Let V = R3 =
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