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EE441_Sample_First_Midterm_091906 - Q flMFLE Eli—441...

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Unformatted text preview: Q flMFLE , Eli—441 D7". Golomb FIRSTMIDTERM (Zea/£4; Mi W2 6/06 7 A. Let M = ( ), a 3X 3 real matrix. OHH Her—a l—‘i—‘O Find each of the following. The determinant of M, lM The trace of M, T7'(M). The characteristic polynomial of M, pM()\). The characteristic roots (eigenvalues) rof M, {A1, A2, A3}. Linearly independent eigenvectors {051,‘a2, 043} corresponding to these eigenvalues. A nonsingular matrix P such that P‘LM P = Afga diagonal matrix. The diagonal matrix of problem 6. PONQQPPJE‘J?‘ The inverse, P"1, of the matrix P in problem 6. l —1 0 B. Let A = «1 0 l , a 3 X 3 real matrix. 0 1 —1 9. What is the domain of A? 10. What is the null-space of A? 11. What is the rangespace of A? 12. What is the order of A? 13. What is the rank of A? 14. What is the nullity of A? .. ‘1 (For problems 9., 10., and 11., you can describe the spaces involved by exhibiting a basis for each one.) . G. Let F5 = {O,1,2,-1,—-2,} be the field of five elements, and let V : Fg‘. Find the number of k-dimensional subspaces of V, for 15.1621 16. k==2 17. k=3 18. k=4 D. Let V = R3 = {(m,y,z) for all real 33,3}, and 2}. For each of the following subsets S,- of V, tell whether or not St- is a subspace of V. If it is a subspace, give its dimension. If it is not a subspace, show how one of the vector space requirements fails. 19. 8'1 2 {all (3:, y, z)with 3; + 2y + 3z = 6}. 20. $2 2 {all (9:, y, z)with at + 33; = 22}. 21. 53 2 {all (:13,y, 2)With m2 + :92 = 22}. 22. S4 = {all (9:,y, z)with m — y = z and 2x + y = 32}. E. TRUE or FALSE. Tell whether each of the following statements is true or false. If it is true, give a general proof. If it is false, exhibit a specific example where it fails. 23. If A and B are n x n real matrices with B = Alg for some positive integer k, then AB 2 BA. 24. Let B r: {071, 062, . . . ,ozn} be a basis for V : R”, and let W be a k-dimensional subspace of V, with 1 g k < n. Thenithere is a k-element subset B’ of B which is a basis for W. ‘ 25. For any n x n real matrix 1%, M M T is a symmetric matrix. ...
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EE441_Sample_First_Midterm_091906 - Q flMFLE Eli—441...

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