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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 nullspace 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 kdimensional 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 kdimensional
subspace of V, with 1 g k < n. Thenithere is a kelement 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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 Fall '08
 Neely

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