Unformatted text preview: Ph.D Preliminary Examination. Fall 1998. Algebra. 1. Let V be a ﬁnite dimensional vector space. Prove that the dimension of V is even if
and only if there is a linear map sz —-> V such that Ketf = Im f . 2. Let V be a ﬁnite dimensional complex vector space and let ¢2V —-> V be a linear map. '
(a) Assume that for each natural number k, trace(¢") = 0 . Prove that 0 is an eigenvalue
of «p. (b)Prove that (1) is nilpotent if and only if for each natural number k, trace(¢") = O. 3. Find two matrices having the same rank and the same characteristic polynomial, but
not similar to each other. ' 4. Let A and B be two self - adjoint matrices. Show that AB is self - adjoint if and only if
AB = BA. 5. Let V be a n-dimensional real vector space, and let q be a quadratic form on V. Let
A = (aij)ls:’.j5u be the symmetric matrix of qin an ordered basis. Show that if the form q is positive deﬁnite, then for each positive integer k, we have detAk > 0, where Ak = (aij)15i. I.“ 6. (a) Show that every n Xn matrix A can be uniquely written as the sum of a symmetric
and a skew-symmetric matrix. (b) Let A andB be two congruent n Xn matrices. Show that A' and B' are also
congruent. (0) Again, letA and B be two congruent n x n matrices, and write A = Al + A2 and B = Bl + 132 ,where Al and B‘ are symmetric and A2 and B2 are skew-symmetric. Show that A,
is congruent to 8,, and that A2 is congruent to 32. . ...
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