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Unformatted text preview: GRADUATE PRELIMINARY EXAM, ALGEBRA SAMPLE A . Let A be the 4 x 4 real matrix 1 1 0 0 —1 —l 0 0 A ‘ ‘—2 -2 2 1 1 1 -—1 0 Show that the characteristic polynomial for A is 32(2 — 1)2 and that it is also the minimal polynomial. Is A similar over the field of complex numbers to a diagonal matrix? . Let N; and N; be 6 x 6 nilpotent matrices over the field F. Suppose that N, and N3 have the same minimal polynomial and the same nullity. Prove that N; and N: are similar. Show that this is not true for 7 x 7 nilpotent matrices. . Let V be an inner product space and B, 7 fixed vectors in V. Show that Ta— - (alfi)7 defines a linear operator on V. Show that T has an adjoint, and describe T’" explicitly. Now suppose V is C“ with the standard inner product, 5- _ (111,... ,yn), and 7 = (2:1,. . . ,2”). What is the j, 1: entry of the matrix of T in the standard ordered basis? What is the rank of this matrix? . Let V be the vector space of the polynomials over. R of degree less than or equal to 3, with the inner product 1 ' my) = / munch. Ift is a real number, find the polynomial g: in V such that (flgg) = f(t) for all F in V. Let D be the differentiation operator on V. Find D". . Let V be a finite-dimensional inner product space, and let W be a subspace of V. Then V = We W'L, that is, each or in V is uniquely expre-sible in the form a = B + 7, with fi in W and 7 in W-L. Define a linear operator U by U(a)— -’ [3 — 7. (a) Prove that U is both self-adjoint and unitary. i (b) If V is R3 with the standard inner product and W is the subspace spanned by (1,0,1), find the matrix of U in the standard ordered basis. . Prove that a normal and nilpotent operator is the zero operator. ...
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