Math 110 - Fall 1996 - Arveson - Final

Math 110 - Fall 1996 - Arveson - Final - THU 15:47 FAX...

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Unformatted text preview: 09/21/2000 THU 15:47 FAX 6434330 1. MOFFITT LIBRARY Math 110, FINAL Exam, W.Arveson Wednesday, 12/ 11/96, 12:30—3:30PM, 306 Latrmm. T here are 100 points total. The point value of each problem is indicated. (10 points ) Give an example of a 3 x 3 matrix A for which A2 75 0 but A3 = 0. {15 points) Let (V, , be a real or complex inner product space and let T e £(V). (a) Define the adjoint operator T*. (b) Let M be an invariant subspace for M. Show that M J- is invariant under T*. (25 points) A real n X 71 matrix A r: (ca-j) is called skew symmetric if an = “‘12:: for all i, j = 1, 2, . . . , 71. Fix such a matrix A, and consider it to be a linear operator acting by multiplication on column vectors in the inner product space R”, Where < may >= $1y1 + 502192 + - ' ' +£11.91;- (a) Let w 6 R” and let G E R. Show that the norm of every vector of the form 3: + cAm satisfies the inequality ||=B + 0A1“ 2 Hit“- (b) Dednce that A has no (real) eigenvalues other than 0. (c) Show that U = (1 + A)(1 — A)‘1 is well—defined, and satisfies the equation U*U = 1. (d) Deduce that U is an isometry: “Um” = for every 3 6 IR“. (10 points) Let T be a linear operator from R2 to R4 such that ranT ={(.’L‘1,332,$3,.’E4) 2:131 = :53,and 1:2 : $4}. Show that T is one-to—one. ( 10 points ) Let V be a complex vector space of dimension n and let T E £(V) be an operator having only a single eigenvalue A. Show that T must have the form T=A1+S 001 09/21/2000 THU 15:47 FAX 6434330 MOFFITT LIBRARY 002 Where S is an operator satisfying S” = O. 6. (10 points) Find a polynomial f E 733 such that f (0) = 0, f’(0) = 0 and 1 / |2 + 3t — f(t)|2 at D is as small as possible. 7. {20 points) Let V be a finite dimensionai complex vector space, let T E £(V) and let p(z)=ao+a1z+-~+anz” with compiex coefiicients. (a) Show that if A E (C is an eigenvalue of T then p()\) is an eigenvalue of p(T). (b) Show that if p, is an eigenvalue of p(T) then there is an eigenvalue A of T such that ,u = 130‘). ...
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