PartAJune05 - 1 1 . 00001 1 1 . 00001 1 1 . 00001 1 1 ....

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DOCTORAL QUALIFYING EXAMINATION SUMMER 2005 NAME : ID # : There are four questions from Linear Algebra and four question from Advanced Calculus. For full credit, answer any THREE questions from Linear Algebra and any THREE questions from Advanced Calculus . Start your answer on each question sheet. Attach all extra sheets you use to the appropriate sheet. Hand in all question sheets. Date: June 13, 2005 Time of Examination: 09:00 – 11:00 Place of Examination: Physics P113
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I.D.# 1. Consider the following four vectors in R 5 : u 1 = (1 , - 2 , 1 , 3 , - 1) , u 2 = ( - 2 , 4 , - 2 , - 6 , 2) , u 3 = (1 , - 3 , 1 , 2 , 1) , u 4 = (3 , - 7 , 3 , 8 , - 1) . Find a subset of the vectors u 1 ,u 2 ,u 3 , and u 4 which gives a basis for the subspace W = span( u 1 ,u 2 ,u 3 ,u 4 ) of R 5 .
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I.D.# 2. Let A = 1 2 3 4 . Show that a real matrix B commutes with A (that is, AB = BA ) if and only if B has the form sI + tA , where s and t are real numbers and I is the 2 × 2 identity matrix.
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I.D.# 3. Let A be a real non-singular matrix. (a) Prove that A T A is symmetric. (b) Prove that A T A is positive definite.
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I.D.# 4. Prove that the matrix
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Unformatted text preview: 1 1 . 00001 1 1 . 00001 1 1 . 00001 1 1 . 00001 1 has one positive eigenvalue and one negative eigenvalue. (Hint: = 0 is the other eigenvalue of the matrix.) I.D.# 5. Consider the polynomial equation a n x n + a n-1 x n-1 + + a = 0, where a ,a 1 ,...a n are integers with a 6 = 0 and a n 6 = 0. Show that if the equation is to have a rational root p/q , then p must divide a and q must divide a n exactly. I.D.# 6. Let a be a positive constant ( a > 0). Prove that the function f ( x ) = 1 /x 2 is uniformly continuous in the interval ( a, ). I.D.# 7. Let a 1 ,a 2 ,a 3 ,... be positive numbers. (a) Prove that if n =1 a n converges then n =1 a n a n +1 converges. (b) Prove that the converse of the statement in part (a) is false. I.D.# 8. Show that Z x 2 e-x 2 dx = 1 2 Z e-x 2 dx....
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PartAJune05 - 1 1 . 00001 1 1 . 00001 1 1 . 00001 1 1 ....

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