com02sA - UIIU') (I? .‘3‘ LJ 'J'F?‘ v p {‘11 mmm...

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Unformatted text preview: UIIU') (I? .‘3‘ LJ 'J'F?‘ v p {‘11 mmm fl. ramp. LIJLII n... TORAL QUALIFYING EXAMINATION SPRING 2002 Advances Calc;;us & Linear Algebra NAME: ID#: “our answer on each question sheet. Attach all extra s you use to the appropriate sheet. Hand in all question JANUARY 23, 2302 :f Exam: I—BPM of Exam: Physics Building, Room P—116 ID#: 1. Calculate The following integral: '” 3: (1'3: —H--—-_——._ a > L n a —- 3111;: ID#: 3. [a] Prove Lhat {l -— Lie—I l. for L"! < 1‘ <1 1. lib} Further. prove that , n! hm — = U, 71—min 1111 If you use Stirlin ’s formulafi vou must derive it." v 3 v ID#: 3. Let fir] and gut) be we reai—valued functions of 3: . which are continuous for —1 :1: ‘2. Prove that \‘2 I, I, \ . . ‘ Ill r. ._ .2. 1, _. \_ I) (“/0 gfl\r__]g§3:j-J {11:} if /U lfLIJ [‘ 0339/0 [9&ch [“ (£33. ID#: .121. .4. be a n. x n—matrix of real numbers. ,- .. 1 {a} Prove that. LAT) : (A '1}? if A is nousingularfi” is the tranSpose.) {le Suppose further than A is symmetric and positive definite,i.e.. ITAEII > O for every vector :3 7i 0. Prove that P“1_»1P is also positive definite. lD#: .1). Calculate the lcngths of the finger and the minor axes of the ellipse 1'31? -12Iy+8y2 =4. ID#: (—__...____ 6. For a. vector u in R”. define |! 11 .i32 ‘5"qu + — - - - + 1151: where u 2 Lu] , U2, - v - 3 an} and for a. matrix if: .4 i532 tsupuu I.“ H Au if . Let A be a real symmetric n x-n~n1atrix. Prove that. 5' A :i2 '15 the ( bsohue value of the eigenvalue of A with the largest magnitude. ...
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com02sA - UIIU') (I? .‘3‘ LJ 'J'F?‘ v p {‘11 mmm...

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