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com01sA - Doctoral Qualifying Examination Spring 2001...

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Unformatted text preview: Doctoral Qualifying Examination Spring 2001 Advanced Calculus and Linear Algebra Name: - 1D# Start your answer on each question sheet. Attach all extra sheets you may have used to the appropriate sheet. Hand in all questions sheets. Time;1-3PM Date: January 24, 2001 Place: SB Union Rm 231 1 . 1D #: [Hun- I.i1e.L| [hp shurursl {Immune from Llw {Ji'igin Lu rim run-v of intmwsmflfim of thv - '4' am I; '- I'm- where u 3:2 U. 3) ‘> U: is: 15 \x’fiflfi 1),!2-5. Hli Ham‘s l D if : ’2. Evaluate .1 f In fig”; + 7,;2 + zédjidydzt where H is the region bounded by the plane 2 : 3 and Lhr- cone 3 : \f’l‘fl +' '93. ID #: 3. Determine the number of positive real roots for the equation ma 2 a", where the constant a. :t> U. 11—)ia’:_ 4. Supposv th n x :1 matrix 51 commuuzs wiLh every nonsingular n x u maLrix. Show that A ' M in: 5011111 «(-aial' L". where'- 1"“ is 11’]? nth order identity matrix. P | ID??? 5. Suppose L.' and W are subspaces of V for which U U W is also a subspace. Show that. either L.' C W or W C L_-'. 1 -—l —J U. Lea. :11 . 3 —-'L —'3 2 :s —2 13.} Find [.110 L1: factorizatioh of A. [h] L-e'ts XA. denote 1.119 SOIULiOI‘L of AX : Bk. Find X1, X3? X3, X4 when Bl : [1, 1‘ it)? and 85—.“ Z 18.;- -I- X;- 101' .‘L' 3’ U ...
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