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actual PrelimSp04

actual PrelimSp04 - Prelim 1 Math 294 Spring 2004 no...

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Prelim 1 Math 294 Spring 2004 no calculators answer + reason = credit condensed printout—there were 5 pages 1. a) (17 points) Find the number a for which the system x 1 + 3 x 3 - x 4 = 20 x 1 + 2 x 2 + 2 x 4 = a x 1 + 4 x 2 - 3 x 3 + 5 x 4 = 60 has solutions, and find all solution vectors x . b) (3 points) Give a definition: “ u 1 , u 2 , . . . , u n are linearly independent iff —-”. 2. (20 points) Find the inverse C - 1 if C = 1 0 0 0 2 1 0 0 0 3 1 0 0 0 4 1 . 3. a) (8 points) Find a basis for the kernel of D , if D is the 5 by 5 matrix having a 1 in every position. b) (8 points) In this part, let v 1 = - 1 - 1 1 1 , v 2 = - 1 1 1 - 1 , e 1 = 1 0 0 0 , e 2 = 0 1 0 0 , e 3 = 0 0 1 0 , and e 4 = 0 0 0 1 . Find a basis for R 4 consisting of v 1 and v 2 together with some of the e k . c) (4 points) Give at least 4 examples of vectors in the image, im A , where
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