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

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Unformatted text preview: Prelim 1 1. Math 294 Spring 2004 no calculators answer + reason = credit condensed printout—there were 5 pages a) (17 points) Find the number a for which the system x1 + 3x3 − x4 = 20 x1 + 2x2 + 2x4 = a x1 + 4x2 − 3x3 + 5x4 = 60 has solutions, and find all solution vectors x. b) (3 points) Give a definition: “ u1 , u2 , . . . , un are linearly independent iff —-”. 1 0 0 0 2 1 0 0 2. (20 points) Find the inverse C −1 if C = . 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. −1 −1 1 0 0 −1 1 0 1 0 b) (8 points) In this part, let v1 = , v2 = , e1 = , e2 = , e3 = , and 1 1 0 0 1 1 −1 0 0 0 0 0 e4 = . Find a basis for R4 consisting of v1 and v2 together with some of the ek . 0 1 c) (4 points) Give at least 4 examples of vectors in the image, imA, where A is the matrix of coefficients in problem 1. 4. Let T : R2 → R2 be rotation clockwise by 45 degrees. a) (10 points) Make a sketch to show why T has the linearity properties T (x + y) = T x + T y and T (cx) = cT x. 1 b) (10 points) Find the matrix for T and evaluate T ( ). 3 5. a) (10 points) Give an example of a 3 by 2 matrix A and a 2 by 3 matrix B for which BA = I, the 2 by 2 identity matrix. b) (5 points) For your matrices of part a), verify that AB = I, the 3 by 3 identity matrix. c) (5 points) Give a reason why AB could not be I no matter what you had picked in part a). 1 ...
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This note was uploaded on 04/04/2008 for the course MATH 2940 taught by Professor Hui during the Spring '05 term at Cornell University (Engineering School).

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