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# 14 - 15 November 2006 1502 CalculusII FALL 2006 EXAM 2 NAME...

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15 November, 2006 1502 ( CalculusII) FALL 2006 EXAM 2. NAME : ............................................................... To receive full credit SHOW ALL your work... Open textbook only...No other material can be used!!! Time allowed : 80 minutes. PROBLEM 1. (10 points(each 2 points)) Decide whether the following statements are true (T) or false (F). If false, give a short argument. a) The map x → | x | from R n R is linear. b) Let A be a 2 by 2 matrix such that A 2 = A . Then either A is the identity matrix or the zero matrix. c) The composition of two nonlinear maps is never linear. d) Let A be an n by n matrix such that A 4 = I . Then A is invertible. e) The sum of two 3 by 3 invertible matrices is invertible. 1

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PROBLEM 2. (20 points) Find the complete solution to 2 y + 2 z + 4 t = 2 x + 3 y + 5 t = 5 - x - y + 2 z + t = 3 You must use proper Gaussian elimination. 2
PROBLEM 3. (5+5+5+5+7+8 = 35 points) Let A = 0 1 2 1 3 - 1 2 7 0 (i) Find a basis for Ker A . (ii) Find a basis for img ( A ) , the image of A. . (iii) Find an equation(s) for all b such that A x = b is solvable (membership problem for img ( A )).

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14 - 15 November 2006 1502 CalculusII FALL 2006 EXAM 2 NAME...

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