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Unformatted text preview: HOMEWORK PROBLEMS FROM STRANGS LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe the intersection of the three planes u + v + w + z = 6 , u + w + z = 4 and u + w = 2 (all in four dimensional space). Is it a line, point, or an empty set? What is the intersection if the fourth plane u = 1 is included? Find a fourth equation so that there is no solution. Page 11: # 22: If ( a,b ) is a multiple of ( c,d ) with abcd 6 = 0 , show that ( a,c ) is a multiple of ( b,d ) . This leads to the observation that if A = a b c d has dependent rows then it has dependent columns. Page 16: # 11: Apply elimination (circle the pivots) and back substitution to solve 2 x 3 y + 0 z = 3 4 x 5 y + 1 z = 7 2 x 1 y 3 z = 5 . List the three operations involved: subtract times row from row . Page 17, # 18: It is impossible for a system of linear equations to have exactly two solutions. Explain why . (a) If ( x,y,z ) and ( X,Y,Z ) are two solutions, what is another one? (b) if 25 planes meet at two points, where else do they meet? 1 2 PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 Page 27, #10: True or false? Give a specific counterexample when false. (a) If columns 1 and 3 of B are the same, so are columns 1 and 3 of AB . (b) If rows 1 and 3 of B are the same, so are rows 1 and 3 of AB . (c) If rows 1 and 3 of A are the same, so are rows 1 and 3 of AB . (d) ( AB ) 2 = A 2 B 2 . Page 27, #12: The product of two lower triangular matrices is lower trian gular. Confirm this with a 3 3 example, and then explain how it follows from the laws of matrix multiplication. Page 28, # 20: The matrix that rotates the x y plane by an angle is A ( ) = cos  sin sin cos . Verify that A ( 1 ) A ( 2 ) = A ( 1 + 2 ) from the identities for cos( 1 + 2 ) and sin( 1 + 2 ) . What is A ( ) times A ( ) ? Page 40, #5: Factor A into LU and write down the upper triangular system Ux = c which appears after elimination, where Ax = 2 3 3 0 5 7 6 9 8 u v w = 2 2 5 . Page 42, # 25: When zero appears in a pivot position, A = LU is not possible. Show why these are both impossible: 0 1 2 3 = 1 0 1 d e f , 1 1 0 1 1 2 1 2 1 = 1 0 0 1 0 m n 1 d e g f h 0 0 i . Page 43, # 29: Compute L and U for the symmetric matrix A = a a a a a b b b a b c c a b c d . Page 44, # 41: How many exchanges will permute (5 , 4 , 3 , 2 , 1) back to (1 , 2 , 3 , 4 , 5) ? How may exchanges to change (6 , 5 , 4 , 3 , 2 , 1) to (1 , 2 , 3 , 4 , 5 , 6) ?...
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This note was uploaded on 11/12/2010 for the course CSCI 271 taught by Professor Wilczynski during the Spring '08 term at USC.
 Spring '08
 Wilczynski

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