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HWStrang4thEd

HWStrang4thEd - HOMEWORK PROBLEMS FROM STRANGS LINEAR...

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HOMEWORK PROBLEMS FROM STRANG’S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. C HAPTER 1: M ATRICES AND G AUSSIAN E LIMINATION 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

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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 θ is A ( θ ) = cos θ - sin θ sin θ cos θ . Verify that A ( θ 1 ) A ( θ 2 ) = A ( θ 1 + θ 2 ) from the identities for cos( θ 1 + θ 2 ) and sin( θ 1 + θ 2 ) . What is A ( θ ) times A ( - θ ) ? 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 0 f , 1 1 0 1 1 2 1 2 1 = 1 0 0 1 0 m n 1 d e g 0 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) ? One is even and one is odd. For ( n, . . . , 1) to (1 , . . . , n ) , show that n = 100
HOMEWORK PROBLEMS FROM STRANG’S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)3 and 101 are even and n = 102 and 103 are odd. Page 44, # 42: If P 1 and P 2 are permutation matrices, so is P 1 P 2 . This still has the rows of I in some order. Give examples with P 1 P 2 6 = P 2 P 1 and P 3 P 4 = P 4 P 3 .

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HWStrang4thEd - HOMEWORK PROBLEMS FROM STRANGS LINEAR...

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