113 HW 3_Math 113 - Mathematics 113 Homework 2 Curtis...

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Mathematics 113: Homework # 2 Curtis Tekell Dr. Poirier September 22, 2010 Exercise 1. (a) Define the orthogonal group O ( n ) = { M GL ( n, R ) : det ( M ) = ± 1 } . Then G = h O ( n ) , ×i is a group. (b) Define the special linear group SL ( n, R ) = { M GL ( n, R ) : det ( M ) = 1 } . Then G = h SL ( n, R ) , ×i is a group. Proof. (a) We need only show that O ( n ) GL ( n, R ). Let A,B O ( n ). Then as det ( B ) = ± 1 6 = 0, B - 1 exists, and clearly AB - 1 GL ( n, R ). Additionally, det ( AB - 1 ) = det ( A ) det ( B - 1 ) = det ( A ) det ( B ) = ± 1 ± 1 = ± 1 , thus AB - 1 O ( n ). It follows O ( n ) GL ( n, R ). (b) We need only show that SL ( n, R ) GL ( n, R ). Let A,B SL ( n, R ). Then as det ( B ) = 1 6 = 0, B - 1 exists, and clearly AB - 1 GL ( n, R ). Additionally, det ( AB - 1 ) = det ( A ) det ( B - 1 ) = det ( A ) det ( B ) = 1 1 = 1 , thus AB - 1 SL ( n, R ). It follows SL ( n, R ) GL ( n, R ).
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113 HW 3_Math 113 - Mathematics 113 Homework 2 Curtis...

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