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# hw3 - Math 113 Homework 3 selected solutions Fraleigh 5.13...

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Math 113 Homework # 3, selected solutions Fraleigh 5.13: Yes, this set of matrices, which is called O( n ), is a subgroup of GL( n, R ). First note that O( n ) is a sub set of GL( n, R ) because if A T A = I , then A is injective, so by the “rank-nullity” theorem in linear algebra, its range has dimension n , so A is invertible. We also note that since left inverses are unique, A T = A - 1 , so AA T = I also. Now O( n ) is closed under multiplication because if A T A = I and B T B = I then ( AB ) T ( AB ) = B T A T AB = B T IB = B T B = I . O( n ) contains the identity because I T I = I 2 = I . O( n ) is closed under inverses because if A T A = I then ( A - 1 ) T A - 1 = ( A T ) - 1 A - 1 = ( AA T ) - 1 = I - 1 = I . (Here we have used the facts that AA T = I as explained above, and also that for any invertible matrix A we have ( A T ) - 1 = ( A - 1 ) T ; this holds because ( A - 1 ) T A T = ( AA - 1 ) T = I T = I .) Fraleigh 5.54: H K is closed under multiplication because if x, y H K , then since x, y H and H is a subgroup we have xy H ; since x, y K and K is a subgroup we have xy K ; so xy H K . H K

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hw3 - Math 113 Homework 3 selected solutions Fraleigh 5.13...

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