Unformatted text preview: Math330 Solutions HW 3 Fall 2008 If you ﬁnd any typos in this, please let Professor Shipley know. 1.10. We know that the composition of two rotations is again a rotation. The composition of a reﬂection and a reﬂection is a rotation, a reﬂection followed by a rotation is a reﬂection and a rotation followed by a reﬂection is a reﬂection. Thus r1 r2 f1 r3 f2 f3 r3 is a reﬂection. 0.16. The symmetries of the string of H s are: 1. translation to the left and the right. 2. ﬂip through the center of the row. 3. an inﬁnite number of vertical reﬂections. The composition of reﬂections through a vertical axis followed by translation is not equal to translation and then reﬂection through a veritcal axis. To see this label the H s...say ..., H−1 , H0 , H1 , ... and ﬂip through the center of H0 . 0.6. For example, let a = [1, 1; 0, 1] and b = [1, 1; 1, 0]. 2.16. We want to show that (ab)−1 = b−1 a−1 . First, we use the deﬁnition of an inverse to show that (ab)(ab)−1 = e. Then we can multiply on the left by a−1 , giving a−1 (ab)(ab)−1 = (a−1 a)b(ab)−1 = b(ab)−1 = a−1 e. We repeat the process, multiplying on the left by b−1 , so that we have b−1 b(ab)−1 = (ab)−1 = b−1 a−1 . 2.18. We want to show that (a−1 )−1 = a. We do this by using the deﬁntion of an inverse, i.e. (a−1 )(a−1 )−1 = e. Then multiplication on the left by a gives a(a−1 )(a−1 )−1 = ae. 1 ...
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- Spring '08