Unformatted text preview: H is already contained in some known group, so we need only check points (1) and (2). We say that a subset H of a group G is closed under multiplication if condition (1) is satisﬁed. We say that H is closed under inverses if condition (2) is satisﬁed. Example 2.2.2. An n-by-n matrix A is said to be orthogonal if A t A D E . Show that the set O .n; R / of n-by-n real–valued orthogonal matrices is a group. Proof. If A 2 O .n; R / , then A has a left inverse A t , so A is invertible with inverse A t . Thus O .n; R / ´ GL .n; R / . Therefore, it sufﬁces to check that the product of orthogonal matrices is orthogonal and that the inverse of an orthogonal matrix is orthogonal. But if A and B are orthogonal, then .AB/ t D B t A t D B ± 1 A ± 1 D .AB/ ± 1 ; hence AB is orthogonal. If A 2 O .n; R / , then .A ± 1 / t D .A t / t D A D .A ± 1 / ± 1 , so A ± 1 2 O .n; R / . n Here are some additional examples of subgroups:...
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- Fall '08
- Algebra, orthogonal matrices, Cyclic Groups, nonempty subset