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Unformatted text preview: 140 2. BASIC THEORY OF GROUPS At least two natural models are available for G=Z . One is as transfor mations of projective .n 1/ dimensional space P n 1 , and the other is as transformations of G itself. Projective .n 1/ dimensional space consists of nvectors modulo scalar multiplication. More precisely, we define an equivalence relation on the set R n n f g of nonzero vectors in R n by x y if there is a nonzero scalar such that x D y . Then P n 1 D . R n nf g /= , the set of equivalence classes of vectors. There is another picture of P n 1 that is a little easier to visualize; every nonzero vector x is equivalent to the unit vector x = jj x jj , and furthermore two unit vectors a and b are equivalent if and only if a D b ; therefore, P n 1 is also realized as S n 1 = , the unit sphere in n dimensional space, modulo equivalence of antipodal points. Write OE x for the class of a nonzero vector x ....
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 Fall '08
 EVERAGE
 Algebra, Transformations

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