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Unformatted text preview: Algebra , Assignment 3, Solutions Due, Friday, Oct 1 1. Let vectorv be any nonzero vector in R n and set H = { A ∈ GL n ( R ) : Avectorv = λvectorv for some real λ } (That is, those A for which vectorv is an eigenvector.) Prove that H is a subgroup of GL n ( R ). There are three parts: Identity: Ivectorv = vectorv so I ∈ H (with λ = 1) Multiplication: If A,B ∈ H then Avectorv = λ 1 vectorv and Bvectorv = λ 2 vectorv (you can’t use the same symbol as the values of λ might be different!) and then ABvectorv = A ( λ 2 vectorv ) = λ 2 A ( vectorv ) = λ 2 λ 2 vectorv so AB ∈ H . Inverse: If A ∈ H then Avectorv = λvectorv for some λ ( λ can’t be zero as A is nonsingular) so that A 1 vectorv = λ 1 vectorv so A 1 ∈ H . 2. Let Ω ⊂ R 2 be any set (e.g.: a square) and let G be the group of symmetries A (by which, formally, we mean a bijective map from Ω to itself which preserves distances) of Ω. For vectorx ∈ Ω let H vectorx = { A ∈ G : A ( vectorx ) = vectorx } . Let B ∈ G with B ( vectory ) = vectorx . Our object (similar to Problem 5 in Assignment 2) is to describe...
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 Fall '08
 JOSEPHS
 Algebra, Normal subgroup, Trigraph, Subgroup, Cyclic group, Coset

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