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Pries: M467 - Abstract Algebra I, Spring 2013 Orthogonal group, rigid motions, group representations: due March 15 Read: Judson 12.2, Handout Problems: 1. Find the characteristic polynomial of this 0 0 1 0 0 1 matrix: −a0 −a1 . −a2 2. Let T be a linear transformation of a vector space V . Let W be the set of eigenvectors of T with eigenvalue λ. Prove that W is a T -stable subspace. 3. Consider the standard representation φ : S4 → Aut(R4 ). Let α = (12) and β = (1234). (a) Prove that α and β generate S4 . Hint: every permutation is a product of transpositions. (b) Find the matrices Mα and Mβ for φ(α) and φ(β) wrt the basis {e1 , e2 , e3 , e4 }. (c) Let w4 = (1, 1, 1, 1). Show that w4 is an eigenvector for φ(g) for all g ∈ S4 . (d) Let w1 = (1, 0, 0, −1) and w2 = (0, 1, 0, −1) and w3 = (0, 0, 1, −1). Show that W = Span(w1 , w2 , w3 ) is stable under φ(g) for all g ∈ S4 . (e) Find the matrices Mα and Mβ for φ(α) and φ(β) wrt the basis {w1 , w2 , w3 , w4 }. 4. Show that SOn is normal in On . 5. What are the eigenvalues for a rotation in R3 by θ around an axis v? 6. Let A ∈ O(3) with det(A) = −1. Show that −1 is an eigenvalue for A. Hint: what do you know about B = −A? 7. What are your top two project topic choices?