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?