MAT067 University of California, Davis Fall 2007
Homework Set Six: Eigenvalues
Directions: Submit your solutions to the Calculational Exercises and the Proof-Writing
Exercises separately at the beginning of lecture on Friday, November 9, 2007.The
two problems sets will be graded by different persons.
Calculational Exercises
Do Problem 1 and 2(a),(b).
1. Let T∈L(F2,F2) be defined by
T(u, v)=(v, u)
for every u, v ∈F. Compute the eigenvalues and associated eigenvectors for T.
2. Find eigenvalues and associated eigenvectors for the linear operators on F2defined by
each given 2 ×2 matrix.
(a) 30
8−1(b) 10 −9
4−2(c) 03
40
(d) −2−7
12
(e) 00
00
(f) 10
01
Hint: Use the fact that, given a matrix A=ab
cd
∈F2×2,λ∈Fis an eigenvalue for
Aif and only if (a−λ)(d−λ)−bc =0.
Proof-Writing Exercises
1. Let Vbe a finite-dimensional vector space over Fwith T∈L(V,V ), and let U1,...,U
m
be subspaces of Vthat are invariant under T. Prove that U1+···+Ummust then also
be an invariant subspace of Vunder T.
2. Let Vbe a finite-dimensional vector space over F, and suppose that the linear operator
P∈L(V) has the property that P2=P. Prove that V=null(P)⊕range(P).