Math 4326 LINEAR ALGEBRA Spring 2008
Dr. Duval Homework
Tuesday, April 15
Follow the separate general guidelines for Parts A,B,C. Be sure to include and label all
four standard parts (a), (b), (c), (d) of Part A in what you hand in.
Generalized Eigenvectors (part II)
pp. 167–168
A: Reading questions. Due by 2pm, Mon., 21 Apr.
1. Verify the claim at the top of p. 167 that the operator N∈ L(F4) defined by
N(z1, z2, z3, z4)=(z3, z4,0,0) satisfies N2= 0.
2. Find a linear operator in L(F4) that is not nilpotent, and show it is not nilpotent.
3. Explain more carefully the following claim made at the beginning of the proof
of Corollary 8.8: “Because Nis nilpotent, every vector in Vis a generalized
eigenvector corresponding to the eigenvalue 0.”
4. Verify both Proposition 8.9 and the displayed equation above it, V= range T0⊃
range T1⊃ · · · ⊃ range Tk⊃range Tk+1, for the linear operator T∈ L(F4) given
by T(z1, z2, z3, z4) = (z1, z3, z4,0).
B: Warmup exercises. For you to present in class. Due by end of class Tue., 22 Apr.
Ch. 8: Exercises 5, 6.
The Characteristic Polynomial (part I)
pp. 168–171
[Part I covers through the end of the proof of Theorem 8.10.]
A: Reading questions. Due by 2pm, Wed., 23 Apr.
1. Answer the question posed in the middle of p. 168, “Could the number of times
that a particular eigenvalue is repeated depend on which basis of Vwe choose?”
2. Demonstrate Theorem 8.10 on on the 4-by-4 upper triangular matrix near the top
of p. 83. In other words, show that dim null(T−λI)dim Vis 2 for λ= 6, since 6
appears twice on the diagonal, and is 1 for λ= 7,8, since 7 and 8 each appear
once on the diagonal. Note that the basis here is the standard basis.
3. Demonstrate the claim, made in the margin of p. 168, that if Thas a diagonal
matrix Awith respect to some basis, then λappears on the diagonal of Aprecisely
dim null(T−λI) times, on the linear operator T∈ L(F3) defined by T(z1, z2, z3) =
(4z1,4z2,5z3) on p. 88. Note that the basis here is the standard basis. Why is
this claim a special case of Theorem 8.10?
B: Warmup exercises. For you to present in class. Due by end of class Thu., 24 Apr.
Ch. 8: 10.