Page 199 - Linear Algebra Done Right

Chapter 8. Operators on Complex Vector Spaces
188
Exercises
2
1. Define T ∈L(C ) by
T(w, z) = (z, 0).
Find all generalized eigenvectors of T.
2
2. Define T ∈L(C ) by
T(w, z) = (−z, w).

Find all generalized eigenvectors of T.

3. Suppose T ∈L(V), m is a positive integer, and v ∈ V is such
m
that T m−1 v = 0 but T v = 0. Prove that
2
(v, Tv, T v,...,T m−1 v)
is linearly independent.
3
4. Suppose T ∈L(C ) is defined by T(z 1 ,z 2 ,z 3 ) = (z 2 ,z 3 , 0). Prove
that T has no square root. More precisely, prove that there does
2
3
not exist S ∈L(C ) such that S = T.
5. Suppose S, T ∈L(V). Prove that if ST is nilpotent, then TS is
nilpotent.

6. Suppose N ∈L(V) is nilpotent. Prove (without using 8.26) that
0 is the only eigenvalue of N.

7. Suppose V is an inner-product space. Prove that if N ∈L(V) is
self-adjoint and nilpotent, then N = 0.

8. Suppose N ∈L(V) is such that null N dim V−1 = null N dim V . Prove
that N is nilpotent and that
j
dim null N = j
for every integer j with 0 ≤ j ≤ dim V.

9. Suppose T ∈L(V) and m is a nonnegative integer such that
range T m = range T m+1 .

k
Prove that range T = range T m for all k>m.