Page 221 - Linear Algebra Done Right

Prove that if T ∈L(V) and j is a positive integer such that
7.
j ≤ dim V, then T has an invariant subspace whose dimension
equals j − 1or j. Exercises 211
7
8. Prove that there does not exist an operator T ∈L(R ) such that
2
T + T + I is nilpotent.
7
2
9. Give an example of an operator T ∈L(C ) such that T + T + I
is nilpotent.
10. Suppose V is a real vector space and T ∈L(V). Suppose α, β ∈ R
2
are such that α < 4β. Prove that
2
null(T + αT + βI) k
has even dimension for every positive integer k.

11. Suppose V is a real vector space and T ∈L(V). Suppose α, β ∈ R
2
2
are such that α < 4β and T + αT + βI is nilpotent. Prove that
dim V is even and
2
(T + αT + βI) dim V/2 = 0.

3
12. Prove that if T ∈L(R ) and 5, 7 are eigenvalues of T, then T has
no eigenpairs.

13. Suppose V is a real vector space with dim V = n and T ∈L(V)
is such that
null T n−2 = null T n−1 .
Prove that T has at most two distinct eigenvalues and that T has
no eigenpairs.

14. Suppose V is a vector space with dimension 2 and T ∈L(V). You do not need to find
Prove that if the eigenvalues of T to

a c do this exercise. As
b d usual unless otherwise

is the matrix of T with respect to some basis of V, then the char- specified, here V may
acteristic polynomial of T equals (z − a)(z − d) − bc. be a real or complex
vector space.
15. Suppose V is a real inner-product space and S ∈L(V) is an isom-
etry. Prove that if (α, β) is an eigenpair of S, then β = 1.