Page 200 - Linear Algebra Done Right

Exercises
10.
V = null T ⊕ range T.
11. Prove or give a counterexample: if T ∈L(V), then 189
Prove that if T ∈L(V), then
n
n
V = null T ⊕ range T ,
where n = dim V.
12. Suppose V is a complex vector space, N ∈L(V), and 0 is the only
eigenvalue of N. Prove that N is nilpotent. Give an example to
show that this is not necessarily true on a real vector space.

13. Suppose that V is a complex 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.
4
14. Give an example of an operator on C whose characteristic poly-
2
2
nomial equals (z − 7) (z − 8) .
15. Suppose V is a complex vector space. Suppose T ∈L(V) is such
that 5 and 6 are eigenvalues of T and that T has no other eigen-
values. Prove that

(T − 5I) n−1 (T − 6I) n−1 = 0,
where n = dim V.

16. Suppose V is a complex vector space and T ∈L(V). Prove that For complex vector
V has a basis consisting of eigenvectors of T if and only if every spaces, this exercise
generalized eigenvector of T is an eigenvector of T. adds another
equivalence to the list
17. Suppose V is an inner-product space and N ∈L(V) is nilpotent.
given by 5.21.
Prove that there exists an orthonormal basis of V with respect to
which N has an upper-triangular matrix.

5
18. Define N ∈L(F ) by
N(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) = (2x 2 , 3x 3 , −x 4 , 4x 5 , 0).

Find a square root of I + N.