Page 108 - Linear Algebra Done Right

Exercises
11.
values.
Suppose T ∈L(V) is such that every vector in V is an eigenvector
12. Suppose S, T ∈L(V). Prove that ST and TS have the same eigen- 95
of T. Prove that T is a scalar multiple of the identity operator.
13. Suppose T ∈L(V) is such that every subspace of V with di-
mension dim V − 1 is invariant under T. Prove that T is a scalar
multiple of the identity operator.
14. Suppose S, T ∈L(V) and S is invertible. Prove that if p ∈P(F)
is a polynomial, then

p(STS −1 ) = Sp(T)S −1 .

15. Suppose F = C, T ∈L(V), p ∈P(C), and a ∈ C. Prove that a is
an eigenvalue of p(T) if and only if a = p(λ) for some eigenvalue
λ of T.

16. Show that the result in the previous exercise does not hold if C
is replaced with R.
17. Suppose V is a complex vector space and T ∈L(V). Prove
that T has an invariant subspace of dimension j for each j =
1,..., dim V.
18. Give an example of an operator whose matrix with respect to These two exercises
some basis contains only 0’s on the diagonal, but the operator is show that 5.16 fails
invertible. without the hypothesis
that an upper-
19. Give an example of an operator whose matrix with respect to triangular matrix is
some basis contains only nonzero numbers on the diagonal, but under consideration.
the operator is not invertible.

20. Suppose that T ∈L(V) has dim V distinct eigenvalues and that
S ∈L(V) has the same eigenvectors as T (not necessarily with
the same eigenvalues). Prove that ST = TS.
2
21. Suppose P ∈L(V) and P = P. Prove that V = null P ⊕ range P.

22. Suppose V = U ⊕W, where U and W are nonzero subspaces of V.
Find all eigenvalues and eigenvectors of P U,W .