2.
(15 points) Give examples, with brief justification, of each of the following.
(a) An operator on
R
2
which is not selfadjoint with respect to the standard inner product.
(b) An isometry on
R
4
with no (real) eigenvalues.
(c) An operator on
C
4
whose characteristic polynomial equals the square of its minimal
polynomial.
3.
(20 points) Suppose that
P
is an operator on an innerproduct space
V
such that
P
2
=
P
.
Prove that
P
is an orthogonal projection if and only if it is selfadjoint.
5.
(20 points) Let
V
be a complex vector space. If you get stuck on part (a) below, assume
it is true and use it in part (b).
(a) Prove that if
N
is a nilpotent operator on
V
, then
N
+
I
has a square root.
(b) Prove that any invertible operator
T
on
V
has a square root. (Hint: Use the generalized
eigenspaces of
T
)
6.
(15 points) Suppose that an operator
T
on a complex vector space has characteristic
polynomial
z
3
(
z

2)
5
(
z
+ 1)
2
and minimal polynomial of the form
z
2
(
z

2)
k
(
z
+ 1)
`
where
k >
2 and
`
≥
1
.
Suppose further that dim range(
T