13)
Let
V
=
C
2
with the standard inner product. Let
T
be the linear operator defined by
T
1
= (1
,

2)
, T
2
=
(
i,

1). Let
α
= (
x
1
, x
2
) and find
T
*
α.
13)
Let
T
be the linear operator on
C
2
defined by
T
1
= (1 +
i,
2) and
T
2
= (
i, i
). Find the matrix of
T
*
in the
standard ordered basis. Does
T
commute with
T
*
?
13)
Show that the range of
T
*
is the orthogonal complement of null
T
, i.e. show
R
=
R
(
T
*
) = (null(
T
))
⊥
=
N
.
13)
Let
V
be a finite dimensional inner product space (fin dim IPS), and
T
a linear operator on
V
.
If
T
is
invertible, show that
T
*
is invertible and that (
T
*
)

1
= (
T

1
)
*
.
13)
Show that the product of 2 selfadjoint operators is selfadjoint iff the two operators commute.
13)
Let
V
be a fin dim IPS over
C
. Let
E
be a projection operator / an idempotent operator on
V
. Prove
E
is
selfadjoint iff
E
is normal, i.e.
E
=
E
*
iff
E
*
E
=
EE
*
.
13)
Let
V
be a fin dim IPS over
C
. Let
T
be a linear operator on
V
. Show that
T
is selfadjoint iff (
Tx

x
) is real
for all
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Linear Algebra, Orthogonal matrix, Hilbert space, α, linear operator

Click to edit the document details