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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
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This document was uploaded on 01/25/2012.
 Spring '09

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