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Kunze Linear Algebra
· Problem 8.1.6
Inner Product Spaces
Mathematics — Methods · inner-products
Problem
Let
(
∣
)
(\ |\ )
(
∣
)
be the standard inner product on
R
2
R^2
R
2
, and let
T
T
T
be the linear operator
T
(
x
1
,
x
2
)
=
(
−
x
2
,
x
1
)
T(x_1, x_2) = (-x_2, x_1)
T
(
x
1
,
x
2
)
=
(
−
x
2
,
x
1
)
. Now
T
T
T
is 'rotation through
90
°
90°
90°
' and has the property that
(
α
∣
T
α
)
=
0
(\alpha|T\alpha) = 0
(
α
∣
T
α
)
=
0
for all
α
\alpha
α
in
R
2
R^2
R
2
. Find all inner products
[
∣
]
[\ |\ ]
[
∣
]
on
R
2
R^2
R
2
such that
[
α
∣
T
α
]
=
0
[\alpha|T\alpha] = 0
[
α
∣
T
α
]
=
0
for each
α
\alpha
α
.
Related theory
Inner Products And Inner Product Spaces
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