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Kunze Linear Algebra
· Problem 8.1.5
Inner Product Spaces
Mathematics — Methods · inner-products
Problem
Let
(
∣
)
(\ |\ )
(
∣
)
be the standard inner product on
R
2
R^2
R
2
. (a) Let
α
=
(
1
,
2
)
\alpha = (1, 2)
α
=
(
1
,
2
)
,
β
=
(
−
1
,
1
)
\beta = (-1, 1)
β
=
(
−
1
,
1
)
. If
γ
\gamma
γ
is a vector such that
(
α
∣
γ
)
=
−
1
(\alpha|\gamma) = -1
(
α
∣
γ
)
=
−
1
and
(
β
∣
γ
)
=
3
(\beta|\gamma) = 3
(
β
∣
γ
)
=
3
, find
γ
\gamma
γ
. (b) Show that for any
α
\alpha
α
in
R
2
R^2
R
2
we have
α
=
(
α
∣
ϵ
1
)
ϵ
1
+
(
α
∣
ϵ
2
)
ϵ
2
\alpha = (\alpha|\epsilon_1)\epsilon_1 + (\alpha|\epsilon_2)\epsilon_2
α
=
(
α
∣
ϵ
1
)
ϵ
1
+
(
α
∣
ϵ
2
)
ϵ
2
.
Related theory
Inner Products And Inner Product Spaces
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