Solutions of Selected Problems in HW4
3.1.12
Let
x,y
be arbitrary elements of
R
+
. Also let
α,β
be arbitrary real numbers.
C1:
x
⊕
y
=
x
·
y
∈
R
+
,
since multiplication of two positive numbers are also positive.
C2:
α
◦
x
=
x
α
∈
R
+
,
since exponential functions with positive base are always positive.
A1:
x
⊕
y
=
x
·
y
=
y
·
x
=
y
⊕
x.
A2: (
x
⊕
y
)
⊕
z
= (
x
·
y
)
·
z
=
x
·
(
y
·
z
) =
x
⊕
(
y
⊕
z
)
.
A3: In this space, 1 plays a role of a zero vector with respect to
⊕
, since 1
⊕
x
= 1
·
x
=
x.
A4: In this space,
1
x
plays a role of an inverse of
x
with respect to
⊕
, since
x
⊕
1
x
=
x
·
1
x
= 1
.
Note that 1 is the zero vector.
A5:
α
◦
(
x
⊕
y
) =
α
◦
(
x
·
y
) = (
x
·
y
)
α
=
x
α
·
y
α
= (
α
◦
x
)
·
(
α
◦
y
) = (
α
◦
x
)
⊕
(
α
◦
y
)
A6: (
α
+
β
)
◦
x
=
x
α
+
β
=
x
α
·
x
β
= (
α
◦
x
)
·
(
β
◦
x
) = (
α
◦
x
)
⊕
(
β
◦
x
)
.
A7: (
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 Spring '08
 KIM
 Linear Algebra, Algebra, Real Numbers, Addition, Multiplication, Vector Space, Closure, arbitrary elements

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