Let ℝ + be the set of positive real numbers. On ℝ + we define the “exotic” operations x ⊕ y = x y (usual multiplication) and k ⊙ x = x k . a. Show that ℝ + with these operations is a linear space; find a basis of this space. b. Show that T ( x ) = ln ( x ) is a linear transformation from ℝ + to ℝ , where ℝ is endowed with the ordinary operations. Is T an isomorphism?
Let ℝ + be the set of positive real numbers. On ℝ + we define the “exotic” operations x ⊕ y = x y (usual multiplication) and k ⊙ x = x k . a. Show that ℝ + with these operations is a linear space; find a basis of this space. b. Show that T ( x ) = ln ( x ) is a linear transformation from ℝ + to ℝ , where ℝ is endowed with the ordinary operations. Is T an isomorphism?
Solution Summary: The author explains that the set R+ with the exotic operations is a linear space and also find the basis of this space.
Let
ℝ
+
be the set of positive real numbers. On
ℝ
+
we define the “exotic” operations
x
⊕
y
=
x
y
(usual multiplication) and
k
⊙
x
=
x
k
. a. Show that
ℝ
+
with these operations is a linear space; find a basis of this space. b. Show that
T
(
x
)
=
ln
(
x
)
is a linear transformation from
ℝ
+
to
ℝ
, where
ℝ
is endowed with the ordinary operations. Is T an isomorphism?
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