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MATH 225
HOMEWORK 8
pp. 351 – 353
#2, 4 – 7, 13 – 19 (odd), 23, 24 – 27, 32 (for all, use the standard bases)
pp. 368 – 369
#1 – 6 (ignore geometric description), 12, 14 – 16
A. From problem 12 on page 249,
V
= ((0, +
∞
),
R
, #,
⋅
) is a vector space over
R
with
a
#
b
=
ab
for any
a, b
∈
R
, and
c
⋅
a
=
a
c
for
a
∈
(0, +
∞
) and
c
∈
R
.
Show that ln(
x
) :
V
→
R
is a linear
transformation.
B. Recall that for functions
f
:
A
→
B
and
g
:
B
→
C
, the composition
g
f
:
A
→
C
is given by
g
f
(
a
) =
g
(
f
(
a
)).
If
T
:
V
F
→
W
F
and
S
:
W
F
→
U
F
are linear transformations, show that
S
T
is
a linear transformation.
C. Let
S
,
T
:
R
1
x
3
→
R
1
x
3
be the linear transformations given by:
T
([
x
,
y
,
z
]) = [
y
+
z
,
x
+
z
,
x
+
y
] and
S
([
x
,
y
,
z
]) = [
–z
,
x
–
z
,
y
+
z
].
Find a similar description of the composition
S
T
.
If
B
= {
ε
i
T
} is the standard basis of
R
1
x
3
,
find the matrices [
S
]
B
and [
T
]
B
, and show that [

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