khan (sak2454) – HW05 – gilbert – (57245)
1
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001
10.0 points
Let
T
:
R
2
→
R
2
be the transFormation
T
(
x
) =
x
1
b
3

1
B
+
x
2
b

4
2
B
.
Determine
T
(
x
) when
x
=
b
2
1
B
.
1.
T
(
x
) =
b
3

1
B
2.
T
(
x
) =
b
1

1
B
3.
T
(
x
) =
b
3
0
B
4.
T
(
x
) =
b
1
0
B
5.
T
(
x
) =
b
2

1
B
6.
T
(
x
) =
b
2
0
B
correct
Explanation:
By matrixvector multiplication,
x
1
u
+
x
2
v
= [
u v
]
b
x
1
x
2
B
.
Thus
T
(
x
) = [
u v
]
b
x
1
x
2
B
=
b
3

4

1
2
Bb
x
1
x
2
B
,
For each
x
.
Consequently, when
x
=
b
2
1
B
,
T
(
x
) =
b
3

4

1
2
2
1
B
=
b
2
0
B
.
002
10.0 points
IF
T
:
R
2
→
R
2
is the linear transFormation
such that
T
pb
x
1
x
2
BP
=
x
1
b

2
2
B
+
x
2
b

1
1
B
,
determine
T
(
x
) when
x
=
b
1
3
B
.
1.
T
(
x
) =
b

5
4
B
2.
T
(
x
) =
b

4
4
B
3.
T
(
x
) =
b

4
5
B
4.
T
(
x
) =
b

5
5
B
correct
5.
T
(
x
) =
b
1
3
B
Explanation:
When
x
=
b
1
3
B
=
b
1
0
B
+ 3
b
0
1
B
,
then
T
(
x
) =
T
pb
1
0
BP
+ 3
T
pb
0
1
BP
=
b

2
2
B
+ 3
b

1
1
B
.
Consequently,
T
(
x
) =
b

2
2
B
+
b

3
3
B
=
b

5
5
B
.
003
10.0 points
IF
A
is an
m
×
n
matrix, then the range oF
the transFormation
T
A
:
x
→
A
x
is
R
m
.
True or ±alse?
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2
1.
FALSE
correct
2.
TRUE
Explanation:
When
A
is
m
×
n
, the matrix product
A
x
is only defned ±or
x
in
R
n
, and the product
is then a vector in
R
m
. Thus
T
A
:
x
→
A
x
maps
R
n
to
R
m
. So the
Range
o±
T
A
is the set
{
T
A
(
x
) :
x
in
R
n
}
o± vectors in
R
m
. But this need not be
all
o±
R
m
.
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 Fall '14
 Linear Algebra, Vector Space, TA, X1, linear transformation

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