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Math 222  200
Solutions Exam 2
April 1, 2005
1. (30) Let
T
:
V
→
V
be a linear transformation. Let
{
~v
1
,
2
,
3
}
be a basis of
V
. Suppose the
matrix representation
A
of
T
with respect to this basis is
A
=
120
01

4

2

50
.
(a) In terms of the basis vectors what does
T
(
1
) equal?
The ﬁrst column of
A
contains the coordinates of
T
(
1
), so
T
(
1
)=
~
v
1

2
~
v
3
(b) Does the equation
T
(
~x
~
v
2

~
v
3
have a solution? If yes, ﬁnd it, if no why not.
In terms of coordinates the question is: does there exist an
~y
∈
R
3
such that
A~y
=
0
1

1
The augmented matrix for this system is
1200

41

2


1
=
⇒
100

2
010 1
001 0
Thus,
~
y
=[

2, 1, 0], and the solution to the original equation is
=

2
1
+
2
2. (50) A linear transformation
E
:
V
→
V
is called a projection if it satisﬁes the equation
E
2
=
E
.
(a) Let
V
=
R
3
.T
h
es
e
t
U
=
{
~
u
1
,
~
u
2
,
~
u
3
}
=
{h
1, 1, 0
i
,
h
1, 0, 1
i
,
h
0, 1, 1
i}
is a basis of
R
3
. Deﬁne
E
by
E
(
λ
1
~u
1
+
λ
2
2
+
λ
3
3
λ
1
~
u
1
. Show that
E
is a projection. Note,
this means you need to show that
E
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 Spring '08
 Stecher
 Math, Linear Algebra, Algebra

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