MAT 202 Midterm Exam, March 10 2004
Average was 51 percent.
1.
(16 points)
The linear transformation
T
transforms vectors in
R
2
in the following way:
first it
rotates the vector through an angle of 30 degrees counterclockwise about the origin, then it doubles
the vector’s length and then it reflects the resulting vector across the line of slope 1 through the
origin. What is the standard matrix of
T
?
Rotation by 30 degrees counterclockwise is given by the matrix
"
cos
π/
6

sin
π/
6
sin
π/
6
cos
π/
6
#
=
"
√
3
/
2

1
/
2
1
/
2
√
3
/
2
#
Doubling the length is accomplished by multiplying this matrix by 2.
Reflection across the line
x
1
=
x
2
is given by the matrix
"
0
1
1
0
#
So the matrix of T is
"
0
1
1
0
# "
√
3

1
1
√
3
#
=
"
1
√
3
√
3

1
#
2.
(16 points)
Let
A
=
1
2
0
2
0
0
1
3

1

2
1
1
2
4

1
1
which row reduces to
rref
(
A
) =
1
2
0
2
0
0
1
3
0
0
0
0
0
0
0
0
.
(a) Find a basis for the image of
A
.
Columns 1 and 3 of rref(A) have lead 1’s so Columns 1 and 3 of A form a basis for the image
of A. Answer:
1
0

1
2
,
0
1
1

1
(b) Find a basis for the kernel of
A
. Answer:

2
1
0
0
,

2
0

3
1
(c) Find a matrix
B
such that the image of
B
is the same as the kernel of
A
.
Answer:
B
=

2

2
1
0
0

3
0
1
(d) Find a matrix
C
such that the kernel of
C
is the same as the image of
A
.
One possible answer is
"
1

1
1
0

2
1
0
1
#
.
3.
(16 points)
Let
V
be the subspace of
R
4
spanned by the vectors
~v
1
=
1
1
1
1
~v
2
=
1
2

1
0
~v
3
=
0
1
1

1
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(a) Find a basis for
V
⊥
.
To compute
V
⊥
we compute
ker
1
1
1
1
1
2

1
0
0
1
1

1
= ker
1
0
0
2
0
1
0

1
0
0
1
0
=

2
t
t
0
t
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 Spring '08
 Staff
 Linear Algebra, Algebra, Vectors

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