Math 217, Linear Algebra, Fall 2002
Exam 1, October 4, 2002
Name:
Solutions
1.(5pts) Let
A
be a 6
×
5 matrix. What must
a
and
b
be in order to define
T
:
R
a
→
R
b
by
T
(
x
) =
A
x
?
If we are trying to compute
A
x
then
x
must be a length 5 vector. The result
of
A
x
is a length 6 vector. So
a
= 5 and
b
= 6.
2.(5pts) Give an example of a 2
×
2 matrix
A
which has the following three properties:
1)
A
6
=
0
, 2)
A
6
=
I
2
, and 3)
A
T
=
A
.
Try
1
0
0
0
on for size. In fact, any matrix
a
b
b
c
will work as long as if
b
= 0
then both
a
and
d
are not both 1 or both zero.
3.(10pts) Note that the matrix
A
=
4
5
6
7
8
7
6
5
6
7
8
9
is similar to the matrix
B
=
4
5
6
7
0

3

6

9
6
7
8
9
.
Write down an elementary matrix
E
such that
EA
=
B
. What is
E

1
?
You can find an elementary matrix
E
such that
EA
=
B
by performing the
same row operation on the identity as you did on
A
. That is, replace Row(2)
of the identity with Row(2)

2Row(1). Doing this yields the matrix
E
=
1
0
0
0

2
1
0
0
0
0
1
0
0
0
0
1
.
The matrix
E

1
is such that
E

1
E
=
I
4
. So we find
E

1
by performing the same
row operation on the identity as we would perform on
E
to obtain
I
4
. That is,
Row(2)
→
Row(2)+2Row(1) transforms
E
to
I
4
, thus Row(2)
→
Row(2)+2Row(1)
transforms
I
4
to
E

1
=
1
0
0
0
2
1
0
0
0
0
1
0
0
0
0
1
.
4.(5pts) Consider the matrix
A
=
1
2
3
· · ·
20
2
3
4
· · ·
21
.
.
.
.
.
.
19
20
21
· · ·
39
.
Are the columns of
A
linearly independent? (Explain).
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We have a theorem which states that a set of
p
vectors in
R
n
for
p > n
is linearly
dependent. Note that
A
has 20 columns and only 19 rows. That means that the
20 column vectors form a set of vectors
R
19
, and we conclude that they must
be linearly dependent.
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 Fall '08
 MINETTI
 Linear Algebra

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