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MA 35100 HOMEWORK ASSIGNMENT #6 SOLUTIONS
Problem 1.
pg. 133; prob. 21
In Exercises 21 through 25, ﬁnd the reduced rowechelon form for the given matrix
A
. Then ﬁnd
a basis for the image of
A
and a basis for the kernel of
A
.
A
=
1 3 9
4 5 8
7 6 3
.
Solution:
First we ﬁnd the reduced rowechelon form for
A
. Subtract 4 times the ﬁrst row from
the second, and 7 times the ﬁrst row from the third:
1
3
9
0

7

28
0

15

60
.
Now divide the second row by

7 and the third row by

15:
1 3 9
0 1 4
0 1 4
FInally, subtract 3 times the second row from the ﬁrst, and subtract the second row from the third:
rref(
A
) =
1 0

3
0 1
4
0 0
0
The pivot columns of
A
are the ﬁrst and second columns. Using Theorem 3.3.5 on page 128 of the
text,
A basis for the image of
A
is
1
4
7
,
3
5
6
.
Using the reduced rowechelon form for
A
, we see that an element in the kernel of
A
is necessarily
in the form
c
1
c
2
c
3
=
3
c
3

4
c
3
c
3
=
c
3
3

4
1
We conclude that
A basis for the kernel of
A
is
3

4
1
.
Problem 2.
pg. 133; prob. 22
1
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View Full DocumentIn Exercises 21 through 25, ﬁnd the reduced rowechelon form for the given matrix
A
. Then ﬁnd
a basis for the image of
A
and a basis for the kernel of
A
.
A
=
2 4 8
4 5 1
7 9 3
.
Solution:
First we ﬁnd the reduced rowechelon form for
A
. Divide the ﬁrst row by 2:
1 2 4
4 5 1
7 9 3
Now subtract 4 times the ﬁrst row from the second, and subtract 7 times the ﬁrst row from the
third:
1
2
4
0

3

15
0

5

25
Divide the second row by

3, and the third row by

5:
1 2 4
0 1 5
0 1 5
Finally, subtract twice the second row from the ﬁrst, and subtract the second row from the third:
rref(
A
) =
1 0

6
0 1
5
0 0
0
The pivot columns of
A
are the ﬁrst and second columns. Using Theorem 3.3.5 on page 128 of the
text,
A basis for the image of
A
is
2
4
7
,
4
5
9
.
Using the reduced rowechelon form for
A
, we see that an element in the kernel of
A
is necessarily
in the form
c
1
c
2
c
3
=
6
c
3

5
c
3
c
3
=
c
3
6

5
1
We conclude that
A basis for the kernel of
A
is
6

5
1
.
Problem 3.
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 Spring '10
 EGoins
 Linear Algebra, Algebra

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