ma253
, Fall
2007
— Problem Set
7
Most of this assignment deals with bases and dimension, both in
R
n
and
in more general vector spaces. Linear transformations also come in, and the
rank and nullity theorem will play role.
This is due on
Wednesday, November 7
.
1.
Problems from the textbook:
a. Section
3
.
3
, problems *
18
, *
22
, *
26
, *
28
, *
30
, *
36
, *
38
.
b. Section
4
.
1
, problems *
18
, *
26
.
c. Section
4
.
2
, problems *
22
, *
24
,
25
, *
27
, *
32
,
33
, *
34
, *
42
, *
50
, *
54
,
*
60
.
*2.
In problem
34
from section
4
.
2
, you are asked to work in the space
V
of
all infinite sequences of real numbers and to consider what is usually called
the
right shift operator
, defined by
R
(
x
0
, x
1
, x
2
, x
3
, . . .
) = (
0, x
0
, x
1
, x
2
, x
3
, . . .
)
.
Show that ker
(
R
) =
{
0
}
, where
0
means the zero vector in
V
, i.e., the se-
quence
(
0, 0, 0, 0, . . .
)
. Is
R
invertible?
*3.
For linear transformations
T
:
R
n
→
R
n
(or, equivalently, for
n
×
n
matrices), we showed that if ker
(
T
) =
{
0
}
then
T
is invertible. Compare this
with the previous problem. What’s going on?
*4.
We can also define the
left
shift operator
L
on the space of all infinite
sequences:
L
(
x
0
, x
1
, x
2
, x
3
, . . .
) = (