ma253
,Fa
l
l
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 in±nite sequences of real numbers and to consider what is usually called
the
right shift operator
, de±ned 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