ma253
, Fall
2007
— Problem Set
4
This problem set deals with linear transformations and matrices, as be-
fore, with special attention to the issue of invertibility and to matrix mul-
tiplication and its meaning. Plus, just a touch on kernels and images. As
usual, solve all the problems, then write up and turn in those marked with
an asterisk. This assignment is due on
Wednesday, October 10
.
1.
Problems from the textbook:
a. Section
2
.
3
, problems
1
, *
2
, *
4
, *
12
,
29
, *
30
,
31
, *
34
, *
43
.
b. Section
2
.
4
, problems
31
, *
32
,
33
, *
36
, *
38
, *
44
,
45
, *
46
, *
86
.
c. Section
3
.
1
, problems *
2
,
4
, *
8
, *
10
.
*2.
Suppose
A
is a
3
×
3
matrix with the property that,
for every other
3
×
3
matrix
X
, we have
AX
=
XA
. Show that
A
=
⎡
⎣
r
0
0
0
r
0
0
0
r
⎤
⎦
for some real number
r
.
(Hint: if it commutes with
every
X
, then it commutes with any matrix
you choose; choose simple ones, such as the elementary matrices we dis-
cussed in class.)
3.
Show that the result in the previous problem is also true for
n
×
n
matrices,
no matter what
n
is.