ma253
, Fall
2007
— Problem Set
3
This problem set deals mostly with linear transformations and matrices, but
there’s also some stuff about more general linear spaces.
As usual, solve all the
problems, then write up and turn in those marked with an asterisk. This assignment
is due on
Wednesday, October 3
.
1.
Problems from the textbook:
a. Section
2
.
1
, problems
1
,
3
, *
4
,
5
, *
6
,
9
, *
10
, *
16
,
18
,
19
, *
20
,
24
–
30
.
40
,
*
49
, *
52
.
b. Section
2
.
2
, problems
1
,
4
,
5
,
7
, *
18
,
19
, *
34
c. Section
2
.
4
, problems
1
–
13
, *
14
,
15
, *
16
,
17
,
19
, *
20
, *
76
2.
The matrix
A
=

0.6
1
0

0.4
0
1
,
corresponds to a linear transformation
T
:
R
3
→
R
2
. Consider the unit cube in
R
3
defined by the standard basis vectors
e
1
=
⎡
⎣
1
0
0
⎤
⎦
e
2
=
⎡
⎣
0
1
0
⎤
⎦
e
3
=
⎡
⎣
0
0
1
⎤
⎦
.
In
R
2
, draw the image under
T
of each of these vectors and of the unit cube they
determine.
3.
The matrix
A
in the previous problem gives one way to represent threedimensional
objects on a twodimensional page. Does
A
send any
nonzero
vectors in
R
3
to the
zero vector in
R
2
? In terms of the interpretation of the matrix as a way to “draw”