Problem Set 4
Math 2270-001 (Summer 2005)
Due: Thursday June 23, 2005
1
Please label the problems with Chapter/Section/Problem (e.g. Exercise 2.3.42, Exercise 2.3.54, etc) in your homework.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Exercises 4.2.34, 4.2.58.
Exercise 4.2.36.
Exercise 4.2.70.
Exercise 4.2.78.
Exercise 4.2.80.
Exercises 4.3.14, 4.3.44
Exercise 4.3.30.
Exercise 4.3.38.
Exercise 4.3.50.
10. Exercise 4.3.58.
Note:
~
x
2
4
1
1
1
3
5 is, by denition, the real number
x1
+
x2
x3
.
11. Exercises 4.3.60 (a), (b).
12. Exercise 4.3.64.
13. Denition An (arbitrary) map : ! from a set to a set is said to be invertible if there exists a map
: ! such that = 1Y and = 1X , where 1X and 1Y are the identity maps on and respectively.
f
g
a)
b)
c)
Y
X
f
g
X
g
Y
X
Y
f
X
Prove that a map is invertible if and only if it is bijective.
Prove that a linear map is invertible if and only if it is an isomorphism.
Suppose :
! is an invertible map. Prove that the map : ! as in the above denition is
unique. This map is called the inverse map of and will be denoted by 1 .
Prove that the inverse map of an invertible linear map is itself linear.
f
X
Y
g
Y
f
d)
Y
X
f
14. Let , ,
be nite-dimensional vector spaces (over F) and E , F , G be bases for them respectively. Let
2 L( ) and 2 L( ).
a) Recall that 2 L(
). Express MG ( (E )) in terms of MF ( (E )) and MG ( (F )).
b) Suppose 2 (
) is invertible. Express MF ( 1 (G )) in terms of MG ( (F ))
U
f
V
W
U; V
g
g
f
V; W
f
L V; W
U; W
g
f
f
f
Please turn over.
g
f
Problem Set 4
Math 2270-001 (Summer 2005)
Due: Thursday June 23, 2005
2
15. Maple Problem Let : R3 ! R2 be dened by
T
02 31 @4 5A := 3
x
T
x
y
x
z
+2 4
5 +3
y
y
z
z
Use Xmaple or Maple to nd MC ( (B)), where
82 1 3 2 1 3 2 1 39
2 <
=
B := :4 1 5 4 1 5 4 0 5; and C := 13
5
1
0
0
as follows: Let E and F be the \standard" bases for R3 and R2 respectively, i.e.
82 1 3 2 0 3 2 0 39
0 <
=
E := :4 0 5 4 1 5 4 0 5; and F := 10
1
0
0
1
Recall that we have the general formula:
MC ( (B)) = MC (F ) MF ( (E )) ME (B)
a) Determine each of the three matrices in the right-hand-side of the above equation, and dene them in Maple
using the command linalg[matrix].
(Reminder: It is easy to read o ME (B) and MF ( (E )), but it is not so easy for MC (F ). However, recall that
MC (F ) = MF (C ) 1 and reading o MF (C ) is again easy. Hence, you may nd the command linalg[inverse]
useful.)
b) Compute the required matrix product using the command evalm. (Do NOT use multiply.)
c) Submit your Maple worksheet along with the rest of your problem set.
Remark: Look up how to use the aforementioned commands in the Maple help pages as necessary. For example,
to call up the Maple help page for, say, evalm, you can simply type at the Maple command prompt the following:
T
;
;
;
;
;
;
T
:
T
T
>?
evalm
and then hit ENTER.
1.
2.
3.
4.
5.
6.
7.
8.
9.
The following are suggested exercises (for developing computational prociency).
Do NOT submit them. They will not be graded.
However, they do contain test and examination material.
Exercise 4.2.8.
Exercise 4.2.18.
Exercises 4.3.24, 4.3.46.
Exercise 4.2.72.
Exercise 4.2.74.
Exercise 4.3.48.
Exercise 4.3.52.
Exercise 4.3.67.
Let V , W be nite-dimensional vector spaces and E , F be bases for them respectively. Let f 2 L(V; W ). Prove that
rank(f ) = rank(MF (f (E )).
10. Let V , W be nite-dimensional vector spaces with dim(V ) = dim(W ) =: n. Let v1 ; : : : ; vn form a basis for V and w1 ; : : : ; wn
form a basis for W . Dene a map f : V ! W by:
f (1 v1 + + n vn ) := 1 w1 + + n wn ; for all scalars 1 ; : : : ; n :
Prove that f is an isomorphism.