Homework List for Math 571 (Summer 2011)
Kiam Heong Kwa
July 16, 2011
Remark 0.
Unless otherwise stated, the exercises are taken from Steven J. Leon, Lin
ear Algebra with Applications 8/e.
Remark 1.
This list will be updated throughout the quarter.
Remark 2.
An exercise with an asterisk (*) is relatively more difﬁcult.
Remark 3.
An exercise in a
box
is an application problem.
Note
that the student is responsible for
making sure that he/she has access to MAT
LAB during the makeup of any quiz or
exam.
5.4
2, 3, 7(c), 9, 10, 11, 17, 18, 20, 21, 22, 24, 26, 28, 29, 30, 32, 33
Exercise 9: Both
m
and
n
are assumed to be integers.
Exercise 17: Use the Pythagorean law. Also, note that
k 
y
k
=
k
y
k
.
Exercise 18: Adapt the proof of the Pythagorean law.
Exercise 20: The nonsingularity of the matrix
A
is to ensure that
k
x
k
A
=
k
A
x
k
2
= 0
if and only if
x
=
0
.
Exercise 22: First show that
k
x
k
2
2
≤ k
x
k
2
1
. Does the result holds for
R
n
, where
n
is an
integer
≥
3
? Yes. But how?
Exercise 26: Simply expand
k
u
+
v
k
2
=
h
u
+
v
,
u
+
v
i
, etc.
Exercise 29: For part (b), if
x
= (
x
1
,x
2
,
···
,x
n
)
, then
k
x
k
2
=
∑
n
i
=1
x
2
i
≤
∑
n
i
=1
max
1
≤
j
≤
n
x
2
j
=
n
max
1
≤
j
≤
n
x
2
j
=
n
(max
1
≤
j
≤
n

x
j

)
2
.
Exercise 30: For part (b), it is the square with vertices
(
±
1
,
0)
and
(0
,
±
1)
. For part (c),
it is the square with vertices
(1
,
±
1)
and
(

1
,
±
1)
.
5.3
1(a), 1(c), 3(a), 4(a), 5, 9, 10, 11, 12, 14
1