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
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Exercise 10: For part (a), recall that
N
(
A
T
) =
R
(
A
)
⊥
by the fundamental subspaces
theorem in section 5.2. Also, recall that the system
A
x
=
b
is consistent
if and only if
b
∈
R
(
A
)
, the column space of
A
. For part (b), the system
A
x
=
b
have infinitely many least squares solutions. To see this, note that
the normal system
A
T
A
x
=
A
T
b
=
0
is a homogeneous system and is
thus consistent because it has the trivial solution
x
=
0
. It has infinitely
many solutions because
rank(
A
T
A
) = rank(
A
) = 3
<
5
, while
A
T
A
is
5
×
5
.
Exercise 11: Recall that an
m
×
m
matrix
M
is called symmetric if
M
T
=
M
.
Exercise 12: First convince yourself that
r
∈
N
(
A
T
) =
R
(
A
)
⊥
. Then recall that
b
x
is a
least squares solution of
A
x
=
b
if and only if
b

A
b
x
∈
R
(
A
)
⊥
.
Exercise 14: Refer to application 3 on pp. 229230 of the text.
5.2
1(c), 1(d), 2, 4, 5, 6*, 7, 9, 11*, 13*, 14, 15, 16*
Exercises 2, 4: Adapt the solutions to exercise 3 in the notes.
Exercise 5: Use the fundamental subspaces theorem (i.e., theorem 5.2.1) and theorem
5.2.3.
Any one dimensional subspace of
R
3
is a line passing through
the origin, while any two dimensional subsapce of
R
3
is a plane passing
through the origin.
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 Summer '08
 KIM
 Linear Algebra, Algebra

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