MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2011, Key to HW#3
1. Problem 2 (e), (f), (i) on Page 41.
Solution
:
(e) Write
a
1
((1
,

1
,
2) +
a
2
(1
,

2
,
1) +
a
3
(1
,
1
,
4) = (0
,
0
,
0)
.
By solving the system of equations
a
1
+
a
2
+
a
3
= 0
,

a
1

2
a
2
= 0
,
2
a
1
+
a
2
+ 4
a
3
= 0
we get
a
1
=

3
,a
2
= 2
,a
3
= 1. So they are linearly dependent.
(f) Write
a
1
((1
,

1
,
2) +
a
2
(2
,
0
,
1) +
a
3
(

1
,
2
,

1) = (0
,
0
,
0)
.
By solving the system of equations, we get
a
1
=
a
2
=
a
3
= 0. So it is linearly
independent.
(i) Write
a
1
(
x
4

x
3
+5
x
2

8
x
+6)+
a
2
(

x
4
+
x
3

5
x
2
+5
x

3)+
a
3
(
x
4
+3
x
2

3
x
+5)
+
a
4
(2
x
4
+ 3
x
3
+ 4
x
2

x
+ 1) +
a
5
(
x
3

x
+ 2) = 0
.
So we get system of equations
a
1

a
2
+
a
3
+ 2
a
4
= 0
,....
,
6
a
1

3
a
2
+ 5
a
3
+
a
4
+ 2
a
5
= 0
.
By solving this, we get
a
1
=
···
=
a
5
= 0. Hence it is linearly independent.
2. Problem 3 on Page 41.
Solution
: Omitted.
3. Problem 13 on Page 42.
Solution
:
(a) This is a ”twoway” statement. We ﬁrst prove ”=
⇒
”, i.e. assume
that
{
u,v
}
is linearly independent, we prove that