Solutions to PS 6 (Math 121)
J. Scholtz
November 15, 2006
Question 1: 5.1/1
The answers are at the back of the book.
Question 2: 5.1/3b
Find the eigenvalues, corresponing eigenvectors, if possible the eigenbasis and in such case also the
diagonal matrix that is equivalent to the following matrix:
A
=
0

2

3

1
1

1
2
2
5
By evaluating det(
A

tI
) =

t
3
+6
t
2

11
t
+6 =

(
t

1)(
t

2)(
t

3), we get that the eigenvalues are 3
,
2
,
1.
Note that since we have three distinct eigenvalues, of which each must have an eigenspace of dimension
at least 1, we now that the matrix is diagonalizable with each eigenspace having dimension exactly 1.
Then by ﬁnding the nullspaces for
A

λI
we can identify the eigenspaces:
E
1
= span
{
(

1
,

1
,
1)
}
,
E
2
= span
{
(

1
,
1
,
0)
}
and
E
3
= span
{
(

1
,
0
,
1)
}
. Therefore the diagonal matrix is:
D
=
1
0
0
0
2
0
0
0
3
and the change of basis matrix is:
Q
=

1

1

1

1
1
0
1
0
1
Question 3: 5.1/4e
For the following linear operator ﬁnd the eigenvalues of
T
and an ordered basis such that [
T
]
β
is diagonal.
let
V
=
P
2
(
R
) and
T
(
f
(
x
)) =
xf
0
(
x
) +
f
(2)
x
+
f
(3).
let us start in the standard basis:
T
(1) =
x
+ 1,
T
(
x
) = 3
x
+ 3,
T
(
x
2
) = 2
x
2
+ 4
x
+ 9. This correpsonds
to a matrix:
A
=
1
3
9
1
3
4
0
0
2
And det(
A

tI
) =

t
(
t

2)(
t

4). So our eigenvalues are
{
0
,
2
,
4
}
and the corresponding eigenvectors
are obtained by solving for the nullspaces of
A

λI
:
v
1
= (

3
,
1
,
0),
v
2
= (

3
,

13
,
4) and
v
3
= (1
,
1
,
0)
and these deﬁne the basis in which [
T
] is diagonal.