MATH 224 LINEAR ALGEBRA
Problem Set 5
Due August 11 in Tutorial
1.
Find an explicit solution to the recursive relation:
a
n
=
a
n

1
+
3
a
n

2
given the initial conditions
a
0
=
1,
a
1
=
2
.
2.
Let
T : V
→
V
be a linear operator over
F
. Recall that a subspace
W
of
V
is called
T
cyclic
if it is a
T
invariant subspace of
V
with the property that it has a nonzero vector
w
∈
W
such that:
W
=
span
F
{
w
,T
w
,...,T
k
w
},
for some number
k
. Also recall the de±nition of an
annihilator subspace
with respect to a polynomial
q
(
x
)
∈
F
[
x
]
:
K
q
(
x
)
=
{
v
∈
V

q
(T)
v
=
0}.
Suppose that
W
is a
T
cyclic subspace of
V
. Show that
W
⊂
K
p
(
x
)
for some polynomial
p
(
x
)
.
3.
Are the matrices
A
=
1

4
0
1

2
1
1

3
2
,
B
=
1

2

2
1
1
1
0

2

1
,
conjugate over
Q
? Show your reasoning.
4.
Let
T : V
→
V
be a linear operator over the ±eld
F
5
. Suppose
A
=
[T]
α
α
is a matrix representation of
T
with respect to some basis and
c
A
(
x
)
=
(1

x
)(
x

3)
2
(
x
2
+
2)
2
. Write a list of all possible
elementary
divisors
,
invariant factors
, and
rational canonical forms
that the matrix
A
can have.