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Math 235 Assignment 8
Due 9:15am, Wednesday March 21, 2007.
1. From the Text
§
5.4 #6 and determine the rank of
T
. Is
T
injective? Is
T
surjective?
§
7.1 #24.
2. Consider the following linear recurrence equation:
x
n

4
x
n

1
+ 5
x
n

2
= 0
for
n
≥
2
,
with initial conditions
x
0
= 1 and
x
1
= 1
.
In this question you are
asked to ﬁnd
x
n
as a function of
n
through linear algebra by taking
the following steps.
No
credit will be given for any other approach.
(a) Let
v
n
= (
x
n
, x
n

1
)
t
for
n
≥
1 (so
v
n
is a column vector). Express
the recurrence equation in the form
v
n
=
Av
n

1
for
n
≥
2
,
where
A
is a 2
×
2 matrix. Hence, express
v
n
in terms
of
A
and
v
1
.
(b) Find an expression for
A
N
of the form
a
(
N
)
A
+
b
(
N
)
I
where
N
≥
2 is an integer,
a
(
N
) and
b
(
N
) are functions of
N,
and
I
is
the 2
×
2 identity matrix.
(c) Hence express
x
n
explicitly in terms of
n
alone.
3. Let
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This note was uploaded on 01/17/2011 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.
 Spring '08
 CELMIN
 Math

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