SECTION 4.8 DIFFERENCE EQUATIONS
Electrical engineers, control systems engineers, biologists, physicists, economists, demog
raphers, etc etc, often measure or sample a process at discrete time intervals. This generates
a signal, or a sequence of numbers
{
y
k
}
= (
. . . , y

2
, y

1
, y
o
, y
1
, y
2
, . . .
)
.
Sometimes the terms
y
k
for
k <
0 are either omitted or assumed to be 0, since sampling
usually begins at a certain time, but we will continue to think about signals that go on
forever in both directions.
We will study the collection
S
of all signals.
We’ve already seen that this is a vector
space with addition and scalar multiplication done naturally. So we have available all the
vector space notions of linear independence, subspaces, bases, etc.
WHAT DOES LINEAR INDEPENDENCE IN
S
MEAN?
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EXAMPLE.
Are the signals 1
k
, 3
k
cos
kπ
2
, and 3
k
sin
kπ
2
linearly independent?
One way to make new signals
{
w
k
}
from old ones
{
y
k
}
is by setting
w
k
=
a
0
y
k
+
n
+
a
1
y
k
+
n

1
+
· · ·
+
a
n

1
y
k
+1
+
a
n
y
k
,
where the
a
i
’s are scalars with
a
0
6
= 0 and
a
n
6
= 0. This is often called a
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 Spring '08
 PAVLOVIC
 Linear Algebra, Vector Space, yk, linear transformation

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