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3.3: LINEAR INDEPENDENCE
AND THE WRONSKIAN
KIAM HEONG KWA
Generally, two functions
f
(
t
) and
g
(
t
) are said to be
linearly indepen
dent
on an interval
I
if the linear relation
(1)
c
1
f
(
t
) +
c
2
g
(
t
) = 0
,
where
c
1
and
c
2
are constant scalars, holds for all
t
∈
I
only if
c
1
=
c
2
= 0. Otherwise,
f
(
t
) and
g
(
t
) are said to be
linearly dependent
on
I
. In the latter case, there are constants
c
1
and
c
2
with the property
that either
c
1
6
= 0 or
c
2
6
= 0 such that the linear relation (1) holds on
I
. Note that if
c
1
6
= 0, then
f
(
t
) =

±
c
2
c
1
²
g
(
t
), while if
c
2
6
= 0, then
g
(
t
) =

±
c
1
c
2
²
f
(
t
) throughout the interval
I
. In other words,
f
(
t
) and
g
(
t
) are linearly dependent on
I
if one of them is a constant multiple
of the other throughout the interval
I
.
A relationship between the Wronskian and the linear independence
of two functions is provided by the following theorem.
Theorem 1.
1
If
f
(
t
)
and
g
(
t
)
are diﬀerentiable functions on an open
interval
I
such that
W
(
f,g
)(
t
0
)
6
= 0
for some point
t
0
in
I
, then
f
(
t
)
and
g
(
t
)
are linearly independent on
I
. Equivalently, if
f
(
t
)
and
g
(
t
)
are linearly dependent on
I
, then
W
(
f,g
)(
t
) = 0
for all
t
in
I
.
Proof.
Let
c
1
and
c
2
be constants such that
c
1
f
(
t
) +
c
2
g
(
t
) = 0 on
I
.
Then, clearly,
c
1
f
0
(
t
) +
c
2
g
0
(
t
) = 0 on
I
as well. In particular,
c
1
f
(
t
0
) +
c
2
g
(
t
0
) = 0
,
(2a)
c
1
f
0
(
t
0
) +
c
2
g
0
(
t
0
) = 0
.
(2b)
Multiplying (2a) and (2b) by
g
0
(
t
0
) and
g
(
t
0
) respectively gives
c
1
f
(
t
0
)
g
0
(
t
0
) +
c
2
g
(
t
0
)
g
0
(
t
0
) = 0
,
(3a)
c
1
g
(
t
0
)
f
0
(
t
0
) +
c
2
g
(
t
0
)
g
0
(
t
0
) = 0
.
(3b)
Date
: January 17, 2011.
1
This is theorem 3.3.1 in the text.
1
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KIAM HEONG KWA
Now substracting (3b) from (3a) yields
c
1
W
(
f,g
)(
t
0
) = 0. Hence
if
W
(
f,g
)(
t
0
)
6
= 0, then
c
1
= 0.
Likewise, we can show that if
W
(
f,g
)(
t
0
)
6
= 0, then
c
2
= 0 as well. Hence if
W
(
f,g
)(
t
0
)
6
= 0, then
f
(
t
) and
g
(
t
) are linearly independent.
±
Remark 1.
The converse of theorem 1 does not hold without additional
conditions on the pair of functions involved. That is to say, it is not
true in general that two diﬀerentiable functions
f
(
t
)
and
g
(
t
)
with a
vanishing Wronskian on an open interval must be linearly dependent on
the same interval. See the following example due to Giuseppe Peano,
a famous Italian mathematician.
Example 1.
Consider the diﬀerentiable functions
f
(
t
) =
t
2

t

and
g
(
t
) =
t
3
on
R
. It can be shown that
f
0
(
t
) =
(

3
t
2
if
t <
0
,
3
t
2
if
t
≥
0
.
Hence
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