MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 23
Linear Independence
Linear Independence.
Let
f
=
f
(
t
) and
g
=
g
(
t
) be two differentiable functions. We say that
f
and
g
are
linearly independent
if the only constants
c
1
and
c
2
such that
c
1
f
(
t
) +
c
2
g
(
t
) = 0
for all
t
are
c
1
=
c
2
= 0. Otherwise, we say that
f
and
g
are
linearly dependent
. Note that linearly dependent means
the functions are scalar multiples of each other i.e.,
g
=
λ f
for some constant
λ
=

c
1
/c
2
.
We can compute the Wronskian of
f
and
g
as a function of
t
:
W
(
f, g
)
(
t
) =
f
(
t
)
g
0
(
t
)

f
0
(
t
)
g
(
t
)
.
We will show that the following statements are equivalent:
i.
W
(
f, g
)
(
t
0
)
6
= 0 for some
t
0
.
ii.
W
(
f, g
)
(
t
)
6
= 0 for all
t
.
iii.
f
and
g
are linearly independent.
The phrase “the following are equivalent” has a precise mathematical meaning. The statement “if
p
then
q
” can be thought of in terms of sets: this means
p
is “contained” in
q
, or “the validity of
p
implies the
validity of
q
.” However note the conditional “if”: proposition
p
might not be true at all! The idea is if
p
is true, then so is
q
. The statement “
p
if and only if
q
” is called a biconditional because it consists of two
statements: “if
p
then
q
” and “if
q
then
p
.” When thinking of these as sets, this statement means logically
p
and
q
are equivalent.
The phrase “the following are equivalent” means propositions
p
1
,
p
2
, . . . ,
p
n
are logically equivalent. That
is, we have the statements “if
p
i
then
p
j
” for all
i
and
j
. When
n
= 2, we have a biconditional. When
n
= 3
we have three propositions to consider, which means we have six conditionals:
p
1
p
2
K
;
p
3
s
f
Consider the following propositions:
p
1
= “
W
(
f, g
)
(
t
0
)
6
= 0 for some
t
0
”
p
2
= “
W
(
f, g
)
(
t
)
6
= 0 for all
t
”
p
3
= “
f
and
g
are linearly independent”
We have already seen that
p
1
=
⇒
p
2
; this is Abel’s Theorem. We show that
p
2
=
⇒
p
3
, so assume that
W
(
f, g
)
(
t
)
6
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 Spring '09
 MA
 Linear Algebra, Linear Independence, Vector Space, If and only if, Logical biconditional

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