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Transcript
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“main”
2007/2/16
page 275
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4.5
Linear Dependence and Linear Independence
275
Solution:
1. Since we have five vectors in R4 , Corollary 4.5.15 implies that {v1 , v2 , v3 , v4 , v5 }
is necessarily linearly dependent.
2. In this case, we have four vectors in R4 , and therefore, we can use the determinant:
1 3 2 −2
4 −5 −1 3
det(A) = det[v1 , v2 , v3 , v4 ] =
= −462.
1 2 6 1
7 3 9 6
Since the determinant is nonzero, it follows from Corollary 4.5.15 that the given
set of vectors is linearly independent.
Linear Independence of Functions
We now consider the general problem of determining whether or not a given set of
functions is linearly independent or linearly dependent. We begin by specializing the
general Definition 4.5.4 to the case of a set of functions defined on an interval I .
DEFINITION 4.5.17
The set of functions {f1 , f2 , . . . , fk } is linearly independent on an interval I if and
only if the only values of the scalars c1 , c2 , . . . , ck such that
c1 f1 (x) + c2 f2 (x) + · · · + ck fk (x) = 0,
for all x ∈ I ,
(4.5.6)
are c1 = c2 = · · · = ck = 0.
The main point to notice is that the condition (4.5.6) must hold for all x in I .
A key tool in deciding whether or not a collection of functions is linearly independent
on an interval I is the Wronskian. As we will see in Chapter 6, we can draw particularly
sharp conclusions from the Wronskian about the linear dependence or independence of
a family of solutions to a linear homogeneous differential equation.
DEFINITION 4.5.18
Let f1 , f2 , . . . , fk be functions in C k−1 (I ). The Wronskian of these functions is
the order k determinant defined by
W [f1 , f2 , . . . , fk ](x) =
f1 (x)
f1 (x)
..
.
f2 (x)
f2 (x)
..
.
(k−1)
(k−1)
f1
(x) f2
...
...
fk (x)
fk (x)
..
.
(k−1)
(x) . . . fk
.
(x)
Remark Notice that the Wronskian is a function defined on I . Also note that this
function depends on the order of the functions in the Wronskian. For example, using
properties of determinants,
W [f2 , f1 , . . . , fk ](x) = −W [f1 , f2 , . . . , fk ](x).
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