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Linear Independence & Wronskian

This document discusses using the Wronskian to determine linear independence of functions that may or may not solve a homogeneous linear system of differential equations. It states that if functions solve such a system, the Wronskian being zero everywhere implies linear dependence, while being nonzero anywhere implies linear independence. However, for general functions not solving a system, a zero Wronskian everywhere does not imply linear dependence, as shown with an example of two linearly independent functions with a zero Wronskian. The document concludes that while a nonzero Wronskian implies linear independence, the Wronskian is not fully reliable on its own for determining linear dependence or independence of general functions.

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Sergio Cubides
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375 views1 page

Linear Independence & Wronskian

This document discusses using the Wronskian to determine linear independence of functions that may or may not solve a homogeneous linear system of differential equations. It states that if functions solve such a system, the Wronskian being zero everywhere implies linear dependence, while being nonzero anywhere implies linear independence. However, for general functions not solving a system, a zero Wronskian everywhere does not imply linear dependence, as shown with an example of two linearly independent functions with a zero Wronskian. The document concludes that while a nonzero Wronskian implies linear independence, the Wronskian is not fully reliable on its own for determining linear dependence or independence of general functions.

Uploaded by

Sergio Cubides
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Math 54: Linear independence and the Wronskian

May 1

Consider n functions x1 (t), x2 (t), . . . , xn (t) which take values in Rn . The Wronskian is
defined
W [x1 , . . . , xn ](t) = det [x1 (t) x2 (t) · · · xn (t)] .
If x1 (t), x2 (t), . . . , xn (t) are solutions of a homogeneous linear system of differential equa-
tions, i.e., of x0 (t) = A(t)x(t), then exactly one of the following cases holds:

(1) W [x1 , . . . , xn ](t) = 0 for all t, in which case x1 , . . . , xn are linearly dependent
(2) W [x1 , . . . , xn ](t) = 0 for NO values of t, in which case x1 , . . . , xn are linearly
independent

This implies a few things:

• If x1 , . . . , xn solve a homogeneous linear system of differential equations and you


can find any value of t for which W [x1 , . . . , xn ](t) = 0, then x1 , . . . , xn are linearly
dependent, hence do not form a fundamental solution set.
• If x1 , . . . , xn solve a homogeneous linear system of differential equations and you
can find any value of t for which W [x1 , . . . , xn ](t) 6= 0, then x1 , . . . , xn are linearly
independent, hence form a fundamental solution set.
• If someone gives you some functions x1 , . . . , xn and the corresponding Wronskian is
zero for at least one value but not all values of t, then x1 , . . . , xn CANNOT all be
solutions of a single homogeneous linear system of differential equations.

Okay now let’s consider what the Wronskian has to say when x1 , . . . , xn are not necessarily
solutions of a homogeneous linear system of differential equations. The following fact holds:

• If x1 , . . . , xn are linearly dependent, then W [x1 , . . . , xn ](t) = 0 for all t.

However, the converse does not hold. To see this, consider


✓ ◆ ✓ ◆
1 t
x1 (t) = , x2 (t) = .
0 0
Notice that the Wronksian is zero for all t, but x1 and x2 are linearly independent. Why?
Suppose that there are constants c1 and c2 such that c1 x1 (t) + c2 x2 (t) = 0 for all t (in other
words, such that c1 x1 + c2 x2 is the zero function). Then in fact c1 + c2 t = 0 for all t, but this
means that c1 = c2 = 0, and we have proved linear independence. Thus even though the
vectors x1 (t) and x2 (t) are linearly dependent for every value of t, the functions
x1 and x2 are linearly independent!

We also have the following fact (the contrapositive of the last one):

• If W [x1 , . . . , xn ](t) 6= 0 for some t, then x1 , . . . , xn are linearly independent.

In summary, the Wronskian is not a very reliable tool when your functions are not solutions
of a homogeneous linear system of differential equations. However, if you find that the
Wronskian is nonzero for some t, you do automatically know that the functions are linearly
independent. (But the Wronskian being zero everywhere does not imply that the functions
are linearly dependent, and linear independence does not imply that the Wronskian can’t
be the zero everywhere.)

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