1. The Wronskian and Linear Independence
MATH 374 Lecture 11
The Wronskian and Linear Independence

2. 4.4: The Wronskian
Using some ideas from linear algebra, we can find a way to check if certain sets of functions are linearly independent!
For a review of the ideas we need, see this handout (Coddington – Introduction to ODE, pp ).

3. Check for Linear Independence
Suppose that each of the functions f1, f2, …, fn are at least (n-1) times differentiable on axb.
If f1, f2, … , fn satisfy
c1 f1 + c2 f2 + … + cn fn = 0 (1)
for all x 2 [a, b], it follows that f1, f2, …, fn also must satisfy: 3

4. More Equations f1, f2, … , fn Satisfies
c1 f1’ + c2 f2’ + … + cn fn’ = 0
c1 f1’’ + c2 f2’’ + … + cn fn’’ = 0
(2)
c1 f1(n-1) + c2 f2(n-1) + … + cn fn(n-1) = 0
for all x 2 [a, b]. 4

5. A System of Equations (for each x)
If we ask what choices of c1, c2, … , cn make (1) true, we are immediately led to a system of n equations in the n unknowns c1, c2, … , cn (for each x 2 [a, b]), because (2) must hold. 5

6. A System of Equations (for each x)
c1 f1(x) + c2 f2(x) + … + cn fn(x) = 0
c1 f1’(x) + c2 f2’(x) + … + cn fn’(x) = 0
(3)
c1 f1(n-1)(x) + c2 f2(n-1)(x)+ … + cn fn(n-1)(x) = 0
for each x 2 [a, b]. 6

7. Matrix Form
An equivalent form for (3) is the matrix equation: 7

8. Matrix Form 8

9. A Sufficient Condition for Linear Independence
If for some x0 2 [a, b], det[A(x0)]  0, it follows from Theorem 4 on the Coddington Handout (page 29) that system (3) with x = x0 has the unique solution c1 = c2 = … = cn = 0.
Hence to have (1) hold for all x 2 [a, b], in particular for x = x0, we need ci = 0 for i = 1, 2, … , n.
Therefore {f1, f2, … , fn } are linearly independent on [a, b] !!
We’ve just proved a sufficient condition for linear independence: 9

10. A Sufficient Condition for Linear Independence
Theorem: Let f1, f2, … , fn be (n-1) times differentiable on axb.
Define the Wronskian of {f1, f2, … , fn } to be the function:
If W(x0)  0 for some x0 2 [a, b], then {f1, f2, … , fn } are linearly independent on [a, b]. 10

11. Notes
W(x0)  0 for some x0 2 [a, b] is not a necessary condition for linear independence.
There exist sets of functions {f1, f2, … , fn } with W(x) ´ 0 on [a, b] and {f1, f2, … , fn } are linearly dependent on [a, b] (see handout HW, problem 10). 11

12. Notes
W(x0)  0 for some x0 2 [a, b] is equivalent to {f1, f2, … , fn } being linearly independent on [a, b], if we impose further conditions on {f1, f2, … , fn}. 12

13. A Necessary and Sufficient Condition for Linear Independence
Theorem 4.3: If on the interval axb, b0(x)  0, b0, b1, … , bn are continuous, and y1, y2, … , yn are solutions of the equation
b0(x) y(n) + b1(x) y(n-1) + … + bn-1(x) y’ + bn(x) y = 0,
then {y1, y2, … , yn} are linearly independent on [a,b] if and only if the Wronskian of {y1, y2, … , yn } differs from zero for at least one point x0 2 [a, b].
Proof: HW exercises for the n = 2 case. Induction for n > 2.  13

14. Example 1
For {1, x, x2},
Therefore these functions are linearly independent on any interval!