Thread: Linear dependence

hi


Determine whether the functions in the following set are linearly dependent or linearly independent. If they are linearly dependent find a linear equation which they satisfy.



{ ln(x - 1), 2ln(x + 1), 3ln(x^2 - 1) } for x > 2


i know the Wronskian is W(ln(x - 1), 2ln(x + 1), 3ln(x^2 - 1))

= 12ln(x^2 - 1)/((x - 1)^2(x + 1)^2) - 12ln(x + 1)/((x - 1)^2(x + 1)^2)

- 12ln(x - 1)/((x - 1)^2(x + 1)^2)


and if i put x = 5 say, then this equals zero, so equations are lin dependent for some interval I.


My problem is finding a linear equation which they satisfy!


I understand that

12ln(x^2 - 1)/((x - 1)^2(x + 1)^2) - 12ln(x + 1)/((x - 1)^2(x + 1)^2)

- 12ln(x - 1)/((x - 1)^2(x + 1)^2) = 0

how do i explain the rest of the question??


help appreciated

I found the Wronskian to be:



This equals 0 for

The function is not 0 over , so it must be
linearly independent. Correct?.

Here's an interesting example of an linearly dependent set we can find by using identities.



Since

This equals 0 for all x. So, it's linearly dependent.

hi

thanks for the insight, i'm gonna have to get to grips with this, may come up on the exam, i am just struggling with the;

If they are linearly dependent (which they are!) find a linear equation which they satisfy part of the question!


ta