
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:
$\displaystyle \frac{12ln(x^{2}1)}{(x1)^{2}(x+1)^{2}}\frac{12ln(x+1)}{(x1)^{2}(x+1)^{2}}\frac{12ln(x1)}{(x1)^{2}(x+1)^{2}}$
This equals 0 for $\displaystyle x\geq{2}$
The function is not 0 over $\displaystyle (\infty,\infty)$, so it must be
linearly independent. Correct?.
Here's an interesting example of an linearly dependent set we can find by using identities.
$\displaystyle sin^{2}(x), \;\ cos^{2}(x), \;\ 5$
Since $\displaystyle 5sin^{2}(x)+5cos^{2}(x)5=5(sin^{2}(x)+cos^{2}(x))=5=0$
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