# Linear dependence

• Oct 10th 2007, 05:49 AM
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
• Oct 10th 2007, 12:18 PM
galactus
I found the Wronskian to be:

$\frac{12ln(x^{2}-1)}{(x-1)^{2}(x+1)^{2}}-\frac{12ln(x+1)}{(x-1)^{2}(x+1)^{2}}-\frac{12ln(x-1)}{(x-1)^{2}(x+1)^{2}}$

This equals 0 for $x\geq{2}$

The function is not 0 over $(-\infty,\infty)$, so it must be
linearly independent. Correct?.

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

$sin^{2}(x), \;\ cos^{2}(x), \;\ 5$

Since $5sin^{2}(x)+5cos^{2}(x)-5=5(sin^{2}(x)+cos^{2}(x))=-5=0$

This equals 0 for all x. So, it's linearly dependent.
• Oct 10th 2007, 02:33 PM