Linearly Dependent? Clarifying Wronskian Results

[kdinser]

I just want to make sure I'm clear on the whole linearly dependent thing.

If I find the Wronskian of a set of functions and it comes out:

12x^2 + 12x

This would indicate that my set of functions is linearly dependent if the interval included x=0 and would be linearly independent if x never equaled 0.

if I find the wronskian of a set of functions and it comes out:

6 + 12x

This would show that my set of functions is linearly independent for all x.

 

[HallsofIvy]

Would it? what if x= -1/2?

Of course, on can show that if the functions involved all satisfy the same linear, homogeneous differential equation, THEN their Wronskian is either always 0 or never 0.

 

[kdinser]

Thanks for pointing that out. That's exactly the kind of thing my professor would put on a test.

 

[xanthym]

I just want to make sure I'm clear on the whole linearly dependent thing.

If I find the Wronskian of a set of functions and it comes out:

12x^2 + 12x

This would indicate that my set of functions is linearly dependent if the interval included x=0 and would be linearly independent if x never equaled 0.

if I find the wronskian of a set of functions and it comes out:

6 + 12x

This would show that my set of functions is linearly independent for all x.

There is an important distinction to be made here regarding the definition of Linear Dependence. Normally, Linear Dependence for an arbitrary differentiable set of Functions is defined on an OPEN INTERVAL "I" and requires the Wronskian to be zero (0) everywhere on "I". Being (0) at 1 point in "I" (or a finite number of points in "I") does not usually indicate Linear Dependence if there exists at least 1 other point in "I" for which the Wronskian is NON-zero.

Most definitions of Linear Dependence would hold that the 2 above Wronskians indicate Linear INdependence on all OPEN INTERVALS, regardless if the interval contained x=(0) in the first case or x=(-1/2) in the second. Again, this results because all such Open Intervals contain at least 1 point for which the Wronskian is NON-zero.


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