Thread: The linearly independent of multivariable function

1. The linearly independent of multivariable function

Dears

We know the definition of linearly independent of functions with one variable, and we know that the Wronskian is used to check the linearly independent.

But what is about the multivariable function? what is the definition of the linearly dependent of multivariable functions? Are there a Wronskian determinant for the multivariable functions.

Thanks allot

2. Re: The linearly independent of multivariable function

The definition of "linearly independent" for a number of functions, say, $f_1$, $f_2$, ..., $f_n$, is that $a_1f_1+ a_2f_2+ \cdot\cdot\cdot+ a_nf_n= 0$ for all possible values of the variables, regardless of how many variables there are, only if $a_1= a_2= \cdot\cdot\cdot= a_n= 0$.

3. Re: The linearly independent of multivariable function

Originally Posted by HallsofIvy
Generalized Wronskians

For n functions of several variables, a generalized Wronskian is the determinant of an n by n matrix with entries Di(fj) (with 0 ≤ i < n), where each Di is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish. .
-Wikipedia Wronskian

So in terms of the Wronskian, I think you would just set up a separate Wronskian matrix for each variable and test it for each to see if it vanishes.

4. Re: The linearly independent of multivariable function

Too late to edit my last post on this thread but I just noticed in my comment that quote was from Wikipedia , not HallsofIvy. I used "reply with quote" button under Ivy's comment but forgot to remove his user name from it so it looks a bit odd...