Given a set of m real functions of n variables, what is a necessary and sufficient condition for the functions to be functionally independent ?(adsbygoogle = window.adsbygoogle || []).push({});

A set a functions [tex] f_i(x_1,...x_n)\quad i=1,...m[/tex] are functionally independent, if the only function [tex]\phi(u_1,...u_m)[/tex] such that [tex]\phi(f_1,....f_m)=0[/tex] is [tex]\phi=0[/tex].

For example if [tex] f(x,y,z)=x^3+y^2+z\quad g(x,y,z)=z^2+y\quad h(x,y,z)=z^4-x^3-2z^2y-z[/tex]

Then clearly f,g,h are lin. indep....

But they are functionally dependent, in the sense that [tex] h(x,y,z)=g(x,y,z)^2-f(x,y,z) [/tex].

One of my problem is that [tex]\phi[/tex] is acting on functions f_i into R (functional), on f_i into functions of (x_1,...x_n) (operator), or on R^m f_i(x1...xn) ???

Because if [tex]\phi[/tex] is a functional, then it suffices for example that a function is some Fourier transform, derivative or iteration of the other functions, instead of just operations on real numbers....

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# Functional independence

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