[Math] Methods to show linear independence of functions, polynomials

algebra-precalculuslinear algebrapolynomials

There are apparently many methods to show that a set of functions are linearly independent.

Forexample, there were some cases, where I saw use of derivative. At least to prove Lgrange polynomials are lin. indep., they were evaluated at certain points and their coordinates formed identity matrix which showed that they are linearly independent.

I guess, the later shows that polynomials are linearly independent iff the set evaluated at a number of points are linearly independent.

Are there other methods to show linear independence of functions/polynomials?

Best Answer

Here some methods can be used:

  • A family of polynomials $(P_n)$ graduated in degree is linearly independant;
  • A family of polynomials $(P_n)$ graduated in valuation is linearly independant;
  • Using the Taylor expansion we can show that a set of functions is linearly independant, for example $(\cos x, \sin x, \tan x^2)$
  • We can show that the family $(\cos nx)_{n\geq 0}$ is linearly independant by calculating $$\int_0^{2\pi}\cos(nx)\cos(mx)dx$$
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