# Prove that polynomial basis functions are linearly independent

I am reading about Basis Function Sets with polynomial functions, and the set $$\{1, x, x^2, x^3\}$$ form a basis set for polynomial functions and so must be linearly independent.

The proof from the definition of linearity, that there must be some set of non-zero coefficients such that $$a_0 + a_1x + a_2x^2 + a_3x^3 = 0$$ is makes sense 'since a polynomial is zero if and only if its coefficients are all zero' from [1] (this itself is discussed in [2]).

However, I have also seen independence defined that one basis vector cannot be written as a linear combination of the others. This is clearly true with vectors i = [0, 1], j = [1, 0] but I cannot see why this is true for the polynomial basis:

e.g. $$x^2 = a_0 + a_1x$$ where $$a_0=0$$ and $$a_1=x$$. This example seems wrong but not sure why!

This is linear independence over the field of real numbers. The $$a_i's$$ have to be real numbers. These are not functions depending on $$x$$.
• Yes, you are correct, the problem makes sense only if the domain of definition of the polynomials is specified. If not specified, it is usually assumed that this is ${\bf R}$. Note that if the polynomials are defined on an interval $[a,b]$, $a<b$, the result also holds, because a non-zero polynomial can have only finitely many roots, the number of roots being bounded by the degree of the polynomial. Feb 16, 2022 at 14:10