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!
[1] https://ltcconline.net/greenl/courses/203/Vectors/basisDimension.htm
[2] Polynomial $p(x) = 0$ for all $x$ implies coefficients of polynomial are zero
Best Answer
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 $x$.