Criterion for linear independence of functions

TL;DR

This paper establishes a criterion for determining the linear independence of functions based on the existence of a nonsingular matrix formed by their evaluations at specific points, generalizing forward elimination.

Contribution

It introduces a generalized forward elimination method to characterize linear independence of functions via nonsingular evaluation matrices.

Findings

Functions are linearly independent iff their evaluation matrix is nonsingular.

Provides a practical criterion for checking linear independence.

Extends classical linear algebra concepts to function spaces.

Abstract

Using a generalization of forward elimination, it is proved that functions $f_1,...,f_n:X\to\mathbb{A}$, where $\mathbb{A}$ is a field, are linearly independent if and only if there exists a nonsingular matrix $[f_i(x_j)]$ of size $n$, where $x_1,...,x_n\in X$.