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Linear Algebra But With Functions
Introducing the Legendre Polynomials
Linear algebra is usually introduced through vectors treated as arrows in space, or columns of numbers. But it is more general than that. Vectors do not have to be vectors in the conventional sense. They just need to support addition, scaling by a constant, and a few other basic rules. Functions satisfy these rules. You can add two functions, scale them by a constant, ask whether a set of functions is linearly independent, and even define a notion of orthogonality between them. Ideas from linear algebra generalize to functions.
This view is powerful. It underpins numerical techniques like the Finite Element Method, and quadrature, and the Fourier decomposition of signals. It shows up in physics, notably in quantum theory, and in fields like control theory and machine learning. I will not dive into any of these here. Instead, I will introduce some concepts and, as an example, the Legendre polynomials.
I assume you have a working knowledge of linear algebra. I start by building up concepts, and close with a numerical example using Python.
Linear Independence of Functions
Consider a set of functions {f₁, f₂, ..fₙ} and a linear combination of them equated to zero as follows.
