Functions are Vectors: Extending Linear Algebra to Infinite Dimensions
2025-07-06
This article explores the concept of functions as infinite-dimensional vectors, demonstrating how the tools of linear algebra can be applied to a wide range of problems, from image and geometry processing to curve fitting, light transport, and machine learning. Starting with finite-dimensional vector spaces, it progresses to infinite dimensions, proving that functions form a vector space. The article then delves into linear operators, differentiation, the Laplacian operator, and the spectral theorem's application in function spaces, culminating in application examples such as Fourier series, image compression, and spherical harmonics.