Functions can be conceptualized as infinite-dimensional vectors, enabling the application of linear algebra tools to diverse problems. This perspective allows treating differentiation and the Laplacian as linear operators that can be diagonalized using eigenfunctions. The Fourier transform emerges as a change of basis into the Laplacian's eigenbasis, with practical applications in image compression, signal processing, and geometry processing. The approach extends to spherical harmonics for functions on spheres and discrete differential geometry for mesh processing.