Lecture 15: Interpolation (part III) | Highlights and Annotations by Gistr.

This segment highlights the evolution of linear algebra from simple dot products between vectors to more complex operations involving functions. It explains how the properties of dot products, such as symmetry and linearity, extend to other operators, creating a flexible framework applicable in various contexts. The analogy of an operating system with different hardware effectively illustrates the adaptability of linear algebra to different spaces. Bilinear and cubic interpolation methods are discussed, emphasizing their symmetry. The lecture then transitions to functional analysis, extending linear algebra concepts (dot products, inner products) to function spaces. Least squares approximation using orthogonal polynomials (e.g., Legendre, Chebyshev) is introduced, highlighting their application in approximating integrals and improving interpolation accuracy. This segment introduces the concept of extending linear algebra to function spaces, where the unknowns are functions rather than numbers. It explains how functions form vector spaces and how the dot product of vectors finds its analog in the inner product of functions, which measures their overlap. The discussion connects this concept to the approximation of functions using polynomials and the use of least squares methods.