Advanced Linear Algebra, Lecture 5.2: Orthogonality

Advanced Linear Algebra, Lecture 5.2: Orthogonality

Two vectors are orthogonal if their inner product is zero. This is the analogue of the concept of being perpendicular in Euclidean space to a general inner product space. Orthogonal bases are particular nice, because the coefficients of any linear combination of the basis vectors are simply a projection, and given by the formula a_i = (v, x_i)/(x_i,x_i). We derive this formula, do some example of what orthogonality means in a number of inner product vector spaces, including applications to Fourier series, and Sturm-Liouville theory, where the Legendre and Chebyshev polynomials arise. Our examples of these infinite dimensional functional spaces are meant to just be a tour, highlighting the diversity of applications of orthogonality in inner product spaces within math, science, and engineering.