We've seen that two vectors a and b are orthogonal if their inner product ab = 0. Two vectors are normalized if their inner product is ab = 1. A basis set or simply basis {e;} is a set of linearly independent vectors. A basis is said to be orthonormal (orthogonal + normalized) if emen = onm = if m=n 0, if men (1) for all i where i is the number of vectors in the basis called its dimension. A basis is complete if it spans the dimensions of its vector space. This is best illustrated with an example. The the unit vectors i = (1,0)" and = (0,1) constitute an orthonormal basis of 2 dimensions. They are complete with w.r.t space R2 because every vector in the plane R2 can be decomposed into them. and are said to span the space R2. In 3 dimension i and i would become i = (1,0,0)7 and j = (0,1,0), the basis set they'd form would still be orthonormal, however would no longer be complete. To complete the basis, we'd need to add = (0,0,1)T. With , the volume of R3 would be spanned. The concepts of orthonormality and completeness aren't limited to array vectors, but can be gen- eralized to include functions themselves. This is fundamental to eigenfunctions and Fourier Analysis and consequently quantum mechanics, molecular orbital theory, and spectroscopy. Functions y1,42,... defined on some interval a 0 if i, if m= n (2) 10, if men Common orthonormal function sets include p(x)ym(x)yn(x)dx = Name Symbol Legendre P(x) Chebyshev(Tchebychef) T(X) Laguerre L, (X) Associated Laguerre LC)(x) Hermite H,x) Interval [a, b] Weight p(x) -1