MATH 415 Study Guide - Final Guide: Khan Academy, Periodic Function, Characteristic Polynomial

• School: University of Illinois
• Department: Mathematics
• Course: MATH 415
• Professor: J Li

Document Summary

Textbook reading: chapter 3. 4, fourier series, 5. 5 spectral theorem. Strang lecture: fourier series, symmetric matrices and positive de niteness. Review: an inner product on v is a map (u, v) v v 7 hu, vi r such that. Let v be the vector space of functions f : rn . Rn that are integrable and periodic of period 2 : f (x + 2 ) = f (x). Examples of functions in v are constants and trig functions like sin(nx), cos(nx). De ne for each pair (f, g) of functions in v a number hf, gi =z 2 . Claim: this de nes an inner product on v . Two vectors are orthogonal if hv, wi = 0, so calculate: (sin(x))2(cid:21)2 cos(x) sin(x)dx = (cid:20) 1. More generally, we will see that 1, cos(x), sin(x), cos(2x), sin(2x), . are all orthogonal to each other. Idea: use this to study signals and sound waves.