Math 415 - Lecture 39ReviewWednesday December 6th 2015Final Information:•Thursday December 17th, 8:00-11:00AM.–101 Armory: AD3,ADG,ADU,ADW–180 Bevier: ADH,ADP,ADQ,ADX–100 Gregory: ADA,ADB,ADJ,ADK,ADV,ADY–151 Loomis: AD4,AD7,AD8,ADI,ADR–103 Mumford: AD9,ADE,ADF,ADN,ADO–100 MSEB: AD1,AD2,ADS,ADT,ADZ–135 THBH: ADC,ADD,ADL,ADM (THBH is Temple Hoyne BuellHall)•Conflict Tuesday, December 15th, 8:00-11:00AM.Bring university ID, pencils and erasers, there will be a part multiple choice.1After Exam 3After Exam 3•Diagonalization,•Discrete Dynamical Systems.•Spectral Theorem and Quadratic forms: each symmetric matrixAgives aquadratic formq(x) =xTAx, and conversely. The eigenvalues ofA(real!)determine if the quadratic form is always positive.•Critical points of functionsf:Rn→Rare described by a quadratic form(Hessian) containing the second derivatives off. Minima, maxima, saddlepoints. Constrained optimization.1
•Singular Value Decomposition ofAfrom spectral theorem forATA, andAAT.•Approximation of a matrixAaccording to the singular values:imagecompression.2Big Topics•Solving SystemsAx=b–Augmented matrix.–Row Operations, Reduced Row echelon form.–Pivots, free variables, parametric form of general solution.–Inconsistent system, unique solution or infinitely many solutions.•Vectors and Matrices–Linear Combinations–Matrix multiplication is linear combination–Row/column calculation of matrix multiplication–Transpose, symmetric matrices.–Elementary row operations and elementary matrices.