Math415_Fa2015_Lecture39.Article - Math 415 Lecture 39...

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Math 415 - Lecture 39 Review Wednesday December 6th 2015 Final 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 Buell Hall) Conflict Tuesday, December 15th, 8:00-11:00AM. Bring university ID, pencils and erasers, there will be a part multiple choice. 1 After Exam 3 After Exam 3 Diagonalization, Discrete Dynamical Systems. Spectral Theorem and Quadratic forms: each symmetric matrix A gives a quadratic form q ( x ) = x T A x , and conversely. The eigenvalues of A (real!) determine if the quadratic form is always positive. Critical points of functions f : R n R are described by a quadratic form (Hessian) containing the second derivatives of f . Minima, maxima, saddle points. Constrained optimization. 1
Singular Value Decomposition of A from spectral theorem for A T A , and AA T . Approximation of a matrix A according to the singular values: image compression. 2 Big Topics Solving Systems A x = 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.

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