09.25.2016
Dear Student,
Our class schedule states that you should complete Lectures 15, 16, 17 and the associated homework problems on or before next Sunday, October 2. These are the last lectures and homework that will be covered on Exam 1 (the week of October 3-7). There is an ‘Exam 1 Review Guide’ under the Syllabus section of our course web site. That guide will give you rather specific information about the kinds of problems that you should be able to solve as well as the more peripheral or theoretical problem types that will not be tested on the exam.
Special Note: If your work or travel schedule will make it difficult for you to prepare adequately for the exam during the scheduled for Exam1, I have authorized the OCEE Course Coordinator Peg Pisel to work with you and your test coordinator to extend the scheduled Exam 1 week by up to 3 days so that you will be able to prepare adequately for the exam.
Lectures 15, 16 and 17 comprise a unit of the course on inner product spaces, orthogonal and symmetric matrices and the orthogonal diagonalization of symmetric matrices. These mathematical structures have many important applications in engineering and applied mathematics including least squares methods in statistics, orthogonal polynomials. Consequently, there are more than the usual number of different mathematical methods discussed in these lectures. However there are also more than the usual number of worked examples in the lectures and more than the usual number of homework problems (with solutions) in these lectures. As a result, I think that you will find this unit useful in your engineering work and study
Week 6 - Content Summary for Lectures 15, 16 and 17
Lecture 15 introduces the idea of an inner product space, which is basically a vector space in which the ideas of angle, length and perpendicularity are meaningful. This leads directly to the meaning and properties of orthogonal and symmetric square matrices, orthonormal sets of vectors and functions, and a stronger and more productive form of matrix diagonalization for symmetric matrices called orthogonal-diagonalization. Inner product spaces will come up in other contexts in the course, including eigenfunction series, discrete and continuous Fourier series, and transforms.
Lectures 16 and 17 apply the ideas introduced in Lecture 15 to a series of important methods in applied mathematics, including least squares methods for fitting data and general methods for computing maximum and minimum values of functions of several variables.
Lecture 16 begins with a quick review of real inner product spaces and the meaning of orthogonal vectors and matrices in n-dimensional space. You should be able to explain how the Gram-Schmidt Orthogonal Process can be implemented for a small number of vectors in two or three dimensional space. The main new topic in Lecture 16 is quadratic forms determined by n by n symmetric matrices and their application to a variety of problems in engineering and applied mathematics. We begin with the geometric interpretation of quadratic forms in two-dimensional space as either an ellipse, or parabola, or hyperbola (or a degenerate form of one of these). Then we do the same geometric analysis for quadratic forms determined by 3 by 3 symmetric matrices and show how ellipsoids, paraboloids and hyperboloids (or degenerate forms of these) are the result. We then introduce quadratic approximation to functions of n variables and use this to find relative maximums, relative minimums and saddle points of such functions.
Lecture 17 and the associated homework problems develop the so-called least squares methods for several iterative methods for solving systems of linear equations, including the Method of Steepest Descent and Least Squares Solutions of inconsistent systems of linear equations. We also apply the Method of Least Squares to find the best polynomial fit for a given data set.
You will find Lecture 17 time consuming because it has a large number and variety of homework problems that discuss several important applications. You should take plenty of time reviewing the homework, examples, and Just Do Its in Lecture 17. This will help you to be well-prepared for Exam 1.
Contact me if you have questions!
My best wishes for an interesting and productive week!
Tony Peressini
Cell PH: 217 840 2871
anthonyperessini@gmail.com