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Ebook Elementary linear algebra (9th edition): Part 2

(BQ) Part 2 book "Elementary linear algebra" has contents: Additional topics, applications of linear algebra, least squares fitting to data, quadratic forms, technology exercises, technology exercises, complex vector spaces,. and other contents. | 9 C H A P T E R Additional Topics I N T R O D U C T I O N : In this chapter we shall see how some of the topics that we have studied in earlier chapters can be applied to other areas of mathematics, such as differential equations, analytic geometry, curve fitting, and Fourier series. The chapter concludes by returning once again to the fundamental problem of solving systems of linear equations . This time we solve a system not by another elimination procedure but by factoring the coefficient matrix into two different triangular matrices. This is the method that is generally used in computer programs for solving linear systems in real-world applications. Copyright © 2005 John Wiley & Sons, Inc. All rights reserved. Many laws of physics, chemistry, biology, engineering, and economics are described in terms of differential equations—that is, equations involving functions and their derivatives. The purpose of this section is to illustrate one way in which linear algebra can be applied to certain systems of differential equations. The scope of this section is narrow, but it illustrates an important area of application of linear algebra. 9.1 APPLICATION TO DIFFERENTIAL EQUATIONS Terminology One of the simplest differential equations is (1) where is an unknown function to be determined, is its derivative, and a is a constant. Like most differential equations, 1 has infinitely many solutions; they are the functions of the form (2) where c is an arbitrary constant. Each function of this form is a solution of Conversely, every solution of must be a function of the form . We call 2 the general solution of . since (Exercise 5), so 2 describes all solutions of Sometimes the physical problem that generates a differential equation imposes some added conditions that enable us to isolate one particular solution from the general solution. For example, if we require that the solution of satisfy the added condition (3) that is, when . Thus , then on substituting these .