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The source of linear algebra is the solution of systems of linear algebraic equations. Linear algebra is the foundation upon which almost all applied mathematics rests. This is not to say that nonlinear equations are less important; rather, progress in the vastly more complicated nonlinear realm is impossible without a ¯rm grasp of the fundamentals of linear systems. Furthermore, linear algebra underlies the numerical analysis of continuous systems, both linear and nonlinear, which are typically modeled by di®erential equations. Without a systematic development of the subject from the start, we will be ill equipped to handle the resulting large systems of linear equations involving many (e.g., thousands of) unknowns. This ¯rst chapter is devoted to the systematic development of directy algorithms for solving systems of linear algegbraic equations in a ¯nite number of variables. Our primary focus will be the most important situation involving the same number of equations as unknowns, although in Section 1.8 we extend our techniques to completely general linear systems. While the former usually have a unique solution, more general systems more typically have either no solutions, or in¯nitely many, and so tend to be of less direct physical relevance. Nevertheless, the ability to con¯dently handle all types of linear systems is a basic prerequisite for the subject. The basic solution algorithm is known as Gaussian elimination, in honor of one of the all-time mathematical greats the nineteenth century German mathematician Carl Friedrich Gauss. As the father of linear algebra, his name will occur repeatedly throughout this text. Gaussian elimination is quite elementary, but remains one of the most important techniques in applied (as well as theoretical) mathematics. Section 1.7 discusses some practical issues and limitations in computer implementations of the Gaussian elimination method for large systems arising in applications. The systematic development of the subject relies on the fundamental concepts of scalar, vector, and matrix, and we quickly review the basics of matrix arithmetic. Gaussian elimination can be reinterpreted as matrix factorization, the (permuted) LU decom- position, which provides additional insight into the solution algorithm. Matrix inverses and determinants are discussed in Sections 1.5 and 1.9, respectively. However, both play a relatively minor role in practical applied mathematics, and so will not assume their more traditional central role in this applications-oriented text... (Applied Mathematics)

by Peter J. Olver and Chehrzad Shakiban