Chapter 6 - Further Directions

Abstract

This chapter discusses linear algebra in function theory, the broad interaction between linear algebra and analysis, and some of the applications. The chapter presents the singular value decomposition and generalized inverses, an area of pure linear algebra growing out of attempts to deal with numerical difficulties. An introduction to abstract linear algebra over an arbitrary field is discussed. This is the proper context for abstract matrix theory, for the interaction between linear algebra and abstract algebra, and for understanding many applications such as coding theory. Infinite sums are series, and they do not always converge or add up. This is where analysis comes in and interacts with linear algebra. The chapter also focuses on orthogonalizing polynomials, Legendre polynomials, and least-squares approximation. The singular value decomposition of a matrix is a very important theoretical and computational tool. The chapter uses this tool to construct the pseudoinverse, and to solve ill-conditioned systems.