@MISC{_1linear, author = {}, title = {1 LINEAR DIFFERENTIAL EQUATIONS}, year = {} }

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Abstract

In this chapter we discuss linear ordinary differentia! equations. For the 1110st part we will not find it necessary to assume that these equation~.; arc also time invariant, although this is an important special casco '0/t are interested in exposing the structure of the solutions and, to a lesser extent, developing actual solution techniques. The presentation relies heavily on vector space methods. This is entirely in keeping with modern methods of computation. The first section covers briefly the background from linear algebra needed in Chapter 1. This policy is repeated in the first section of the remaining three chapters. Taken together these four sections are intended as a convenient reference to make the book more nearly self contained. I. LINEAR INDEPENDENCE AND LINEAR MAPPINGS The theory of linear dynamical systems is so completely entwined with the study of basic linear algebra that any attempt to relegate the latter to appendices is, in our view, out of the question. 011 the other hand, excellent books devoted entirely to the many facets of the subject already exist and there is no need to duplicate here material that is widely available. Our policy will be to steer a middle course. At the start ofeach chapter, we give some background in those aspects of linear algebra that arc most germane and try to point out in the subsequent sections the most informative relationships. In this section we discuss the algebraic structure ofthe set R " of all n-tuples of real numbers," and linear transformations of sets of z-ruplcs into sets of »-tuplcs. By R " we mean the set of all objects of the form (x, , x "..., x,,) with the Xi real numbers. This is a specific example of a finite dimensional vector space. We define the following operations for members of R".