Elements of Linear and Multilinear Algebra

Linear Algebra and its Applications, 1983

This unusual little book, which is intended to serve both as a text and as a reference, is a survey of a broad portion of the theory of linear spaces, with particular attention to the solution of linear equations. The book is in four chapters, entitled "Nontopological linear space theory," "Finite systems of linear algebraic equations and their generaliTations," "Topological linear spaces: some comparisons," and "Current research problems." It differs from other brief books on these subjects in two principle ways. First, the author takes a very broad view, putting into 168 pages the beginnings or the essentials of a wide variety of topics, including the general theory of linear spaces, systems of linear equations, integral operators and integral equations, compact operators, the Fredholm theorems and the theories of Banach spaces, of Hflbert spaces, and of topological vector spaces. Second, the book is built on a very systematic and explicit comparison of the finite and infinite dimensional theories. In much of the book, this comparison is undertaken on a theorem by theorem basis and most results are presented in two different versions, with comments on their differences. Where no reasonable generalization to the infinite dimensional case exists, the book generally provides relevant examples in considerable detail. The author's goal seems to have been that of illuminating a large area of mathematics by means of this systematic comparison. Generalizing a theorem is usually an excellent way of coming to a firm understanding of it, so Professor Jfi_rvinen's approach has much to recommend it. Moreover, this book contains an accessible compilation of a large number of results, from different but related areas, which otherwise would not be easy to find in a single source. Unfortunately, no treatment of so much in so short a space can possibly be complete. Professor Jfi_rvinen is thus obliged to replace most of the proofs by references, and to limit himself largely to supplying a context and to describing the flow of the results in his subjects. This approach poses the danger that the exposition can easily turn into a loosely connected list of definitions and results. On the other hand, it makes for a very clear view of broad outlines, and it places on the reader the burden of supplying or

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Proceedings of the American Mathematical Society, 1974

Various theorems on positive matrices are shown to be corollaries of one general theorem, the proof of which bears on Legendre functions, as used in Rockafellar's Convex analysis. 1. Introduction: The main theorem. Let X be a finite set, ([¿(x))xeX strictly positive numbers, and H a fixed linear subspace of Rx. We shall prove the following : Theorem 1. There exists a unique {nonlinear) map from Rx to H, denoted ft->hf, such that 2 texp/(x)-exp hf(x)]g(x)/¿(x) = 0 xeX for all g in H. 2. A first application: Matrices with prescribed marginals. Given an «-sequence /• = (ri)"=1 and an /w-sequence s=(sj),J!=1 of nonnegative numbers such that 22-i ^i=Yj=\ •*/> denote by <J?(r, s) the set of (n, m) matrices (aiS) with a^O such that ri=^JL1aij and Sj-^X-iO^ for all 1 = 1,2, •••,« and j=l,2, • • • , m. Also, let E={1,2, • ■ ■ ,n} and F={1, 2, ■ • •, m}. We define the linear map c from RE®RF to RExF by: c[(b¡)ieE> (t>'j)jeF] = (bi + b'i)(i.i)eExF-If A' is a subset of Ex F, tt denotes the canonical map from RExF to Rx, i.e. 7T[{ai})tí,i)eExF\-(aii)(i.i)eX-We say also that X is an (r, s) pattern if there exists (ai}) in ^#(r, s) such thatX={(ij);a«>0)-Now the first corollary to the Theorem 1 is : Corollary 1. Let two sequences r=(r$=1 ands=(s,)™=l of nonnegative numbers with 2?=i r¿=2í=i sn M an (n,m) matrix with /¿¿3-=0 and

These notes were written as a part of a graduate level course on transform theory offered at King's College London during 2002 and 2003. The material is heavily indebt to the excellent textbook by Gilbert Strang [1], which the reader is referred to for a more complete description of the material; for a more in-depth coverage, the reader is referred to [2–6].

2009

Linear Algebra and its Applications, 1988

This book is the rewritten second edition of the very successful textbook by Peter Lancaster of the same title.

2003

INDEX 169