Linear algebra : a pure mathematical approach
Linear algebra : a pure mathematical approach
- : softcover
- : hardcover
大学図書館所蔵 件 / 全12件
Includes bibliographical references (p. -240) and indexes
: softcover ISBN 9783764367923
In algebra, an entity is called linear if it can be expressed in terms of addition, and multiplication by a scalar; a linear expression is a sum of scalar multiples of the entities under consideration. Also, an operation is called linear if it preserves addition, and multiplication by a scalar. For example, if A and Bare 2 x 2 real matrices, v is a (row) vector in the real plane, and c is a real number, then v(A + B) = vA + vB and (cv)A = c(vA), that is, the process of applying a matrix to a vector is linear. Linear Algebra is the study of properties and systems which preserve these two operations, and the following pages present the basic theory and results of this important branch of pure mathematics. There are many books on linear algebra in the bookshops and libraries of the world, so why write another? A number of excellent texts were written about fifty years ago (see the bibliography); in the intervening period the 'style' of math- ematical presentation has changed. Also, some of the more modern texts have concentrated on applications both inside and outside mathematics. There is noth- ing wrong with this approach; these books serve a very useful purpose. But linear algebra contains some fine pure mathematics and so a modern text taking the pure mathematician's viewpoint was thought to be worthwhile.
1 - Algebraic Preamble.- Groups, Rings and Fields.- Permutation Groups.- Problems 1.- 2 - Vector Spaces and Linear Maps.- Vector Spaces and Algebras.- Bases and Dimension.- Linear Maps.- Direct Sums.- Addendum - Modules.- Problems 2.- 3 - Matrices, Determinants and Linear Equations.- Matrices.- Determinants.- Systems of Linear Equations.- Problems 3.- 4 - Cayley-Hamilton Theorem and Jordan Form.- Polynomials.- Cayley-Hamilton and Spectral Theorems.- Jordan Form.- Problems 4.- 5 - Interlude on Finite Fields.- Finite Fields.- Applications - Linear Codes and Finite Matrix Groups.- Problems 5.- 6 - Hermitian and Inner Product Spaces.- Hermitian and Inner Products, and Norms.- Unitary and Self-adjoint Maps.- Orthogonal and Symmetric Maps.- Problems 6.- 7 - Selected Topics.- The Geometry of Real Quadratic Forms.- Normed Algebras, Quaternions and Cayley Numbers.- to the Representation of Finite Groups.- Problems 7.- Appendix A - Set Theory.- Sets and Maps.- Problems A.- Appendix B - Answers and Solutions to the Problems.- Notation Index.- Definition Index.- Theorem Index.
: hardcover ISBN 9783764369057
Linear algebra is one of the most important branches of mathematics - important because of its many applications to other areas of mathematics, and important because it contains a wealth of ideas and results which are basic to pure mathematics. This book gives an introduction to linear algebra, and develops and proves its fundamental properties and theorems taking a pure mathematical approach - linear algebra contains some fine pure mathematics. Its main topics include: vector spaces and algebras, dimension, linear maps, direct sums, and (briefly) exact sequences; matrices and their connections with linear maps, determinants (properties proved using some elementary group theory), and linear equations; Cayley-Hamilton and Jordan theorems leading to the spectrum of a linear map - this provides a geometric-type description of these maps; Hermitian and inner product spaces introducing some metric properties (distance, perpendicularity etc.) into the theory, also unitary and orthogonal maps and matrices; applications to finite fields, mathematical coding theory, finite matrix groups, the geometry of quadratic forms, quaternions and Cayley numbers, and some basic group representation theory; and, a large number of examples, exercises and problems are provided. It gives answers and/or sketch solutions to all of the problems in an appendix -some of these are theoretical and some numerical, both types are important. No particular computer algebra package is discussed but a number of the exercises are intended to be solved using one of these packages chosen by the reader. The approach is pure-mathematical, and the intended readership is undergraduate mathematicians, also anyone who requires a more than basic understanding of the subject. This book will be most useful for a 'second course' in linear algebra, that is for students that have seen some elementary matrix algebra. But as all terms are defined from scratch, this book can be used for a 'first course' for more advanced students.
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