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Linear Algebra

by

Linear Algebra Cover

 

Synopses & Reviews

Publisher Comments:

Lie group theory, developed by M. Sophus Lie in the 19th century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a textbook for graduate courses.

Chapter I introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Carlan's criterion and its consequences, and split semi-simple Lie algebras. Chapter 5, on universal enveloping algebras, provides the abstract concepts underlying representation theory. Then the basic results on representation theory are given in three succeeding chapters: the theorem of Ado-Iwasawa, classification of irreducible modules, and characters of the irreducible modules. In Chapter 9 the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic zero are determined. These results are applied in Chapter 10 to the problems of sorting out the simple Lie algebras over an arbitrary field. The reader, to fully benefit from this tenth chapter, should have some knowledge about the notions of Galois theory and some of the results of the Wedderburn structure theory of associative algebras.

Nathan Jacobson, presently Henry Ford II Professor of Mathematics at Yale University, is a well-known authority in the field of abstract algebra. His book, Lie Algebras, is a classic handbook both for researchers and students. Though it presupposes a knowledge of linear algebra, it is not overly theoretical and can be readily used for self-study.

Synopsis:

Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers.

Table of Contents

chapter 1

  DETERMINANTS

  1.1. Number Fields

  1.2. Problems of the Theory of Systems of Linear Equations

  1.3. Determinants of Order n

  1.4. Properties of Determinants

  1.5. Cofactors and Minors

  1.6. Practical Evaluation of Determinants

  1.7. Cramer's Rule

  1.8. Minors of Arbitrary Order. Laplace's Theorem

  1.9. Linear Dependence between Columns

    Problems

chapter 2

  LINEAR SPACES

  2.1. Definitions

  2.2. Linear Dependence

  2.3. "Bases, Components, Dimension"

  2.4. Subspaces

  2.5. Linear Manifolds

  2.6. Hyperplanes

  2.7. Morphisms of Linear Spaces

    Problems

chapter 3

  SYSTEMS OF LINEAR EQUATIONS

  3.1. More on the Rank of a Matrix

  3.2. Nontrivial Compatibility of a Homogeneous Linear System

  3.3. The Compatability Condition for a General Linear System

  3.4. The General Solution of a Linear System

  3.5. Geometric Properties of the Solution Space

  3.6. Methods for Calculating the Rank of a Matrix

    Problems

chapter 4

  LINEAR FUNCTIONS OF A VECTOR ARGUMENT

  4.1. Linear Forms

  4.2. Linear Operators

  4.3. Sums and Products of Linear Operators

  4.4. Corresponding Operations on Matrices

  4.5. Further Properties of Matrix Multiplication

  4.6. The Range and Null Space of a Linear Operator

  4.7. Linear Operators Mapping a Space Kn into Itself

  4.8. Invariant Subspaces

  4.9. Eigenvectors and Eigenvalues

    Problems

chapter 5

  COORDINATE TRANSFORMATIONS

  5.1. Transformation to a New Basis

  5.2. Consecutive Transformations

  5.3. Transformation of the Components of a Vector

  5.4. Transformation of the Coefficients of a Linear Form

  5.5. Transformation of the Matrix of a Linear Operator

  *5.6. Tensors

    Problems

chapter 6

  THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR

  6.1. Canonical Form of the Matrix of a Nilpotent Operator

  6.2. Algebras. The Algebra of Polynomials

  6.3. Canonical Form of the Matrix of an Arbitrary Operator

  6.4. Elementary Divisors

  6.5. Further Implications

  6.6. The Real Jordan Canonical Form

  *6.7. "Spectra, Jets and Polynomials"

  *6.8. Operator Functions and Their Matrices

    Problems

chapter 7

  BILINEAR AND QUADRATIC FORMS

  7.1. Bilinear Forms

  7.2. Quadratic Forms

  7.3. Reduction of a Quadratic Form to Canonical Form

  7.4. The Canonical Basis of a Bilinear Form

  7.5. Construction of a Canonical Basis by Jacobi's Method

  7.6. Adjoint Linear Operators

  7.7. Isomorphism of Spaces Equipped with a Bilinear Form

  *7.8. Multilinear Forms

  7.9. Bilinear and Quadratic Forms in a Real Space

    Problems

chapter 8

  EUCLIDEAN SPACES

  8.1. Introduction

  8.2. Definition of a Euclidean Space

  8.3. Basic Metric Concepts

  8.4. Orthogonal Bases

  8.5. Perpendiculars

  8.6. The Orthogonalization Theorem

  8.7. The Gram Determinant

  8.8. Incompatible Systems and the Method of Least Squares

  8.9. Adjoint Operators and Isometry

    Problems

chapter 9

  UNITARY SPACES

  9.1. Hermitian Forms

  9.2. The Scalar Product in a Complex Space

  9.3. Normal Operators

  9.4. Applications to Operator Theory in Euclidean Space

    Problems

chapter 10

  QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES

  10.1. Basic Theorem on Quadratic Forms in a Euclidean Space

  10.2. Extremal Properties of a Quadratic Form

  10.3 Simultaneous Reduction of Two Quadratic Forms

  10.4. Reduction of the General Equation of a Quadratic Surface

  10.5. Geometric Properties of a Quadratic Surface

  *10.6. Analysis of a Quadric Surface from Its Genearl Equation

  10.7. Hermitian Quadratic Forms

    Problems

chapter 11

  FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS

  11.1. More on Algebras

  11.2. Representations of Abstract Algebras

  11.3. Irreducible Representations and Schur's Lemma

  11.4. Basic Types of Finite-Dimensional Algebras

  11.5. The Left Regular Representation of a Simple Algebra

  11.6. Structure of Simple Algebras

  11.7. Structure of Semisimple Algebras

  11.8. Representations of Simple and Semisimple Algebras

  11.9. Some Further Results

    Problems

*Appendix

  CATEGORIES OF FINITE-DIMENSIONAL SPACES

  A.1. Introduction

  A.2. The Case of Complete Algebras

  A.3. The Case of One-Dimensional Algebras

  A.4. The Case of Simple Algebras

  A.5. The Case of Complete Algebras of Diagonal Matrices

  A.6. Categories and Direct Sums

HINTS AND ANSWERS

BIBLIOGRAPHY

INDEX

Product Details

ISBN:
9780486635187
Translator:
Silverman, Richard
Publisher:
Dover Publications
Translator:
Silverman, Richard
Author:
Shilov, G. E.
Author:
Mathematics
Author:
Shilov, Georgi E.
Location:
New York :
Subject:
Algebra - General
Subject:
Algebra - Linear
Subject:
Algebras, linear
Subject:
General Mathematics
Subject:
Mathematics-Linear Algebra
Subject:
Linear Algebra
Subject:
Linear Spaces
Subject:
Bilinear and quadratic forms
Edition Number:
Rev. English ed. /
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
19770631
Binding:
TRADE PAPER
Language:
English
Pages:
400
Dimensions:
8.5 x 5.38 in 1 lb

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Product details 400 pages Dover Publications - English 9780486635187 Reviews:
"Synopsis" by ,
Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers.
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