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An Introduction to Linear Algebra (Dover Books on Mathematics)

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An Introduction to Linear Algebra (Dover Books on Mathematics) Cover

 

Synopses & Reviews

Publisher Comments:

"The straight-forward clarity of the writing is admirable." — American Mathematical Monthly.

This work provides an elementary and easily readable account of linear algebra, in which the exposition is sufficiently simple to make it equally useful to readers whose principal interests lie in the fields of physics or technology. The account is self-contained, and the reader is not assumed to have any previous knowledge of linear algebra. Although its accessibility makes it suitable for non-mathematicians, Professor Mirsky's book is nevertheless a systematic and rigorous development of the subject.

Part I deals with determinants, vector spaces, matrices, linear equations, and the representation of linear operators by matrices. Part II begins with the introduction of the characteristic equation and goes on to discuss unitary matrices, linear groups, functions of matrices, and diagonal and triangular canonical forms. Part II is concerned with quadratic forms and related concepts. Applications to geometry are stressed throughout; and such topics as rotation, reduction of quadrics to principal axes, and classification of quadrics are treated in some detail. An account of most of the elementary inequalities arising in the theory of matrices is also included. Among the most valuable features of the book are the numerous examples and problems at the end of each chapter, carefully selected to clarify points made in the text.

Synopsis:

Rigorous, self-contained coverage of determinants, vectors, matrices and linear equations, quadratic forms, more. Elementary, easily readable account with numerous examples and problems at the end of each chapter.

Description:

Includes bibliographical references (p. [434]-435) and index.

Table of Contents

PART I

"DETERMINANTS, VECTORS, MATRICES, AND LINEAR EQUATIONS"

I. DETERMINANTS

  1.1. Arrangements and the Î-symbol

  1.2. Elementary properties of determinants

  1.3. Multiplication of determinants

  1.4. Expansion theorems

  1.5. Jacobi's theorem

  1.6. Two special theorems on linear equations

II. VECTOR SPACES AND LINEAR MANIFOLDS

  2.1. The algebra of vectors

  2.2. Linear manifolds

  2.3. Linear dependence and bases

  2.4. Vector representation of linear manifolds

  2.5. Inner products and orthonormal bases

III. THE ALGEBRA OF MATRICES

  3.1. Elementary algebra

  3.2. Preliminary notions concerning matrices

  3.3. Addition and multiplication of matrices

  3.4. Application of matrix technique to linear substitutions

  3.5. Adjugate matrices

  3.6. Inverse matrices

  3.7. Rational functions of a square matrix

  3.8. Partitioned matrices

IV. LINEAR OPERATIONS

  4.1. Change of basis in a linear manifold

  4.2. Linear operators and their representations

  4.3. Isomorphisms and automorphisms of linear manifolds

  4.4. Further instances of linear operators

V. SYSTEMS OF LINEAR EQUATIONS AND RANK OF MATRICES

  5.1. Preliminary results

  5.2. The rank theorem

  5.3. The general theory of linear equations

  5.4. Systems of homogeneous linear equations

  5.5. Miscellaneous applications

  5.6. Further theorems on rank of matrices

VI. ELEMENTARY OPERATIONS AND THE CONCEPT OF EQUIVALENCE

  6.1. E-operations and E-matrices

  6.2. Equivalent matrices

  6.3. Applications of the preceding theory

  6.4. Congruence transformations

  6.5. The general concept of equivalence

  6.6. Axiomatic characterization of determinants

PART II

FURTHER DEVELOPMENT OF MATRIX THEORY

VII. THE CHARACTERISTIC EQUATION

  7.1. The coefficients of the characteristic polynomial

  7.2. Characteristic polynomials and similarity transformations

  7.3. Characteristic roots of rational functions of matrices

  7.4. The minimum polynomial and the theorem of Cayley and Hamilton

  7.5. Estimates of characteristic roots

  7.6. Characteristic vectors

VIII. ORTHOGONAL AND UNITARY MATRICES

  8.1. Orthogonal matrices

  8.2. Unitary matrices

  8.3. Rotations in the plane

  8.4. Rotations in space

IX. GROUPS

  9.1. The axioms of group theory

  9.2. Matrix groups and operator groups

  9.3. Representation of groups by matrices

  9.4. Groups of singular matrices

  9.5. Invariant spaces and groups of linear transformations

X. CANONICAL FORMS

  10.1. The idea of a canonical form

  10.2. Diagonal canonical forms under the similarity group

  10.3. Diagonal canonical forms under the orthogonal similarity group and the unitary similarity group

  10.4. Triangular canonical forms

  10.5. An intermediate canonical form

  10.6. Simultaneous similarity transformations

XI. MATRIX ANALYSIS

  11.1 Convergent matrix sequences

  11.2 Power series and matrix functions

  11.3 The relation between matrix functions and matrix polynomials

  11.4 Systems of linear differential equations

PART III

QUADRIATIC FORMS

XII. "BILINEAR, QUADRATIC, AND HERMITIAN FORMS"

  12.1 Operators and forms of the bilinear and quadratic types

  12.2 Orthogonal reduction to diagonal form

  12.3 General reduction to diagonal form

  12.4 The problem of equivalence. Rank and signature

  12.5 Classification of quadrics

  12.6 Hermitian forms

XIII. DEFINITE AND INDEFINITE FORMS

  13.1 The value classes

  13.2 Transformations of positive definite forms

  13.3 Determinantal criteria

  13.4 Simultaneous reduction of two quadratic forms

  13.5 "The inequalities of Hadamard, Minkowski, Fischer, and Oppenheim"

  MISCELLANEOUS PROBLEMS

  BIBLIOGRAPHY

  INDEX

Product Details

ISBN:
9780486664347
Author:
Mirsky, L.
Author:
Mirsky, Lawrence
Author:
Mathematics
Publisher:
Dover Publications
Location:
New York :
Subject:
Algebra
Subject:
Algebra - General
Subject:
Algebra - Linear
Subject:
Algebras, linear
Subject:
General Mathematics
Subject:
Algebra of Matrices
Subject:
Vector Spaces and Linear Manifolds
Subject:
matrix theory
Subject:
Mathematics-Linear Algebra
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
20111131
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
462
Dimensions:
8.5 x 5.38 in 1.03 lb

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Related Subjects


Science and Mathematics » Materials Science » General
Science and Mathematics » Mathematics » Algebra » General
Science and Mathematics » Mathematics » Algebra » Linear Algebra

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Product details 462 pages Dover Publications - English 9780486664347 Reviews:
"Synopsis" by ,
Rigorous, self-contained coverage of determinants, vectors, matrices and linear equations, quadratic forms, more. Elementary, easily readable account with numerous examples and problems at the end of each chapter.
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