Linear Algebra.

Format:

Book

Author/Creator:

Shilov, G. E. (Georgiĭ Evgenʹevich)

Series:

Dover Books on Mathematics

Language:

English

Subjects (All):

Algebras, Linear.

Physical Description:

1 online resource (670 p.)

Edition:

1st ed.

Place of Publication:

Newburyport : Dover Publications, 2012.

Language Note:

English

Summary:

<DIV> <DIV>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.</DIV></DIV>

Contents:

DOVER BOOKS ON MATHEMATICS; Title Page; Copyright Page; PREFACE; Table of Contents; chapter I - DETERMINANTS; I.I. Number Fields; I.2. Problems of the Theory of Systems of Linear Equations; I.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; 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 Spaceschapter 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 Compatibility 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; chapter 4 - LINEAR FUNCTIONS OF A VECTOR ARGUMENT; 4.1. Linear Forms; 4.2. Linear Operators; 4.3. Sums and Products of Operators; 4.4. Corresponding Operations on Matrices; 4.5. Further Properties of Matrix Multiplication

4.6.The Range and Null Space of a Linear Operator4.7. Linear Operators Mapping a Space Kn into Itself 2&; 4.8.Invariant Subspaces; 4.9.Eigenvectors and Eigenvalues; 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; 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 Polynomials6.3. 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; 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; 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; 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

Notes:

Description based upon print version of record.

Description based on publisher supplied metadata and other sources.

ISBN:

0-486-13504-7

1-62198-583-0

OCLC:

868273744