Linear Algebra

Description

recommended by Goodfellow et al. (ref:11.4) as a good introduction to linear algebra.

Interestingly, this e-book library's only other linear algebra book has been Shafarevich, Linear Algebra and Geometry.

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Introductory treatment covers basic theory of vector spaces and linear maps — dimension, determinants, eigenvalues, and eigenvectors — plus more advanced topics such as the study of canonical forms for matrices. 1992 edition.

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This book is the result of numerous major changes made in the author's earlier work [Introduction to the theory of linear spaces\/ (Russian), GITTL, Moscow, 1952; MR0054162; English translation, Prentice-Hall, Englewood Cliffs, N.J., 1961; MR0126450]. The present treatise gives a thorough and reliable exposition of material which is, in the main, classical. The long chapter on Hilbert space in the earlier book has been omitted and the entire discussion is now concerned with spaces of finite dimension. However, the underlying field of scalars is an arbitrary number field, and the complex and real fields are treated simply as special cases.
The arrangement of the material is fairly standard. Chapter 1 deals with determinants. Chapters 2–6 are devoted to linear algebra proper and include, in particular, a discussion of canonical forms. The subject matter of Chapter 7–10 is the theory of bilinear and quadratic forms and of euclidean and unitary spaces. The most distinctive sections of the work are to be found in the last two chapters. Chapter 11 treats finite-dimensional and matric algebras and Chapter 12 categories of finite-dimensional spaces. All in all, this is a useful and solid if not an enormously exciting book; but perhaps it would be unreasonable to look for excitement in a subject as mature as linear algebra.

Reviewer:Mirsky, L.

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Machine derived contents note: 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